| Date post: | 11-Jan-2016 |
| Category: |
Documents |
| Upload: | neal-green |
| View: | 219 times |
| Download: | 5 times |
CH101 GENERAL CH101 GENERAL CHEMISTRY I CHEMISTRY I SPRING 2013SPRING 2013
bull Textbook lsquoChemical Principles The Quest for Insightrsquo by P Atkins and L Jones Freeman New York 2010 (International Edition)
1
CHEMISTRYCHEMISTRY The science of matter and the changes it can undergo
A science that deals with the
composition structure and
properties of substances and
with the transformations that
they undergo
F1
Chemistry A Science at Three Levels
Macroscopic levelThe level dealing with the propertiesof large visible objects
Microscopic levelAn underworld of changeat the level of atoms and molecules
Symbolic levelThe expression of chemical phenomena in terms of chemical symbols and mathematical equations
F2
3
How Science Is DoneF3
4
The Branches of ChemistryF4
Traditional areas
Organic chemistry (carbon compounds)Inorganic chemistry (all other elements and their compounds)Physical chemistry (principles of chemistry)
Biochemistry (chemistry in living systems)Analytical chemistry (techniques for identifying substances)Theoretical (computational) chemistry (mathematical and computational)Medicinal chemistry (application to the development of pharmaceuticals)
Specialized areas
Interdisciplinary branches
Molecular biology (chemical basis of genes and proteins)Materials science (structure and composition of materials)Nanotechnology (matter at the nanometer level)
5
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
INVESTIGATING ATOMS
QUANTUM THEORY
11 The Nuclear Model of the Atom12 The Characteristics of Electromagnetic Radiation13 Atomic Spectra
14 Radiation Quanta and Photons15 The Wave-Particle Duality of Matter16 The Uncertainty Principle17 Wavefunctions and Energy Levels
6
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
CHEMISTRYCHEMISTRY The science of matter and the changes it can undergo
A science that deals with the
composition structure and
properties of substances and
with the transformations that
they undergo
F1
Chemistry A Science at Three Levels
Macroscopic levelThe level dealing with the propertiesof large visible objects
Microscopic levelAn underworld of changeat the level of atoms and molecules
Symbolic levelThe expression of chemical phenomena in terms of chemical symbols and mathematical equations
F2
3
How Science Is DoneF3
4
The Branches of ChemistryF4
Traditional areas
Organic chemistry (carbon compounds)Inorganic chemistry (all other elements and their compounds)Physical chemistry (principles of chemistry)
Biochemistry (chemistry in living systems)Analytical chemistry (techniques for identifying substances)Theoretical (computational) chemistry (mathematical and computational)Medicinal chemistry (application to the development of pharmaceuticals)
Specialized areas
Interdisciplinary branches
Molecular biology (chemical basis of genes and proteins)Materials science (structure and composition of materials)Nanotechnology (matter at the nanometer level)
5
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
INVESTIGATING ATOMS
QUANTUM THEORY
11 The Nuclear Model of the Atom12 The Characteristics of Electromagnetic Radiation13 Atomic Spectra
14 Radiation Quanta and Photons15 The Wave-Particle Duality of Matter16 The Uncertainty Principle17 Wavefunctions and Energy Levels
6
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Chemistry A Science at Three Levels
Macroscopic levelThe level dealing with the propertiesof large visible objects
Microscopic levelAn underworld of changeat the level of atoms and molecules
Symbolic levelThe expression of chemical phenomena in terms of chemical symbols and mathematical equations
F2
3
How Science Is DoneF3
4
The Branches of ChemistryF4
Traditional areas
Organic chemistry (carbon compounds)Inorganic chemistry (all other elements and their compounds)Physical chemistry (principles of chemistry)
Biochemistry (chemistry in living systems)Analytical chemistry (techniques for identifying substances)Theoretical (computational) chemistry (mathematical and computational)Medicinal chemistry (application to the development of pharmaceuticals)
Specialized areas
Interdisciplinary branches
Molecular biology (chemical basis of genes and proteins)Materials science (structure and composition of materials)Nanotechnology (matter at the nanometer level)
5
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
INVESTIGATING ATOMS
QUANTUM THEORY
11 The Nuclear Model of the Atom12 The Characteristics of Electromagnetic Radiation13 Atomic Spectra
14 Radiation Quanta and Photons15 The Wave-Particle Duality of Matter16 The Uncertainty Principle17 Wavefunctions and Energy Levels
6
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
How Science Is DoneF3
4
The Branches of ChemistryF4
Traditional areas
Organic chemistry (carbon compounds)Inorganic chemistry (all other elements and their compounds)Physical chemistry (principles of chemistry)
Biochemistry (chemistry in living systems)Analytical chemistry (techniques for identifying substances)Theoretical (computational) chemistry (mathematical and computational)Medicinal chemistry (application to the development of pharmaceuticals)
Specialized areas
Interdisciplinary branches
Molecular biology (chemical basis of genes and proteins)Materials science (structure and composition of materials)Nanotechnology (matter at the nanometer level)
5
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
INVESTIGATING ATOMS
QUANTUM THEORY
11 The Nuclear Model of the Atom12 The Characteristics of Electromagnetic Radiation13 Atomic Spectra
14 Radiation Quanta and Photons15 The Wave-Particle Duality of Matter16 The Uncertainty Principle17 Wavefunctions and Energy Levels
