Atomic Structure and Periodic Trends. n Lectures: week1: W9 am; week2: W9 am (ICL) & 11 am (DP), F 9...

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Atomic Structure and Periodic Atomic Structure and Periodic TrendsTrends

Atomic Structure and Periodic Atomic Structure and Periodic TrendsTrends Lectures:

week1: W9 am; week2: W9 am (ICL) & 11 am (DP), F 9 am (DP); week 3: 9 am (ICL) Books:• Inorganic Chemistry by Shriver and Atkins• Physical Chemistry by P.W.Atkins , J. De Paula • Essential Trends in Inorganic Chemistry D Mingos • Introduction to Quantum Theory and Atomic Structure by P. A. Cox Other resources:

– Web-pages: http://timmel.chem.ox.ac.uk/lectures/– Ritchie/Titmuss, Quantum Theory of Atoms and Molecules, Hilary Term

Why study atomic electronic Why study atomic electronic structure?structure?

All of chemistry (+biochemistry etc.) ultimately boils down to molecular electronic structure.

Reason: electronic structure governs bonding and thus molecular structure and reactivity

Before it is possible to understand molecular electronic structure we need to introduce a number of concepts which are more easily demonstrated in atoms.

atomic structure molecular structure chemistry

The Periodic TableThe Periodic Table

Mendeleyev was the first chemist to understand that all elements are related members of a single ordered system. From his table he predicted the properties of elements then unknown, of which three (gallium, scandium, and germanium) were discovered in his lifetime.

Mendeleyev, Dmitri Ivanovich (1834-1907)

This course......This course......

will introduce new concepts gradually starting with the “simplest”:

H-Atom Energy levels, Wavefunctions, Born Interpretation, Orbitals

Many electron atoms Effects of other electrons, Penetration, Quantum Defect

The Aufbau Principle Electronic Configuration of atoms and their ions

Trends in the PT Ionisation Energy, Electron Affinity, Size of atoms and ions

The H-AtomThe H-Atom

The Hydrogen AtomThe Hydrogen Atom

The simplest possible system and the basis for all others:

vElectron orbits the proton under the influence of the Coulomb Force:

H-Atom: consider the energyH-Atom: consider the energy

ETotal =

Energy Levels?Energy Levels?

Consider 3 approaches: Classical Bohr Model (old quantum theory) Full Quantum: Schrödinger Equation

Classically: There is no theoretical restriction at all on v or r

(there are infinitely many combinations with the same energy).

Hence the energy can take any value - it is

. As a result transitions should be possible everywhere

across the electromagnetic spectrum. So lets see the spectrum...

The H-atom Emission SpectrumThe H-atom Emission Spectrum

Infra-red

Ultra-violet XUVvisible

lens lens

Principles of Quantum MechanicsPrinciples of Quantum Mechanics Quantization Quantization

Energy levels

The Rydberg FormulaThe Rydberg Formula

1c yRv

In 1890 Rydberg showed that the frequencies of all transitions could be fit by a single equation:

Bohr Theory (old quantum)Bohr Theory (old quantum) Bohr explained the observed frequencies by

restricting the allowed orbits the electron could occupy to particular circular orbits(by quantizing the angular momentum).

His theory gives energy levels:

e where

The problem with Bohr TheoryThe problem with Bohr Theory

Bohr theory works extremely well for the H-atom.

However; it provides no explanation for the

quantization of energy -it just happens to fit the observed spectrum

But, more seriously: it just doesn’t work at all for any other atoms

Quantum mechanical PrinciplesQuantum mechanical Principlesand the Solution of the and the Solution of the Schrödinger EquationSchrödinger Equation

Principles of Quantum MechanicsPrinciples of Quantum Mechanics Quantization Quantization

Energy levels2

Quantum mechanicsClassical mechanics

is continuous

Principles of Quantum MechanicsPrinciples of Quantum MechanicsIt’s all about probabilityIt’s all about probability

In classical mechanicsPosition of object specified

In quantum mechanicsOnly

of object at a particular location

Principles of Quantum MechanicsPrinciples of Quantum Mechanics

How do we describe the How do we describe the electrons in atoms?electrons in atoms?

