Dr. Murad A. AlDamenInorganic Chemistry 221

Inorganic Chemistry

Chemistry is like the story about the group of blind men who encountered an elephant? Each one moved his hands over a different part of the elephant's body— the trunk, an ear, or a leg— and came up with an entirely different description of the beast. In this elephant (chemistry), the part we see is Inorganic Chemistry.

Fundamental particles of an atom

Atomic number, mass number and isotopes

Successes in early quantum theory

An introduction to wave mechanics

Atomic orbitals

Many-electron atoms

The periodic table

The aufbau principle

Ionization energies and electron affinities

Chapter 1: Basic Concepts: atoms

Chapter 1: Basic Concepts: atoms

An atom: the smallest unit quantity of an element that capable of existence

fm=femtometer

Fundamental Particles of an Atom

ProtonsFound in nucleusRelative charge of +1Relative mass of 1.0073 amu

NeutronsFound in nucleusNeutral chargeRelative mass of 1.0087 amu

ElectronsFound in electron cloudRelative charge of -1Relative mass of 0.00055 amu

Relative atomic massThe mass of a single atom is very small → non integral number → we define the atomic mass 1/12 th of a 12

6C atom, the atomic mass of 126C=1.660x10-27 kg,

1 amu (atomic mass unit)=1.660x10 -27 kg,

The mass on an atom=(the summation of protons)/(Avogadro’s number)

Atomic Number, Mass Number and Isotopes

Atomic number, ZThe number of protons in nucleusThe number of electrons in a neutral atomThe integer on the periodic table of each element

Mass Number, AInteger representing the approximate mass of an atom equal to the sum of

the number of protons and neutrons in the nucleus=A-Z

IsotopesAtoms of the same element which differ in the number of neutrons in the

nucleus designated by mass numberSome isotopes occur naturally others produce artificially,

Naturally elements (e.g. 3115 P, 129 F) are monotopic → naturally occurs with one isotope only. With more than one isotope e.g. 12C, 13C and 14C.

Isotopes can be separated by mass spectrometry.

Allotropes: Each of two or more different physical forms in which an element can exist. Graphite, charcoal, diamond and fullerenes... are all allotropes of carbon (see page 384 for more information).

Caution! Do not confuse isotope and allotrope

EAZ

Element symbol

Successes in early quantum theory

Quantum theory: An introduction

• Electrons in an atom occupy a region of space around the nucleus.

• Understanding of the electronic structures leads to understand the properties of atoms, ions and molecules including bonding and further interactions.

• A way of understanding the electronic structures is through quantum theory and wave mechanics.

• The quantum theory is necessary to describe electrons. It predicts discrete allowed energy levels and wavefunctions, which give probability distributions for electrons. Wavefunctions for electrons in atoms are called atomic orbitals.

• Although the electron is a wave and a particle at the same time, it easier and more convenient to consider the electron as a particle rather than a wave.

We with Quantum Chemistry is like Alice in marvel land.

Classical Quantum Mechanics

Observation!At low temperatures, the radiation emitted by a hot body is mainly of low energy and occurs in the

infrared, but as the temperature increases, the radiation becomes successively dull red and white.

Explanation!In the end of nineteenth century, much experimental work had been done on analysis this discrete spectrum

of radiation. attempts to explain this observation failed until, Planck suggested that energy could be

observed or emitted only in quanta of magnitude ∆E related to the frequency of the radiation, ν.

ΔE ∝ ν

∆E =constant*ν

∆E =hν

c =λν ν=c/λ

Wavelength (λ)

Speed of light (c)=2.998x108 m s-1

Energy (E)

Frequency (v)

Planck’s constant (h)=6.626x10-34 J s∆E =hc/λ Planck relationship

wavenumber (v ) 1wavelength(v)

cm-1

Planck

Very precise spectroscopic measurements showed that hydrogen had the simplest spectrum of all the

elements. It was found that various lines in the optical and nonoptical regions where systematically spaced in

various series. Amazingly, it turned out that all the wavelengths of atomic hydrogen were given by a single

empirical relation Rydyberg formula.

where n1 =1 and n2 =2,3,4... gives the Lyman series (ultraviolet region)where n1 =2 and n2 =3,4,5... gives the Balmer series (visible region)where n1 =3 and n2 =4,5,6... gives the Paschen series (infrared region)where n1 =4 and n2 =5,6,7... gives the Brackett series (far infrared region)

RH =1.097x107 m-1

The existence of discrete spectroscopic frequencies is evidence that the

energy of the electron in the hydrogen atom is quantized. A photon

of light is emitted when the electron moves from a higher to a lower

energy level separated by energy difference, ∆E=hv.

v 1 RH ( 1

n12

1n2

2 )

364.7n2

n2 4 (n 3, 4,5...) Balmer’s equation

Experimentally!