6
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
The Branches of ChemistryF4
Traditional areas
Organic chemistry (carbon compounds)Inorganic chemistry (all other elements and their compounds)Physical chemistry (principles of chemistry)
Biochemistry (chemistry in living systems)Analytical chemistry (techniques for identifying substances)Theoretical (computational) chemistry (mathematical and computational)Medicinal chemistry (application to the development of pharmaceuticals)
Specialized areas
Interdisciplinary branches
Molecular biology (chemical basis of genes and proteins)Materials science (structure and composition of materials)Nanotechnology (matter at the nanometer level)
5
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
INVESTIGATING ATOMS
QUANTUM THEORY
11 The Nuclear Model of the Atom12 The Characteristics of Electromagnetic Radiation13 Atomic Spectra
14 Radiation Quanta and Photons15 The Wave-Particle Duality of Matter16 The Uncertainty Principle17 Wavefunctions and Energy Levels
6
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
INVESTIGATING ATOMS
QUANTUM THEORY
11 The Nuclear Model of the Atom12 The Characteristics of Electromagnetic Radiation13 Atomic Spectra
14 Radiation Quanta and Photons15 The Wave-Particle Duality of Matter16 The Uncertainty Principle17 Wavefunctions and Energy Levels
6
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
INVESTIGATING ATOMS (Sections 11-13)
11 The Nuclear Model of the Atom11 The Nuclear Model of the Atom Discovering the electron
JJ Thomson (English physicist 1856-1940) in 1897 discovers the electron and determines the charge to mass ratio (eme) as ldquocathode raysrdquo In 1906 he wins the Nobel Prize
7
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Robert Millikan designed an ingeniousapparatus in which he could observetiny electrically charged oil droplets
Fundamental charge the smallest increment of charge
e = 1602times10-19 C
From the value of eme measuredby Thomson
me = 9109times10-31 kg
8
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Brief Historical Summary
Atomic TheoryJ Dalton 1807
Discovery ofelectron (cathoderays) value of emeJ J Thomson 1897
Implies internalstructure of sub-atomic particleselectron + proton= neutral
How are electrons andprotons arrangedin the atom
Value of e(1602 x 10-19 C)
me = 9109 x 10-31 kgR Millikan andH A Fletcher 1906
9
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Pudding model(J J Thomson)
Nuclear model(E Rutherford)
Two Models of the Atom
10
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Ernest Rutherford (1871-1937) and his students in England studied α emission from newly-discovered radioactive elements
Nuclear Model
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atomNucleus contains particles called proton (+e) and neutron (uncharged)
11
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Nuclear Model of the Atom
In the nuclear model of the atom all the positive charge and almost all the mass is concentrated in the tiny nucleus and the negatively charged electrons surround the nucleus The atomic number is the number of protons in the nucleus
12
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
13
Some Questions Posed by the Nuclear Model
1 How are the electrons arranged around the nucleus2 Why is the nuclear atom stable (classical physics predicts instability)3 What holds the protons together in the nucleusAtomic spectroscopy (involving the absorption by or emission of electromagnetic radiation from atoms) provided many of the clues needed to answer questions 1 and 2
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
12 The Characteristics of Electromagnetic 12 The Characteristics of Electromagnetic RadiationRadiation
Spectroscopy ndash the analysis of the light emitted or absorbed by substances
- Light is a form of electromagnetic radiation which is the periodicvariation of an electric field (and a perpendicular magnetic field)
amplitude the height of the waveabove the center lineintensity the square of the amplitudewavelength peak-to-peak distance
wavelength times frequency = speed of light
= c
c = 29979 x 108 ms-1
14
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
visible light = 700 nm (red light) to 400 nm (violet light)
infrared gt 800 nm the radiation of heat
ultraviolet lt 400 nm responsible for sunburn
The color of visible light depends on its frequency and wavelengthlong-wavelength radiation has a lower frequency thanshort-wavelength radiation
15
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
16
Color Frequency and Wavelength ofElectromagnetic Radiation
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
17
Self-Test 11A
Calculate the wavelengths of the light from traffic signalsas they change Assume that the lights emit the following frequencies green 575 x 1014 Hz yellow 515 x 1014 Hz red 427 x 1014 Hz
Solution Green light =2998 x 108 ms
575 x 1014 s
= 521 x 10-7 m = 521 nm
Similarly yellow light is 582 nm and red light is702 nm wavelength
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
13 Atomic Spectra13 Atomic Spectra
White light
Discharge lamp ofhydrogen (emission
spectrum)
spectral linesdiscrete energy
levels
18
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Johann Rydbergrsquos general empirical equation
R (Rydberg constant) = 329times1015 Hz
an empirical constant
n1 = 1 (Lyman series) ultraviolet region
n1 = 2 (Balmer series) visible region
n1 = 3 (Paschen series) infrared region
For instance n1 = 2 and n2 = 3
= 657times10-7 m
19
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
20
Self-Test 12A