You know: Electrons can be described as (characterised by mass, momentum, position…)

p = h/ De Broglie

h = Planck’s constant = 6.626 10-34 Js

However: Electrons can also be described as

(characterised by wavelength, frequency, amplitude)

and show properties such as interference, diffraction

The Davisson Germer ExperimentThe Davisson Germer Experiment

Proving the wave properties of electrons (matter!)Intensity variation in diffracted beam shows constructive and destructive interference of wave

Principles of Quantum Mechanics Principles of Quantum Mechanics

The WavefunctionThe Wavefunction

(position, time

In quantum mechanics, an electron, just like any other particle, is described by a

Contains all information there is to know about the particle

The Results of Quantum MechanicsThe Results of Quantum Mechanics

Schrödinger equation:

where ,zyx 2

is the wavefunction,V(r) the potential energy andE the total energy

Spherical Polar CoordinatesSpherical Polar Coordinates

Instead of Cartesian (x,y,z) the maths works out easier if we use a different coordinate system:

z x = r sin cosy = r sin sin z = r cos

(takes advantage of the spherical symmetryof the system)

So Schrödinger’s Equation So Schrödinger’s Equation becomes.....becomes.....

2as before

But now2

sinsin

with r4ππ

We separate the wavefunction into 2 parts: a radial part R(r) and an angular part Y(,),

such that =

The solution introduces 3 quantum numbers:

....which can be solved exactly for the H-....which can be solved exactly for the H-atom atom with the solutions called with the solutions called orbitalsorbitals, more , more specifically, specifically, atomic orbitals.atomic orbitals.

We separate the wavefunction into 2 parts: a radial part R(r) and an angular part Y(,),

such that =R(r)Y(,)

The solution introduces 3 quantum numbers:

The quantum numbers;The quantum numbers;

quantum numbers arise in the solution; R(r) gives rise to:

the principal quantum number, n Y(,) yields:

the orbital angular momentum quantum number, l and the magnetic quantum number, ml

i.e., =Rn,l(r)Yl,m(,)

The values of The values of nn,, l l, &, & m mll

n = 1, 2, 3, 4, .......

l = 0, 1, 2, 3, ......(n-1)

ml = -l, -l+1, -l+2,..0,..., l-1, l

You are familiar with these.....You are familiar with these.....

n is the integer number associated with an orbital

Different l values have different names arising from early spectroscopy

e.g., l =0 is labelled

l =1 is labelled

l =2 is labelled

l =3 is labelled etc...

Looked at another way.....Looked at another way.....

n =1 l=0 ml=0 1s orbital (1 of)n =2 l=0 ml=0 2s orbital (1 of)

l=1 ml=-1, 0, 1 2p orbitals (3 of)n =3 l=0 ml=0 3s orbital (1 of)

l=1 ml=-1, 0, 1 3p orbitals (3 of)l=2 ml=-2,-1,0,1,2 3d orbitals (5 of)

etc. etc. etc.Hence we begin to see the structure behind

the periodic table emerge.

Exercise 1;Exercise 1;

Work out, and name, all the possible orbitals with principal quantum number n=5.

How many orbitals have n=5?

The Radial WavefunctionsThe Radial Wavefunctions

The radial wavefunctions for H-atom are the set of Laguerre functions in terms of n, l

a0 (=0.05292nm)is the-the most probable

orbital radius of an H-atom 1s electron.

3s 3p3d

Revisit: The Born InterpretationRevisit: The Born Interpretation

The square of the wavefunction,

at a point is proportional to the

of finding the particle at that point.

In 1D: If the amplitude of the wavefunction of a particle at

some point x is xthen the probability of finding the

particle between x and x +dx is proportional to

(x)(x)dx (x)

is the complex conjugate of

R(r)2R(r)1s

Radial Wavefunctions and the Radial Wavefunctions and the Born InterpretationBorn Interpretation

At long distances from the nucleus, all wavefunctions decay to .

Some wavefunctions are zero at the nucleus (namely, all but the l=0 (s) orbitals). For these orbitals, the electron has a zero probability of being found .

Some orbitals have nodes, ie, the wave function passes through zero; There are such radial nodes for each orbital.

In 3D: If the amplitude of the wavefunction of a particle

at some point (x,y,z) is x,y,zthen the probability of

finding the particle between x and x+dx, y+dy and z+dz,

ie, in a volume dV = dx dy dz is given by

(x,y,zdV

It follows therefore that

2(x,y,z)

is a probability density.