Bohr’s equation

R. Grinter, THE QUANTUM IN CHEMISTRY: An Experimentalist’s View, , John Wiley & Sons Ltd, 2005.

1. The electron in the hydrogen atom orbits the nucleus in a circular path, like a planet around the sun.

2. Of the infinite number of possible orbits, only those are allowed for which the orbital angular

momentum (mvr) of the electron is an integral multiple of Planck’s constant divided by 2π .

3. Contrary to classical electromagnetic theory, electrons in these allowed orbits do not radiate energy.

4. When an electron changes its orbit a quantum of energy (photon) is emitted or absorbed in accordance

with the equation hν, where is the difference in the energy of the two orbits.

Bohr’s Laws of the Hydrogen Atom Structure

v

+e

M

-e me

from eq.2 and eq.3 we obtain

E=potential energy+kinetic energy

E= e2/(4πε0 v2)+me v2/2 by eq.1 and eq.2 and after some algebraic rearrangements: =-e2/(4πε0 ) × (e2πme )/(ε0 n2h2) +me /2(e2/(2ε0 nh))2 =(-e4me )/(8ε02n2h2)

E(n)=-1/n2 ×(e4m)/(8ε02h2)

Bohr’s 1st law: The electron in the hydrogen atom orbits the nucleus in a circular path, like a planet around the sun.

The electron does not fall into the nucleus because of its velocity at right-angles to the line between the two particles. the radial vector.

The coulombic force does cause an acceleration of the electron towards the nucleus equal to -v2/r.

According to Newton, force=mass x acceleration so: e2/(4πε0 r2)=me v2/r → r= e2/(4πε0 v2) ....eq.1

Bohr’s 2nd law: This law introduces quantization with the postulate that only those orbits for which the angular momentum is an integer multiple of h/2π are allowed, i.e. the angular momentum=nh/2π or me vr=nh/2π where n=1,2,3...,∞ ...eq.2

to eliminate the v we use from eq.1. r= e2/(4πε0 v2)= or r=e2/(4πε0 ) × 1/v2 and from eq 2. v=nh/(2πme r) → 1/v2=(2πme r)2/(nh)2 then me r=(ε0 n2h2)/(e2π) .....eq.3

v=nh/(2πme r)=nh/2π × e2π/(ε0 n2h2)= e2/(2ε0 nh)

∆E(n)=hv=hc/λ Bohr’s equation

Amazingly, when a viable theory was found, two apparently very different theories were announced at almost the same time. Karl Werner Heisenberg described his matrix mechanics in 1925 and Erwin Schrödinger his wave mechanics in 1926. But the two approaches were soon shown to be different formulations of the same theory (compare the Newtonian, Lagrangian and Hamiltonian forms of classical mechanics), and we shall deal only with Schrödinger’s version here.

de Broglie (1892 – 1987) descended from a noble family who had served many French kings. He studied history before taking up physics and in his doctoral dissertation (1924) he reasoned as follows:

According to Einstein’s theory of relativity: E=mc2

but Einstein, following Planck also suggested that: E=hv=hc/λ

therefore, if the momentum (p) of the photon as a particle is mc, then: p = mc = E/c = h/λ

and if this result applies to other particles as well as to photons, then all moving particles should be associated with a wavelength (λ) given by the equation:

λ= h/p = h/mv

where m and v are the mass and velocity of the particle respectively.

The wave-nature of electrons

Ein

stei

n liv

ed h

ere

The uncertainty principle

Electrons moving in circles around the nucleus, as in Bohr's theory, can

be thought of as forming standing waves that can be described by the de

Broglie equation. However, we no longer believe that it is possible to

describe the motion of an electron in an atom so precisely. This is a

consequence of another fundamental principle of modern physics,

Heisenberg's uncertainty principle, which states that there is a

relationship

px h4

Δx = uncertainty in the position of the electron Δpx = uncertainty in the momentum of the electron

This concept means that we cannot treat electrons as simple particles

with their motion described precisely, but we must instead consider the

wave properties of electrons, characterized by a degree of uncertainty in

their location. In other words, instead of being able to describe precise

orbits of electrons, as in the Bohr theory, we can only describe orbitals, regions that describe the probable location of electrons. The probability of finding the electron at a particular point in space (also called the

electron density) can be calculated, at least in principle.