Calculate the wavelength of the radiation emitted by a hydrogen atom for n1 = 2 and n2 = 4 Identify the spectralline in Fig 110b
= R1
221
42_ =
316
R
Now =c =
16c
3R=
16 x 2998 x 108 m s-1
3 x 329 x 1015 s-1
= 486 x 10-7 m or 486 nm
It is the second (greenblue) line in the spectrum
Solution
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
21
Absorption Spectra
When white light passes through a gas radiation isabsorbed by the atoms at wavelengths that correspondto particular excitation energies The result is an atomic absorption spectrum
Above is an absorption spectrum of the sun elements canbe identified from their spectral lines
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
QUANTUM THEORY (Sections 14-17)
14 Radiation Quanta and Photons14 Radiation Quanta and Photons This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra) black body radiation and the photoelectric effect
Black body radiation is the radiation emitted at different wavelengths by a heated black body for a series of temperatures Two empirical laws are associated with it (below) and plots of emitted radiation energy density versus wavelength give bell-shaped curves (right)
Stefan-Boltzmann law
Total intensity = constant times T4
Wienrsquos law T max = constant
22
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
23
Self-Test 13A
In 1965 electromagnetic radiation with a maximum of105 mm (in the microwave region) was discovered topervade the universe What is the temperature oflsquoemptyrsquo space
Use Wiens law in the form T = constantmax
T =29 x 10-3 m K
105 x 10-3 m= 28 K
Solution
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
24
Black Body Radiation Theories
bull Rayleigh-Jeans theory was based on classical physics It assumed the black body atoms behave like mechanical oscillators that absorb and emit energy continuously
bull The Rayleigh-Jeans equation did not agree with experimental data except at high wavelengths
bull Classical physics predicts intense UV or higher energy radiation from hot black bodies
Radiant energy density =8kBT
4
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
25
Planckrsquos Quantum Theory
bull Max Planck (1900) made the assumption that black body atoms could absorb and emit energy only in multiples of a fundamental quantity (a quantum) whose value is
E = h (4) bull The constant h became known as Planckrsquos constant
(= 6626 x 10-34 Js)bull The Planck equation describing the black body
radiation profile agreed well with experiment
Radiant energy density =8hc
5
1
e
hckBT
_
_ 1
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Ultraviolet catastrophe Classical physics predicted that any hot body should emit intense ultraviolet radiation and even X-rays and -rays Assumes continuous exchange of energy
Quantization of electromagnetic radiationsuggested by Max Planck
E = h
Radiation of frequency can be generated only if an oscillator of that frequency has acquired the minimum energy required tostart oscillating
26
Assumes energy can be exchangedonly in discrete amounts (quanta)
Summary
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
The Photoelectric effect ndash ejection of electrons froma metal when its surface isexposed to ultraviolet radiation
1 No electrons are ejected lt a certain threshold value of frequency which depends on the metal
2 Electrons are ejected immediately at that particular value
3 The kinetic energy of ejected electrons increases linearly with the frequency of the radiation
27
Photoelectric effect observations
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Albert Einstein proposed that electromagnetic radiation consists of particles called ldquophotonsrdquo
ndash The energy of a single photon is proportional to the radiation frequency by E = h
work function
Bohr frequency condition relates photon energy to energy difference between two energy levels in an atom
28
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
29
Self-Test 15A (and part of 15B)
The work function of zinc is 363 eV (a) What is thelongest wavelength of electromagnetic radiation that could eject an electron from zinc(b) What is the wavelength of the radiation that ejectsan electron with velocity 785 km s-1 from zinc
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
30
Solution
(a) =hc
0
hence 0 =hc
Firstly the work function should convertedfrom eV to J (1 eV = 1602 x 10-19 J)= 363 eV x (1602 x 10-19 J1 eV) = 582 x 10-19 J
0 = 6626 x 10-34 J s x 300 x 108 m s-1
582 x 10-19 J
= 342 x 10-7 m = 342 nm
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
31
(b) Ek =12
x 9109 x 10-31 kg x (785 x 105 m s-1)2
= 281 x 10-19 J
From Einsteins equation hc = Ek +
=6626 x 10-34 J s x 300 x 108 m s-1
(281 x 10-19 J + 582 x 10-19 J)
= 230 x 10-7 m = 230 nm
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
32
Summary of 14Black bodyradiation
Plancksquantumhypothesis
Particulate natureof electromagneticradiation
Photoelectriceffect
Bohrs frequency condition h = Eupper - Elower
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
15 The Wave-Particle Duality of Matter15 The Wave-Particle Duality of Matter
ndash Wave behavior of light diffraction and interference effects of superimposed waves (constructive and destructive)
ndash Louis de Broglie proposed that all particles have wavelike properties
is the de Broglie wavelength ofan object with linear momentum p = mv
electron diffraction reflected from a crystal
33
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
34
De Broglie Wavelengths for Moving Objects
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
16 The Uncertainty Principle16 The Uncertainty Principle
Complementarity of location (x) and momentum (p)
ndash uncertainty in x is x uncertainty in p is p
Heisenberg uncertainty principle