In spherical coordinates2(r

More important to know probability of finding More important to know probability of finding

electron at a given distance from nucleuselectron at a given distance from nucleus!!

dxdydz

dV=dxdydz

Cartesian coordinates not very useful to describe orbitals!

…..in a shell of

dA = r2sindddA=dxdydV = r2sindddr

Surface element

Volume element

The Surface area of a The Surface area of a spheresphere is hence: is hence:

( cos cos0)(2 0)

For spherically symmetric orbitals:

the radial distribution function is defined as

P(r)= r2 (r)

and P(r)dr is the probability of finding the electron in a shell of radius r and thickness dr

Construction of the radial Construction of the radial distribution functiondistribution function

For spherically symmetrical orbitals

Radial distribution function P(r)Radial distribution function P(r)Im

Born interpretationprob = d

Hence plot P(r) = r2R(r)2

P(r)=4r

for spherical symmetry)

Pn(r) has

So what do we So what do we learn?learn?

The 2p orbital is on average closer to the nucleus, but note the 2s orbital’s high probability of being

3d vs 4s3d vs 4sP(r)

The Angular WavefunctionThe Angular Wavefunction

The Ylm(,) angular part of the solution form a set of functions called the

n.b. These are generallyimaginary functions -we usereal linear combinations topicture them.

The Shapes of Wavefunctions The Shapes of Wavefunctions (Orbitals)(Orbitals)

Shapes arise from combination of radial R(r) and angular parts Ylm(,) of the wavefunction.

Usually represented as boundary surfaces which include of the probability density. The electron does not undergo planetary style circular orbits.

These are the familiar spherical s-orbitals and dumbell-shaped p-orbitals, etc...

Electron densities representationsElectron densities representations

of corresponding wavefunctions (white areas)

Signs of corresponding wave functions

*Please note that R and R2 are arbitrarily scaled.

The s-orbitals:R2

Boundary model of an s orbital within which there is 90% probability of finding the electron

Boundary ModelBoundary Model

The p-orbitals: boundary surfacesThe p-orbitals: boundary surfaces- actually imaginary functions but linear combinations give the familiar dumbells

px pzpy

nt Different shades denote

different signs of the wavefunction

One Nodal plane

The d-orbitals : boundary surfacesThe d-orbitals : boundary surfacesIm

Again, different shades denote different signs of the wavefunction

dxz dxydyzdxz

dz2dx-y

The d-orbitals : boundary surfacesThe d-orbitals : boundary surfaces

Again, different shades denote different signs of the wavefunction

dxz dxydyzdzx

TwoNodal planes

dxz dxydyzdxz

Two Nodal planes which split orbital into 4 lobes, orbitals lie in a plane perpendicular to the two nodal panes and point between the axes

The d-orbitals : boundary surfacesThe d-orbitals : boundary surfaces

Two Nodal planes which split orbital into 4 lobes, orbitals lie in xy-plane, pointing along the x and y axes, nodal planes at 45o to xz- and yz-planes

Cylindrical symmetry, two angular nodes which take the form of cones at 54.7o and 125.3o to the z-axis.

125.3o

The energies of orbitalsThe energies of orbitals

In the case of the H-atom (& only the H-atom) the energy is determined exclusively by the principal quantum number, n:

yRwhere Ry is the Rydberg constant

Ry for H-atom = 109 677 cm-1 (= Ionization energy, 13.6eV)

H-atom Energy H-atom Energy LevelsLevels

i) All levels with the same n

i.e., E(3s)=E(3p)=E(3d)

ii) All energies are negative(because the electron is bound)iii) n= by definition has energy zero, hence E- E1= ionization energy (13.6 eV)= Ry (109 677 cm-1)

The Ionization EnergyThe Ionization Energy

Corresponds to the total removal of an electron (i.e., transition to n=)

Since in the ground state H-atom n=1 is the highest occupied atomic orbital, the ionization energy is given by:

11hcEnergyIonization yy RR

See later for Koopman’s theorem

Other AtomsOther Atoms

A) A) Other single electron Other single electron atoms/ions (H-like atoms)atoms/ions (H-like atoms)

The ions He+, Li2+, Be3+, B4+,... etc. are said to be with the H-atom (same electron configuration).