The probability of finding an electron at a given point in space is determined from the function ψ2 where ψ is the wavefunction

Imagine a car in a circle, when this ball has zero velocity then its momentum will be zero, and so we can determine where is the ball exactly. If this ball moves by very high speed, then we will see the car in all the circle, it means that our accuracy in determining the position of the car will decreases and the energy increases.

v=0p=0

v=v0

p=mv0

The Schrödinger equation describes the wave properties of an electron in terms of its position, mass, total energy, and potential energy. The equation is based on the wave function, ψ, which describes an electron wave in space; in other words, it describes an atomic orbital. In its simplest notation, the equation is:

The Schrödinger wave equation

HΨ=EΨH=The Hamiltonian operatorΨ= The wave functionE= The energy of the electron

H h2

8 2m(

2

x2 )VIn the form used for calculating energy levels, the Hamiltonian operator is

The image above is not of an atom, but shows an alternative electron corral pattern, predicted by the Schrödinger wave equation and created by electrons in experiment

This equation can exactly be solved only for hydrogen-like atom: An atom contains a nucleus and only one electron

Courtesy D. Eigler, IBM research division

Particle in a Box

1. The wave function ψ is a solution of the Schrödinger equation and describes the behavior of an electron in a region of space called atomic orbital.2. We can find the energy values that are associated with particular wave functions.3. The quantization of energy levels arises naturally from the Schrödinger equation.

2x2

8 2mh2 (E V ) 0

ψ=Asin(rx)+Bcos(sx) substitution into Schrödinger equation leads: r=s=√(2mE) ×2π/h

ra=±nπ or r=±nπ/a=√(2mE) ×2π/hE=n2h2/(8ma2)The wave function ψ=√(2/a) sin (nπx/a)

where n: principal quantum number

A number of conditions required for a physically realistic solution for Ψ in atoms:

1. The wave function ψ must be single valued.2. The wave function ψ and its derivatives must be continuous.3. The wave function ψ must be quadratically integrable. This means that ∫ψψ*.dτ=1. This is called normalizing the wave function.4. Any two solutions to the wave function must be orthogonal, which means ∫ψA ψB .dτ=0

see Box 1.3

V=∞

V=0

V=∞

Potential Energy Node Node

x=rsin y=rsin sin z=rcos dx.dy.dz=r2sin().dddr

ψ(r,R(r)Θ()Φ()radial factor angular factors

Atomic Orbitals

The wavefunctions of the hydrogen atom are not easy to describe in a Cartesian co- ordinate system of three, mutually

perpendicular axes. The polar co-ordinate system with the nucleus at the origin is much more suitable. The mathematical functions

for atomic orbitals may be written as a product of two factors:

The radial wavefunction describes the behavior of the electron as a function of distance from the nucleus; the angular wavefunction

shows how it varies with the direction in space. (Describe the electron density)

Angular wavefunctions do not depend on n and are characteristic features of s, p, d,...orbitals. (Describe the shape and orientation)

Quantum numbers and their properties

SymbolSymbol NameName ValuesValues RoleRole

n Principal 1,2,3,... Determine the size and so the major part of energy

l Angular momentum 0,1,2,3,...n-1 Describes angular dependence (shape) and contributes to the energy

ml Magnetic 0,±1,±2,±3,.. .±l Describes orientation in space

ms Spin ±½ Describes orientation of the electron spin in space

Orbitals with different l values are usually known by the following labels, which are derived from early terms for different families of spectroscopic lines:

l=label

0 1 2 3 4 5, ...s, p, d, f, g, continuing alphabetically

For a given value of n (n≥1) there is one s atomic orbital For a given value of n (n≥2) there are three degenerate p atomic orbitalsFor a given value of n (n≥3) there are five degenerate d atomic orbitals For a given value of n (n≥4) there are seven degenerate f atomic orbitals

s sharp, p principal, d diffuse, f fundamental

+

-

21 3p

4pnodesnode

+-

n-1 Totaln-l-1 Radiall Angular

Radial wavefunctions depend on n and l but not on m; thus each of the three 2p orbitals has the same radial form. The wavefunctions may have positive or negative regions, but it is more instructive to look at how the radial probability distributions for the electron depend on the distance from the nucleus.

• Radial distributions may have several peaks, the number being equal to n-1. • The outermost peak is by far the largest, showing where the electron is most likely to be found. The distance of this peak from the nucleus is a measure of the radius of the orbital, and is roughly proportional to n2 (although it depends slightly on l also).

Radial distributions determine the energy of an electron in an atom. As the average distance from the nucleus increases, an electron becomes less tightly bound. The subsidiary maxima at smaller distances are not significant in hydrogen, but are important in understanding the energies in many-electron atoms.