where ħ = h 2 = 10546times10-34 Jmiddots
ndash x and p cannot be determined simultaneously
35
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
36
Example Calculation
If the Bohr radius of the H atom is 0529 A and we know theposition of the electron to within 1 uncertainty calculatethe uncertainty in the electrons velocity
x =100
0529 x 10-10 (m) = 529 x 10-13 m
From the Heisenberg uncertainty principle xp gt h4
p gt6626 x 10-34 (J s)
4 x 3142 x 529 x 10-13 (m)or gt 996 x 10-23 kg m s-1
Because p = mv the uncertainty in the velocity is v =
996 x 10-23 (kg m s-1)
9110 x 10-31 (kg)
= 109 x 108 m s-1 an uncertainty that is the magnitude ofthe speed of light
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
17 Wavefunctions and Energy Levels17 Wavefunctions and Energy Levels
Erwin Schroumldinger introduced a central concept of quantum theory
particle trajectory wavefunction
ndash Wavefunction ( psi) a mathematical function with values that vary with position
ndash Born interpretation probability of finding the particle in a region is proportional to the value of 2
ndash Probability density (2) the probability that the particle will be found in a small region divided by the volume of the region
37
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
38
Schroumldinger EquationSchroumldinger Equation WaveWave Equation for ParticlesEquation for Particles
The classic differential equation describing a standing wave in one dimensionis
d2dx2 +
42
2= 0
From de Broglies hypothesis 2 = h22mK = h2[2m(E-V)]
( = wavefunction = wavelength) (1)
Substituting for in (1) gives
d2dx2 +
[82m(E-V)]
h2= 0 or d2
dx2_ h2
82m_ V = E (2)
h2i
(K is kinetic and Vispotential energy)
ddx
instead of p = mv = mdxdt
This implies that the (electron wave) momentum p is
Hence K = p 2
2m=
12m
h
2i
2=
ddx
2 d2
dx2_ h2
8p2m(3)
Combining (2) and (3) gives K + V =K + V = E
That is H = E
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Schroumldinger equation for a particle of mass m moving in one dimension in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schroumldinger equation
If the wavefunction is described as(x) = A sin 2x
d2(x)dx2
= - 2
2 (x) d2(x)dx2
= - 2h
2 (x)p = hp
d2(x)dx2
= (x)ħ2
2m- p2
2m kinetic energy V(x) potential energy
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
The lsquoparticle in a boxrsquo scenario is the simplest application of the Schroumldinger equation
ndash Mass m confined between two rigid walls a distance L apart ndash = 0 outside the box and at the walls (boundary condition) ndash Potential energy is zero within and infinite outside the box
n = quantum number
Particle in a Box
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m
Whole-number multiples of half-wavelengthscan follow the boundary condition
When this expression for is inserted intothe energy formula
41
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
More General Approach
From the Schroumldinger equation with V(x) = 0 inside the box
Solution
k2 = 2mEħ2 and it follows
From the boundary conditions of (0) = 0 and (L) = 0
0 L
42
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Wavefunction obtained so far is (with just A leftto identify)
The normalization condition determines A
n(x) = A sin nxL
n(x) = sin nxL
2
L
Hence
n2
= A2L
0
sin2 nxL dx = 1 A = 2L
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Energies of a particle of mass m in a one dimensional box of length L
Energy of the particle is quantized and restricted to energy levels
- Energy quantization stems from the boundary conditions on the wavefunction
- Energy separation between two neighboring levels with quantum numbers n and n+1
n = quantum number
- L (the length of the box) or m (the mass of the particle) increases the separation between neighboring energy levels decreases
44
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Zero-point energy
The lowest value of n is 1 and the lowest energy is E1 = h28mL2 not zero
According to quantum mechanics aparticle can never be perfectly stillwhen it is confined between two wallsit must always possess an energy
consistent with the uncertaintyprinciple
ndash The shapes of the wavefunctions of a particle in a box
E1 = h28mL2 E2 = h22mL2
45
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
EXAMPLE 18
Treat a hydrogen atom as a one-dimensional box of length 150 pmcontaining an electron Predict the wavelength of the radiationemitted when the electron falls to the lowest energy level from thenext higher energy level
n = 1 n+1 = 2 m = me and L = 150 pm
= h = hc
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
THE HYDROGEN ATOM
18 The Principal Quantum Number
19 Atomic Orbitals
110 Electron Spin
111 The Electronic Structure of Hydrogen
47
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
THE HYDROGEN ATOM (Sections 18-111)
18 The Principal Quantum Number18 The Principal Quantum Number
A particle in a box An electron held within the atom bythe pull of the nucleus
For a hydrogen atom V(r) = coulomb potential energy
Solutions of the Schroumldinger equation lead to the expressionfor energy
R (Rydberg constant) = 329times1015 Hz in good agreement with experiment
48
n is the principalquantum number
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
For other one-electron ions such as He+ Li2+ and even C5+
- Z = atomic number equal to 1 for hydrogen
- n = principal quantum number
- As n increases energy increases the atom becomes less stable and energy states become more closely spaced (more dense)
- Ground state of the atom the lowest energy state E = ndashhR when n = 1
- Ionization the bound electron reaches E = 0 and freedom and has left the atom Ionization energy the minimum energy needed to achieve ionization