Again ns, np, nd etc. levels are degenerate. However the attractive Coulomb force is much

bigger due to the Z+charge of the nuclei. The energy levels are given by a similar

expression to that for the H-atom with the inclusion of a :

Note that exponential function decays faster as Z

Orbitals contract with increasing Z

H-like atoms/ions : SummaryH-like atoms/ions : Summary The orbitals contract with increasing Z The effect of the Z2 term is to increase the energy level

spacing. E.g., The energy level spacings in the He+ spectrum are approx.

4 times those in the H-atom whilst in Be3+ they are 16 times larger (Z=4).

This is only approximate because of the slight mass dependence of the Rydberg constant.

Ry(H) = 109 677cm-1 Ry(mass) = 109 737 cm-1

Compare ionisation energies for IE(H) = 13.6eV, IE(He+) = 54.4eV, IE(Li2+)=122.4eV

B) Multi-electron Atoms B) Multi-electron Atoms

Schrödinger equation cannot be solved analytically anymore (apart from He)

Need to develop an approximate picture for multi-electron atoms

Helium atom

The orbital approximation in The orbital approximation in quantum mechanicsquantum mechanics

r1, r2, …) = r1)(r2)….

Total wavefunction of many electron atom

Each electron is occupying its individual orbital with nuclear charge modified to take account of all other electrons’ presence (repulsion!)

The orbital approximation in wordsThe orbital approximation in wordsMake multi electron atom look like a one electron atomAssumes every electron to be on its own experiencing an effective nuclear charge, Zeff

Then the orbitals for the electrons take the form of those in hydrogen but their energies and sizes are modified by simply using an effective nuclear charge

One electron atom

- - - -Zeff

Two-electron atom Orbital Approximation

The Concept of Electron SpinThe Concept of Electron Spin

The solution of the Schrödinger equation above accounts very well for the structure of the H-atom spectrum and, as we will see, for other atoms too.

However under very high resolution “fine structure” is observed in the transitions which is not explained using this approach.

Dirac extended wave mechanics to include Special relativity and an extra coordinate - time.

Solution of the Dirac equation yields a new intrinsic angular momentum -

The Electron SpinThe Electron Spin

An electron has The projection of this spin is also quantized

(by analogy with orbital angular momentum; l and ml) such that the spin projection quantum number, ms=½, (spin up or ) and ms=-½, (spin down or )

This is the final theoretical plank behind the structure of the periodic table.

Pauli Pauli ExclusionExclusion Principle Principle

(as all electrons necessarily have the same s=1/2) Hence, no individual orbital may be occupied by more

than 2 electrons Electrons occupying the same orbital must be “paired up”.

no two electrons in the same atom can have the same 4 quantum numbers n, l, ml, ms

The Pauli Principle and SymmetryThe Pauli Principle and Symmetry

The Pauli Exclusion Principle applies to any

pair of identical (spin quantum

number s = half integer) – protons, electrons,

neutrons have s = ½ and are fermions.

BUT, any number of (s = integer)

can occupy the same orbital 12C (s = 0) and

the photon (s = 1)are bosons

Now, as we proceed from H to Now, as we proceed from H to other atoms, we need to considerother atoms, we need to consider

1. The Pauli Exclusion Principle

2. The coulombic repulsions between the electrons

Two-electron atom

B)B) Atoms with one outer electron Atoms with one outer electron

Easiest: alkali metals Li, Na, K, Rb but also Be+, Mg+, Ag (4d105s1) as only one outer electron.

But still, the situation is no longer as simple as the H-atom due to the inner electrons - the “core”;

The core electrons will

tend to the outer

electron from the full

nuclear charge.

Shielding and PenetrationShielding and Penetration If an electron is always outside the core it experiences

only a net charge of nucleus and core. If, however, the electron spends much of its time close to

the nucleus (within the core) it will experience a larger nuclear attraction and have a lower energy (more tightly bound).

Hence the energy of the outer electron depends on how much it the core region.

This in turn depends on the type (s, p, d, f etc.) of orbital it is in.

Recall the radial distribution Recall the radial distribution functions.....functions.....