49
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
50
19 Atomic Orbitals19 Atomic Orbitals
h2
82m
_
x2
2
y2
2
z2
2
+ + + V(xyz) (xyz) = E (xyz)
2 _ h2
2m+ V = Eor
2becomes
1
r2
2
r2
rr +1
r2sin sin +
1
r2sin2 2
or H = E
in spherical polar coordinates
The Schroumldinger equation for the H atom is often written as below
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
51
Spherical Polar Coordinate System
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Atomic orbitals the wavefunctions () of electrons in atoms they are meaningful solutions of the Schroumldinger equation
ndash The square of a wavefunction (2) is the probability density of an electron at each point
ndash Expressing wavefunctions in terms of spherical polar coordinates (r is more convenient
radialwavefunction
depends on two quantumnumbers n and l
angularwavefunction
depends on two quantumnumbers l and ml
52
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
53
For the ground state of the hydrogen atom (n = 1)
Bohr radius = 529 pmHere Y is a constant (independent of angle) the wave-function is the same in all directions it is spherically symmetrical)This is the 1s orbital It is the only wavefunction for n = 1
Its spherical electron cloud representationand plot of 2 versus r are shown opposite
2
r
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
54
Hydrogenlike Wavefunctions
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
2012 General Chemistry I 5555
Boundary Conditions
Boundary conditions are constraints that reality places on the solutions to a physically relevant equation (eg a quadratic equation and here the Schrodinger equation for the H atom)
The solutions of the Schrodinger equation (wavefunctions ) are thus seen to be lsquowell-behavedrsquo
Ψ must be smooth single-valued and finite everywhere in space
Ψ must become small at large distances r from the nucleus (proton)
r cosr cosΘΘ= z= z
Boundary Condition yields quantum numbers
Ψ is just the embodiment of de Brogliersquos hypothesis of matter waves)
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Three quantum numbers (n l ml) specify an atomic orbital
n principal quantum number (n = 1 2 3hellip)
bull It is related to the size and energy of the orbital
bull It defines shell AOs with the same n value
56
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
l orbital angular momentum quantum number (l = 0 1 2 hellip n-1)
bull It defines subshell AOs with the same n and l values n subshells with a principal quantum number n
Value of l 0 1 2 3
Orbital type s p d f
bull It is related to the orbital angular momentum of the electron
(Orbital angular momentum = )
ml magnetic quantum number (ml = +l l-1 l-2 hellip 0 hellip -l)
bull There are 2l+1 different values of ml for a given value of leg when l = 1 ml = +1 0 -1
bull It is related to the orientation of the orbital motion of the electron
57
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
A summary of the three quantum numbers
58
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
DegeneracyDegeneracy
- ldquoNormallyrdquo the energy should depend on all three quantum
numbers
- Hydrogen atom is special in that the energy depends only on
principal quantum number n
- Two or more sets of quantum numbers corresponds to the same energy are referred as ldquodegeneraterdquo
Eg 200 211 210 21-1 (for n = 2) states have the same energy
- Each n given total energy rarr n2 possible combinations of
quantum numbers (total number of orbitals) rarr degeneracy
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Shape of the s-Orbitals (l = 0)
Radial distribution function the probability that the electron will be found anywhere in a thin shell of radius r and thickness r is given by P(r)r with
For s orbitals = RY = R212
Visualizing AO as a cloud of points proportional to probability of finding the electron in that volume (probability density 2 plot versus distance r)
60
httpwwwmpcfacultynetron_rinehartorbitalshtm
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
2012 General Chemistry I 61Department of Chemistry KAIST
61
RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
27s
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
62
a function of r only
spherically symmetric
exponentially decaying
no nodes
- H-atom (The only atom for which the Schroumldinger Eq can be solved exactly a model for bigger and many-electron atoms)
- 1s orbital (n = 1 ℓ = 0 m = 0) rarr R10(r) and Y00(θФ)
- 2s orbital (n = 2 ℓ = 0 m = 0)
zero at r = 2a0 = 106Aring
nodal sphere or radial node
[rlt2ao Ψgt0 positive] [rgt2ao Ψlt0
negative]
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
63
Self-Test 19ACalculate the ratio of the probability density forthe 1s orbital at r = 2a0 and r = 0
Probability density at r = 2a0Probability density at r = 0
= 2(2a0)
2(0)
From Table 2 2 = e -2ra0
a03
Hence2(2a0)
2(0)=
e -4a0a0
a03
_ e 0
a03
= e-4 = 00183
e-4
1
Solution
The probability is just under 2 of that findingthe electron in a region of the same volumelocated at the nucleus
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Visualizing AO as a boundary surface that encloses most of the cloud
The 95 boundary surface
Surface enclosing volume where probability of finding an electron is 95
radial wavefunction versus radius
64
A radial node exists where the curve crossesthe x-axis
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Shape of the 2p-Orbitals
Boundary surface Radial function
- Two lobes with + and - signs
- Nodal plane ( = 0) separating the two lobes Also called angular node No radial node for 2p orbitals
- l = 1 and ml = +1 0 -1triply degenerate in energy
- three p-orbitals px py pz
65
+
+
+_ _
_
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
66
-n = 2 ℓ = 1 m = 0 rarr 2p0 orbital R21 Y10
middot Ф = 0 rarr cylindrical symmetry about the z-axismiddot R21(r) rarr ra0 no radial nodes except at the originmiddot cos θ rarr angular node at θ = 90o x-y nodal plane
(positivenegative)
middot r cos θ rarr z-axis 2p0 rarr labeled as 2pz
The 2p0 or 2py Orbital
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Department of Chemistry KAIST
- px and py differ from pz only in the
angular factors (orientations)
The 2px and 2py orbitals
Here n = 2 ℓ = 1 ml = plusmn1 rarr 2p+1 and 2p-1
Their wave functions contain the complex term eplusmniφ
which makes description difficult
Since eplusmniφ = cosφ plusmn isinφ (Eulerrsquos formula) a linear
combination of 2p+1 and 2p-1 gives two real orbitals
2px and 2py that complement 2pz
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Shape of the 3d- and 4f-Orbitals
3d-orbitals
l = 2 and ml = +2 +1 0 -1 -2
Each has two angularNodes No radial nodes for 3d orbitals
4f-orbitals
l = 3 and seven ml values
68
+
+
++
++ +
++
+_
__ _ _
_
_ _
_
Each has three angular nodes No radial nodes for 4f orbitals
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
69
A Note on Radial (Spherical) and Angular Nodes
bull Nodes are regions of space (spheres planes or cones) where = 0
bull The total number of nodes possessed by a given orbital = n ndash1
bull The number of angular nodes for a given orbital = l
bull The remainder (n ndash 1 ndash l) are radial or spherical nodes
bull Example 1 4s orbital (n = 4 l = 0) has zero angular nodes and 3 radial nodes
bull Example 2 5d orbital (n = 5 l = 2) has 2 angular nodes and 2 radial nodes
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
70
110 Electron Spin110 Electron Spin
bull Schroumldingerrsquos theory although successful has several inadequacies
1 The time-dependent equation is 2nd order with respect to space but only 1st order in time
2 It ignores relativity
3 It does not account for electron spin bull The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920) See nextbull Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the doublet at 5892 and 5898 nm (the lsquoD-linersquo) in the emission spectrum of sodium
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Electron Spin States
- An electron has two spin states as uarr(up) and darr(down) or a and b
ms Spin magnetic quantum number
- The values of ms only +12 and -12 for the electron
71
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Department of Chemistry KAIST
Diracrsquos Relativistic Electron
Dirac (1928)Dirac (1928) Using the Schroumldingerrsquos idea and Einsteinrsquos (1905) special theory of
relativity rarr 4 coupled Schroumldinger-like equations (Dirac Eq)
Dirac equations rarr 4th quantum number ms
(Electrons are required to have spin a law of nature NOT a postulate)
Dirac predicted rarr antimatter (antiparticle negative energy)
Dirac Equation for spin -12 particles(relativistic modification of the Schroumldinger wave equation)
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
73
Summary
Inadequacies ofSchrodingers theory
Electronspin
Stern and Gerlachexperiments (1920)
Uhlenbeck andGoudsmit explanationof sodium D-line doublet(1925)
Diracs equation(combination ofspecial relativity withwave mechanics 1928)Spin is a fundamentalproperty of electronsSpin magnetic quantumnumber ms = +12 and-12
THEORY
EXPERIMENTAL
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
111 The Electronic Structure of Hydrogen111 The Electronic Structure of Hydrogen
Ground state n = 1 and there is only the 1s orbitalThe 1s electron has the following quantum numbers
n = 1 l = 0 ml = 0 ms = +12 or -12
Excited states are achieved by absorption of photons
- The first excited state with n = 2 has four orbitals 2s 2px 2py or 2pz
The average distance of an electron from the nucleus increases
- The next excited state with n = 3 has nine orbitals one 3s three 3p and five 3d orbitals with larger distances
- Eventually the electron can escape the atom by absorbing enough energy and thus ionization of hydrogen occurs
74
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
75
Orbital Energy Diagram for Hydrogen
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
76
Self-Test 110A
The three quantum numbers for an electron in a hydrogenatom in a certain state are n = 4 l = 2 ml = -1 In what typeof orbital is the electron located
= 2 d type n = 4 4d
Solution
l
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Chapter 1Chapter 1ATOMS THE QUANTUM WORLDATOMS THE QUANTUM WORLD
2012 General Chemistry I
MANY-ELECTRON ATOMS
THE PERIODICITY OF ATOMIC PROPERTIES
112 Orbital Energies113 The Building-Up Principle114 Electronic Structure and the Periodic Table
115 Atomic Radius116 Ionic Radius117 Ionization Energy118 Electron Affinity119 The Inert-Pair Effect120 Diagonal Relationships121 The General Properties of the Elements
77
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
MANY-ELECTRON ATOMS (Sections 112-114)
112 Orbital Energies112 Orbital Energies- In many-electron atoms Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions- There are no exact solutions of the Schroumldinger equation
- For a helium atom
r1 = the distance of electron 1 from the nucleus
r2 = the distance of electron 2 from the nucleus
r12 = the distance between the two electrons
78
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
The hydrogen atom no electron-electron repulsionAll the orbitals of a given shell are degenerate
Many-electron atoms electron-electron repulsionsThe energy of a 2p-orbital gt that of a 2s-orbital
Shielding
Each electron attracted by the nucleusand repelled by the other electrons