An e- in the 3s orbital spends more time close to the nucleus than an electron in 3p and is thus more tightly bound (lower energy).

nucleusImp

RamificationsRamifications

The energy of a given quantum state is now no longer simply a function of its principal quantum number but also of its penetration into the core region which depends on the orbital shape (and thus l).

i.e., E=En,l

In general the energies of sub-shells of the same principal quantum number n lie in the order

ZZeffeff – the effective nuclear charge – the effective nuclear charge To account for the effects of penetration and shielding we

nt an effective nuclear charge Zeff such that

2 hc)(hcE

RZ yyeffnl

where is the shielding parameter and Z is the charge of the nucleus.

Zeff is a function of n and l as electrons in different shells and subshells approach nucleus to different extents

Trends in Trends in ZZeff eff

The Grotrian diagram for NaThe Grotrian diagram for Na

•Different l levels have different energy.

•The H-atom levels are marked on the RHS

•Note more rapid stabilisation of 4s with respect to 3d due toH-Atom

energylevels

Radial Distribution FunctionRadial Distribution Function3d vs 4s3d vs 4s

The Aufbau Principle and the The Aufbau Principle and the Structure of the Periodic TableStructure of the Periodic Table

Electron ConfigurationsElectron Configurations

To obtain a ground state configuration for an atom we apply the Pauli exclusion and the Aufbau principle which states that electrons are added to orbitals in increasing order of energy.

n lPossible # of e

n = 22p

l = 1 (p)

l = 0 (s)

l = 1 (p)

l = 2 (d)

# of orbitals2l+1

n = 1 l = 0 (s) 1s

Principles of how to build up electron Principles of how to build up electron configurationsconfigurations

The Aufbau Principle - “The building-up”principle

When establishing the ground state configuration of an atom start at the energetic bottom and work your way up

2sNB: The energy ordering of the orbitals changeswith the number of electrons.

Principles of how to build up electron Principles of how to build up electron configurationsconfigurations

(1) The Pauli Exclusion Principle - No two electrons in one atom may have the same set of four quantum numbers (that is they must differ in one or more of n, l, ml , ms)

Differ in Differ in

Differ in

Remember, when two electrons share Remember, when two electrons share one orbital, their magnetic spin one orbital, their magnetic spin

quantum numbers must bequantum numbers must be up down

Paired electron spins

Principles of how to build up Principles of how to build up electron configurationselectron configurations

(2) HUND’s Rule - When electrons occupy orbitals of the same energy, the lowest energy state corresponds to the configuration with the greatest number of unpaired electrons

E(Paired) E(Unpaired)

Energetically favored

Maximizing the number of parallel spins - Maximizing the number of parallel spins - The exchange interactionThe exchange interaction

Quantum mechanical in origin Arguments based on the fact that total

wavefunction has to be with respect to exchange of the electrons (Pauli)

Nothing to do with the fact that electrons are charged!

Result is that each electron pair with parallel spins leads to a lowering of the electronic energy of the atom

Now, let’s start rememberingNow, let’s start remembering

(1) The Pauli Exclusion Principle (only two anitparallel spins in one orbital)

(2) Hund’s Rule (parallel spin configuration of lower energy for degenerate orbitals)

(3) Aufbau Principle (from energetic bottom to energetic top)

2HeConfiguration 1s2

Full ShellInertNoble GasUnlikely to form bondsor ions

1HConfiguration 1s1

Can lose one electron to from stable ion H+

Can form single bond

H2O, H2

3Li – Lithium – remember: can’t have s3 (Pauli)•Configuration 1s22s1

•Easily ionised to Li+ (1s2) •Alkali Metal•Under standard conditions: lightest metal and least dense element

4Be- Beryllium•Configuration 1s22s2

•Be2+ (1s2) stable ion•Alkaline Earth Metal•Toxic: replaces Magnesium from Magnesium activated enzymes due to stronger coordination ability

5B - Boron

Config 1s22s2 2p1

Forms stable covalently bonded molecular networksMainly tervalentLewis acidity of many of its compounds and multicentre bondingChemistry highly diverse and complex

Number of Electrons: 5

6C - CarbonConfig: 1s22s2 2p2

Diamond, graphite, amorphous, fullerenes (e.g., C60)Forms (usually) four bonds (tetravalent), see CH4

Non-metallicCarbon's unique characteristic of bonding to itself is responsible for complex molecules composed of long chains of carbon atoms, the skeleton of life

7N - Nitrogen

Config 1s22s2 2p3

In compounds typically forms three bonds (trivalent), see NH3, N2

Non-metal

N2: •Gaseous, odourless, tasteless•78% of air is N2

•Very inert at room temperature (and below) due to strong triple bond

8OConfig 1s22s2 2p4

member of chalcogen groupNormally considered divalent but other oxidations states vary widelyhugely electronegative (see later)

Colourless, odourless, highly reactive gas,Created biologically from CO2 by green photosynthesizing plantsParamagnetic (attracted by a magnetic field)

Exercise 3;Exercise 3;

Name the electron configurations of the F and Ne atoms. Which is more stable

and why?

9F•Config

•Member of the halogen group•Most reactive of all elements•Forms F readily•highly electronegative!•Does not exist in nature in the elemental state at all because of high reactivity

10Ne - NeonConfig

Full Outer shellNoble GasUnlikely to form bonds or ions

And now the cycle repeats itself….

Remember:

Li: 1s22s1 Na: 1s22s2 2p63s1Ne: 1s22s2 2p6

s1 s2 p1 p2 p3 p4 p5 p6

s-block p-block

The Third PeriodThe Third Period

1s22s22p63s1Na: cf. Li (second period) 1s22s1

1s22s22p63s2Mg: cf. Be (second period) 1s22s2

1s22s22p63s2 3p1Al: cf. B (second period) 1s22s22p1

1s22s22p63s2 3p2Si: cf. C (second period) 1s22s22p2

1s22s22p63s2 3p3P: cf. N (second period) 1s22s22p3

1s22s22p63s2 3p4S: cf. O (second period) 1s22s22p4

1s22s22p63s2 3p5Cl: cf. F (second period) 1s22s22p5

1s22s22p63s2 3p6Ar: cf. Ne (second period) 1s22s22p6

And what now?And what now?

3s 3p 3d

Here, again, is the picture in the H-atom

The 3d electron is more firmly bound than the 4s because of its lower principle quantum number

Energy

Remember: Grotrian diagram for NaRemember: Grotrian diagram for Na

•Note more rapid stabilisation of 4s with respect to 3d due to difference in penetration!

H-Atomenergylevels

Remember: Remember: Radial Distribution FunctionRadial Distribution Function

3d vs 4s3d vs 4s

4s through the inner shells to some extent and from N on all the way to Ca, it is more stable (stronger bound) than 3d

The energy of individual (singly The energy of individual (singly occupied orbitals)occupied orbitals)

Hence, from N Hence, from N

Energy

4s is lower in energy than 3d

1s22s22p6

Careful though – the actual energy ordering is now:Careful though – the actual energy ordering is now:

1s22s22p6

1s22s22p63s23p64s23d1 1s22s22p63s23p64s23d2

And now we start filling the d-block!

1s22s22p63s23p64s23d1

1s22s22p63s23p64s23d2

Order in which they were filled

Energy Ordering

And all the way to Zn

Order in which they were filled

1s22s22p63s23p6

Energy Ordering

1s22s22p63s23p63d14s2

Ti 1s22s22p63s23p63d24s2

Important for formation of ions!Important for formation of ions!

Ti3+ 1s22s22p63s23p63d24s2

Ti3+ 1s22s22p63s23p63d1

1s22s22p63s23p63d14s2

1s22s22p63s23p63d24s2

V 1s22s22p63s23p63d34s2

Cr 1s22s22p63s23p6

Mn 1s22s22p63s23p63d54s2

1s22s22p63s23p63d64s2

1s22s22p63s23p63d74s2

Ni 1s22s22p63s23p63d84s2

Cu 1s22s22p63s23p6

Zn 1s22s22p63s23p63d104s2

Now, we should really understand the Now, we should really understand the Structure of the periodic tableStructure of the periodic table

s1 s2 p1 p2 p3 p4 p5 p6

s-block p-block

d1 d2 d3 d4 d5 d6 d7 d8 d9 d10

d-block

Electron Configuration of BaElectron Configuration of Ba1s22s22p63s23p64s23d104p65s24d105p66s2