rarr shielded from the full nuclear attraction by the other electrons
- effective nuclear charge Zeffe lt Ze
the energy of electron
79
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Penetration
s-electron ndash very close to the nucleuspenetrates highly through the inner shell
p-electron ndash penetrates less than an s-orbital effectively shielded from the nucleus
In a many-electron atom because of the effects ofpenetration and shielding the order of energies of orbitals in a given shell is s lt p lt d lt f
80
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
81
The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurationsof the elements
1 Assume electrons lsquooccupyrsquo orbitals in such a way as to minimize the total energy (lowest energy first)
2 Assume a maximum of two electrons can lsquooccupyrsquo an orbital and these must have opposite spins (Paulirsquos Exclusion Principle no two electrons in an atom can have the same set of quantum numbers)
3 Assume electrons occupy unoccupied degenerate subshell orbitals first and with parallel spins (Hundrsquos rule)
In building-up electron configurations in order of energy subshell energy overlap must be taken into account (next slide)
113 The Building-Up Principle113 The Building-Up Principle
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
82
Subshell Energy Overlap
bull The complex shieldingrepulsion effects that occur when more and more electrons are lsquofedrsquo into orbitals during the building-up process leads to subshell energy overlap beyond n = 3
bull See Figs 141 and 144bull For example the 4s subshell is slightly lower in energy
than the 3d subshell and hence fills firstbull Likewise the 5s subshell fills before the 4d or 3f subshells
for the same reason See next slide
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
83
Alternative Pictorial Representation of Orbital Energy Order
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Electronic configuration of an atom is a list of all its occupied orbitals with the numbers of electrons that each one contains
H 1s1
84
Electron Configurations of the Elements in Period 1 (H and He) where n = 1
Element
Electronconfiguration
Orbital withelectron occupancy
He 1s2
- Closed shell a shell containing the maximum number of electrons allowed by the exclusion principle
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Li 1s22s1 or [He]2s1
- core electrons electrons in filled orbitals valence electrons electrons in the outermost shell
Be 1s22s2 or [He]2s2
- stable ionic form Be2+
- stable ionic form losing valence electrons Li+
85
Electron Configurations of the Elements in Period 2 (from Li to Ne) the valence shell with n = 2
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
B 1s22s22p1 or [He]2s22p1
C 1s22s22p2 or [He]2s22p2
C is first element to illustrate Hundrsquos rule If more than one orbital in a subshell is available add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals
parallel spinsrarr 1s22s22px
12py1
- Excited state An atom with electrons in energy states higher than predicted by the building-up principle
In carbon ground state [He]2s22p2 rarr excited state [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
87
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
All atoms in a given period have the same type of core with the same n
All atoms in a given group have analogous valence electron configurationsthat differ only in the value of n
Period 2
Period 3
Period 4
Group IA Group 18VIIIA
Period 5
Period 6
Period 7
88
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
n = 3 Na [He]2s22p63s1 or [Ne]3s1 to Ar [Ne]3s23p6
n = 4 from Sc (scandium Z = 21) to Zn (zinc Z = 30) the next 10 electrons enter the 3d-orbitals
The (n + l ) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n + l) first Subshells in a group with the same value of (n + l) are filled in the order of increasing n
order 1s lt 2s lt 2p lt 3s lt 3p lt 4s lt 3d lt 4p lt 5s lt 4d lt hellip
- There are exceptions two of which are Cr [Ar]3d54s1 instead of [Ar]3d44s2
and Cu [Ar]3d104s1 instead of [Ar]3d94s2 because of lower energy of half-filled (d5) and filled (d10) 3d subshell
n = 6 5s-electrons followed by the 4f-electrons Ce (cerium [Xe]4f15d16s2)
89
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
2012 General Chemistry I 90
Anomalous ConfigurationsAnomalous Configurations
Exceptions to the Aufbau principle
36
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
91
Self-Test 112A
Write the ground-state configuration of a bismuth
atom
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 6
these will be 6s26p3 The nearest noble gas is Xe (Z
= 54) leaving 24 electrons to be accounted for in
filled 4f and 5d subshells
The configuration is [Xe]4f145d10 6s26p3
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
92
Self-Test 112B
Write the ground-state configuration of an arsenic
atom
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell As it is in period 4
these will be 4s24p3 Also because As is in period 4
it will have an argon (Ar Z = 18) core The remaining
10 electrons are held in the filled 3d subshell The
configuration is [Ar]3d10 4s24p3
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
114 Electronic Structure and the Periodic 114 Electronic Structure and the Periodic TableTable
- H and He unique properties
93
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
- Main group s- and p-blocks (groups IA- VIIIA)-The Roman numeral group number tells us how many valence-shell electrons are present
Group IA Group 18VIIIA
The modern nomenclature is groups 1 2 and 13-18
94
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
-Period 1 H He 1s orbital- Period 2 Li through Ne 8 elements one 2s and three 2p orbitals- Period 3 Na through Ar 8 elements 3s and 3p- Period 4 K through Kr 18 elements 4s 4p and 3d- Period 5 Rb through Xe 18 elements 5s 5p and 4d- Period 6 Cs through Rn 32 elements 6s 6p 5d and 4f
95
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
THE PERIODICITY OF ATOMIC PROPERTIES(Sections 115-121)
96
The variation of effective nuclear charge throughout the periodic tableplays an important part in the explanation of periodic trends See below (Fig 145) Zeff refers to outermost valence electron
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
97
Atomic radius defined as half the distance between the centers of neighboring atoms
-For metals r in a solid sample is the metallic radius- For nonmetal r for diatomic molecules is the covalent radius- For a noble gas r in the solidified gas is the Van der Waals radius
- r decreases from left to right across a period (effective nuclear charge increases)
- r increases from top to bottom down a group (change in n and size of valence shell effective nuclear charge decreases)
General trends
115 Atomic Radius115 Atomic Radius
- See Figs 146 and 147 (next slides)
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
98
186
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
99
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
116 Ionic Radius116 Ionic Radius
Ionic radius its share of the distance between neighboring ions in an ionic solid
- Cations are smaller than parent atoms ie Zn (133 pm) and Zn2+ (83 pm)- Anions are larger than parent atoms ie O (66 pm) and O2- (140 pm)
Isoelectronic atoms and ions are atoms and ions with the same number of electrons
eg Na+ F- and Mg2+
radius Mg2+ lt Na+ lt F-
due to different nuclear charges
100
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Self-Tests 113A and 113B
Arrange each of the following pairs in order of increasing ionic radius (a) Mg2+ and Al3+ (b) O2- and S2-
Arrange each of the following pairs in order of increasing ionic radius (a) Ca2+ and K+ (b) S2- and Cl-
(a) Al lies to the right of Mg hence r(Al3+) lt r(Mg2+)
Solution
(b) O lies above S in group VIA hence r(O2-) lt r(S2-)
Solution
(a) Ca lies to the right of K therefore r(Ca2+) lt r(K+)
(b) Cl lies to the right of S therefore r(Cl-) lt r(S2-)
In both cases check agreement with Fig 148
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
117 Ionization Energy117 Ionization Energy Ionization Energy I is the minimum energy needed to remove an electron from an atom in the gas phase
The first ionization energy I1
The second ionization energy I2
- I1 typically decreases down a group (change in n of valence electron Zeff increases)- I1 generally increases across a period (Zeff increases)
- Metals lower left of the periodic table low ionization energies
- Nonmetals upper right of the periodic table high ionization energies
I1 (746 kJmiddotmol-1)
I2 (1958 kJmiddotmol-1)
102
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
103
The periodic variation of the first ionization energies of the elements (Fig 151)
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
104
For a particular element second (I2) third (I3) and higher ionization energies are greater than I1 because of the increasing ionic chargeVery high ionization energies occur when an electron is removed from an inner shell See Fig 152 (opposite)Blue outline denotesionization from thevalence shell
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
118 Electron Affinity118 Electron Affinity
Electron Affinity Eea
The energy released when an electron is added to a gas-phase atom
eg
- Eea generally decreases down a group (change in n of valence electron Zeff decreases)
- Eea generally increases across a period (Zeff increases)
ndash Group 17VIIA the highest 1st Eea (F-) strongly negative 2nd Eea (F2-)
ndash Group 16VI positive 1st Eea (O-) negative 2nd Eea (O2-)
105
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
Self-Test 115B
Account for the large decrease in electron affinity between fluorine and neon
Solution
For F (1s22s22p5) F (1s22s22p6) an electronenters the 2p subshell and completes the octet
For Ne (1s22s22p6) Ne ([Ne]3s1) the electronenters the energetically higher 3s subshell
__
__e
e
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
119 The Inert-Pair Effect119 The Inert-Pair Effect
ndash Tendency to form ions two units lower in chargethan expected from the group number
ndash Due in part to the different energies of the valence p- and s-electrons
120120 Diagonal RelationshipsDiagonal Relationships
ndash A similarity in properties between diagonal neighbors in the main groups of the periodic table
ndash Similarity in atomic radius ionization energy and chemical property
107
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
121 The General Properties of the Elements121 The General Properties of the Elements s-Block elements low ionization energy easy to lose electrons likely to be a reactive metal
Left side of p-block (heavier elements) relatively low ionization energy metallicmetalloid but less reactive than those in s-block
Right side of p-block high electron affinities tend to gain electrons mostly nonmetals forming molecular compounds
s-blockright
p-block
leftp-block
108
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
d-block metals transitional between the s- and p-block elements called ldquotransition metalsrdquo they have similar properties and form ions with different oxidation states (Fe2+ and Fe3+) They have catalytic properties facilitating subtle changes in organisms They form alloys
Lanthanoids metals incorporated in superconducting materials Actinides radioactive many do not naturally exist on earth
f-Block
d-Block
109