The LanthanidesThe Lanthanides

TheActinidesTheActinides

Inner Transition Metals

(f1-14)

Transition Metals

(d1-10)

Noble Gases

Halogens(p5)

AlkalineEarth Metals (s2)

Alkali Metals (s1)

The Periodic Table of the Elements

Periodic TrendsPeriodic Trends

Periodic TrendsPeriodic Trends1) Effective Nuclear Charge - Zeff

2s – 3s: Zeff greater for 3s probably due to actual higher overall charge

Periodic Trends – (Periodic Trends – (ZZeffeff//n)n)22

Increase across periods unchanged(but note that relative slopes are different)for (Zeff/n)2)

BUT (Zeff/n)2

H(1)Li(2s)Na(3s)

11.261.85

10.400.38

M(g) M+(g) + e(g)

Ionization is the process that removes an electron from the neutral gas phase atom, eg,

Li(g) Li+(g) + e(g) I1 = 520 kJ/mol

M+(g) M2+(g) + e(g)

Li+( g) Li2+(g) + e(g) I2 = 7300 kJ/mol

1st IE

2nd IE

These reactions require energy (endothermic).

Periodic Trends

2) The Ionization Energies (I) within the Periodic Table

I1 = E(M+, g) – E(M, g)

I2 = E(M2+, g) – E(M+, g)

a) Electron Impact

Light (Eh) M e ejected from Mif Eh high enough

I = Eh - Eelkin

Determining Ionisation Energies

e accelerated through potential

b) Photo Electron Spectroscopy

M(g) + e(g) M2+(g) + 2 e(g)

Trends in Ionisation Energies

General increase across periods

But kinks?Let’s recall….

Rcall: The orbital energies

2 hc)(hcE

RZ yyeffnl

Koopman’s Theorm

I orbital energy

Together with our trend in (Zeff/n)2 across the period, that explains the general trend beautifully

But the kinks?

11stst Ionization Energies in the 1 Ionization Energies in the 1stst Period PeriodH He Li Be N O F NeB C

Recall: Maximizing the number of parallel Recall: Maximizing the number of parallel spins - The exchange interactionspins - The exchange interaction

Quantum mechanical in origin Arguments based on the fact that total

wavefunction has to be antiparallel with respect to exchange of the electrons (Pauli)

Nothing to do with the fact that electrons are charged!

Result is that each electron pair with parallel spins leads to a lowering of the electronic energy of the atom

Space for extra Notes

Al GaIn Tl

Ionisation Energy /eV

2 3 4 5 6n

Moving on through the Periodic Table

Group 13

Group 17I

Note large increase in IE from K to Cu (10 extra units of nuclear charge badly shielded by d electrons and again from Rb to Ag). Between Cs and Au the 4f shell is filled giving a total increase of 24 units of nuclear charge!

14121086420

IE(eV)

K Rb Cs

Cu AgAu

Moving on through the Periodic Table

1 3 5n

More on Ionization EnergiesMore on Ionization EnergiesIonization Potentials tend to for the successively heavier elements within a period as the number of protons in the nucleus increases and electrons are successively added to the same shell (Zeff increases at constant n).However, some irregularities (penetration of p vs s and due to exchange energy contributions) occur.Ionization Potentials tend to for the successively heavier elements in a group in the periodic table (as Zeff increases but n also increases and does so faster) .Note transition metal and lanthanide contraction affect these trends. The trends in second IE are similar but shifted by one atomic number:

The second IEs are than the first IE for that element.Also, note particularly large increases as we start to take electrons from inner shells, e.g., the first, second and third ionisation energies of beryllium are: 899 kJ mol-1, 1756 kJ mol-1 and 14846 kJ mol-1.

More Periodic TrendsMore Periodic Trends3) Electron Affinities (EAs) within the Periodic Table

X -(g) X(g) + e

Cl- (g) Cl(g) + e E = 348 kJ/mol

The amount of energy needed to remove an electron from a negative ion = amount of energy released when a neutral atom in its ground state gains an electron.

Together with IEs, EAs tell us about chemical bonding: if M has a low ionization energy an X a high EA, then it is likely that

+ X - M + X M+

MX will be ionic

A positive electron affinity tells us that X -(g) has a lower (more favorable) energy than the neutral atom, X(g).

3) The Electron Affinities

1s22s2 2p51s22s2 2p6

1s22s2 2p31s22s2 2p2

Note: EA always less than IE due to extra electron repulsion on adding an electron!

4) Atomic Radii

2s22p6

3s23p6

Decrease along period

Increase down group

Again, the Lanthanide Contraction

Nb(Z=41) and Ta(Z=73) have identical atomic radii

4) Atomic Radiia) Atomic radii generally decrease moving from left to right within the periods

(nuclear charge keeps on increasing but electrons are added to the same shell),

eg, going from Li (1s2 2s1) 157pm to F (1s2 2s22p6) 64pm; for both n = 2

b) Atomic radii generally increase down the group with increasing atomic

number as electrons are occupying more and more distant electron shells, eg,

going from Li (1s2 2s1) 157pm to Cs (1s2 2s22p63s23p64s23d104p65s24d105p66s2)

c) There is a large increase as electrons go into next shell (like

between He and Li or Ne to Na)

d) All anions are larger than their parent atoms and all cations are smaller,

compare Be2+(27 pm) and Be (112pm), I(206pm) and I(133) – please note that

ionic radius depends on coordination number of ion

e) Ionic radii generally decrease with increasing positive charge

on the same ion (Tl+, 164pm > Tl3+, 88pm)

5) Electronegativity

•The electronegativity of an atom is a measure of its power when in chemical combination to attract electrons to itself•With few exceptions, electronegativity increases across the periodic table and decreases down a group,•F is far more electronegative than I•F is far more electronegative than Li

Appendices

1) Revisit: The Born Interpretation1) Revisit: The Born Interpretation

2 1d The wave function is normalised so that:

where the integration is over all space accessible to the electron. This expression simply shows that the probability of finding the electron somewhere must be 1 (100%).

rdrsind

rsinThe radius of the latitude is

Remember that the arc length, s, is given by s = r with in radians)

The Volume Element follows hence as dV = rsindrddrr2sindddr

3) Volume Element in spherical coordinates3) Volume Element in spherical coordinates

The Surface Element follows hence as dA = rsindrdr2sindd

4) Pauli Principle4) Pauli Principle

fermion2,1) = fermion (1,2)

boson2,1) = boson (1,2)

When the labels of any two identical fermions are exchanged, the total wavefunction changes sign.

When the labels of any two identical bosons are exchanged the total wavefunction retains the same sign.

Two particles (fermions)Two particles (fermions)

1,2) =(1)2

Total wave function of particles 1 and 2

Space wave functions of particles 1 and 2 residing in the same orbital (characterized by the same n, l, ml)

Total Spin wave function of particles 1

Now exchanging labelsNow exchanging labels

As it is just a product and a x b = b x a!

1,2) =(1)2

And in that is easy: (1)2(2)1

2,1) =(2)1exch

But in the spin wave functions But in the spin wave functions ??

antisymmetric

symmetric

In analogy:In analogy:

symmetric

Hence, the only allowed overall Hence, the only allowed overall wavefunction is:wavefunction is:

With an antiparallel arrangement of spins

1,2) =(1)2

5) Slater’s Rules5) Slater’s Rules

Approximate method for estimating the effective nuclear charge

Zeff = Z - S

Where Z is the actual nuclear charge and S is a shielding constant.

Computing SComputing S

1. Divide orbitals into groups

(1s) (2s2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f)

Note s and p with same n grouped together

2. There is no contribution to S from electron to the right of the one being considered.

3. A contribution is added to S for each electron in the same group as the one being considered – except in the (1s) group where the contribution is 0.30.

3. If the electron being considered is in an ns or np orbital, the electrons in the next lowest shell (n-1) each contribute 0.85 to S. Those electrons in lower shells ((n-2) and lower) contribute 1.00 to S.

4. If the electron being considered is an nd or nf orbital, all electrons below it in energy contribute 1.00 to S.

ExampleExampleFor P:

Zeff = 15 – 4 x 0.35 – 8 x 0.85 – 2 x 1.00 = 4.8

Trends right, actual values bads and p orbitals treated the same – huge differences for the orbitals in terms of penetration!!!!!

Atomic Structure and Periodic Trends. n Lectures: week1: W9 am; week2: W9 am (ICL) & 11 am (DP), F 9...

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