Radial Part

Radial Distribution Function

1 atomic unit=Bohr’s radius=5.293pm

Probability of finding the electron at r = 4πr2R2

Example

Nodal Planes

dxy

The five degenerate d orbitals

Angular Part

Orbital sign key

0 1 2 3 4 5 6

1 2 3 4 5 6

Total number of nodes=n-1

s

pz

2 3 4 5 6

2 3 4 5 6

dz2

dxz

nodes

nodes

nodes

nodes

n=1

l=0m=0

n=2 n=3 n=4 n=5 n=6 n=7

l=1m=0

l=2m=0

l=2m=1

Orbital angular momentum= h2

l(l 1)

Spin angular momentum= h2

s(s 1)

Example: l=2, ml=-2,-1,0,+1,+2

Orbital angular momentum=

Actual magnitude of z component of l=

mlh

2

h2

6Total angular momentum=

h2

j( j 1)

Actual magnitude of z component of s=

msh

2

j l 12

or l 12

l=0 no angular momentum

Total angular momentum= h2

8.75 or h2

3.75

j possible orientations

Angular momentum, the inner quantum number, j, and spin-orbit coupling

R Rydberg constant=1.097107m-1

h Planck constant=6.62610-34 J sc= Speed of light=3.000108m s-1

Z is the atomic number

Hydrogen-like

Hydrogen→1s→spherical→only depends on n (l doesn't affect the energy)

En RhcZ 2

n2r n2a0

Z

3s

2s

1s

2p

3p 3d 3d 3d 3d3s

3s

3s

2s2s

2s

1s1s

1s

3p3p

3p

2p

2p2p

Ene

rgy

(not

to sc

ale →

Three multielectron atomsHydrogen atom

Li (Z=3) Na (Z=11) K (Z=19)

The aufbau Principle

1- Orbitals are filled in order of energy

2- Hund’s rule of maximum multiplicity: electrons be placed in orbitals so as to give the maximum total spin possible.

3- Pauli exclusion principle: each electron in an atom have a unique set of quantum numbers.

degenerate for Hydrogen atoms

degenerate for Hydrogen atoms

1s

2s

3s

4s

5s

6s

7s

2p

3p

4p

5p

6p

7p

3d

4d

5d

6d

4f

5f

First Rule

adding to 3d (4s13dn)

adding to 4f (4fn6d2)

adding to 4f (4fn6s25d1)

adding to 6d (6dn7s2)

adding to 5f (5fn7s26d1)

adding to 3d (5s03d10)adding to 4d (5s14dn)

adding to 5f (5fn7s2)

Alk

ali m

etal

sA

lkal

i ear

th m

etal

sd-block elements

Pnic

toge

nsC

halc

ogen

sH

alog

ens

Nob

le g

ases

Hund’s Rule

Step 1 Step 2 Step 3

∏c

∏e

total pairing energy=∏=∏c +∏e

Oxygen

Or

∏=3∏e +∏c ∏=2∏c +2∏e

∏e add stability ∏c add instability

When two electrons (e.g. Helium atom) occupy the same part of the space around an atom, they repel each other because of their mutual negative charges, with a Coulombic energy of repulsion, ∏c.

In addition, there is an exchange energy, ∏e , which arises from purely quantum mechanical considerations; the number of electrons at the same energy that could be exchanged while retaining the same overall spins. see Box 1.8

In this atom, the electrons of 1s are the core electrons (lower energy quantum levels)and those of 2s, 2p are valence electrons (outer)

2s and 2p ...etc penetrate the 1s2s more penetrating than 2p

The core electrons shield the valence electrons from the nuclear charge, i.e. 1s shield the nucleus and the atomic number will replace with effective charge Zeff

Zeff can be calculated by Slater’s rules, which is based on experimental data.

Effective charge(Z*)=number of protons(Z)-Shielding(S)1. The electronic structure of the atom is written in groupings as follows: (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d, 4f) (5s, 5p), etc.

2. Electrons in higher groups (to the right in list above) don not shield those in lower groups.

3. For ns or np valence electrons:a. Electrons in the same ns, np group contribute 0.35, except the 1s, where 0.30 works better.b. Electrons in the n-1 group contribute 0.85.c. Electrons in the n-2 or lower groups contribute 1.00.

4. For nd and nf valence electrons:a. Electrons in the same nd or nf group contribute 0.35.b. Electrons in groups to the left contribute 1.00.

At maximum 1s, 2s>2p, and so the electron will insert in the orbital that is closer to the nucleus.

Oxygen (1s2) (2s2p4) Z=8For outmost electron Z*=Z-S=8-2*0.85-5*0.35=4.55Z (1s) (2s,2p)

4.55/8*100%=57% force expected for 8 nucleus and -1 electron

Shielding (S) (Screening)

Ionization energy The general increases in IE1 across a given period is a consequence of an increase in Zeff. A group 15 element has a ground state electronic configuration ns2np3 and the np level is half- occupied. A certain stability is associated with such configurations and it is more difficult to ionize a group 15 element than its group 16 neighbor. In going from Be (group 2) to B (group 13), there is a marked decrease in IE1 and this may be attributed to the relative stability of the filed shell 2s2 configuration compared with the 2s22p1 arrangement; similarly, in going from Zn (group 12) to Ga (group 13), we need to consider the difference between 4s23d10 and 4s23d104p1 configurations.

Electron Affinity The attachment of an electron to an atom is usually exothermic. Two electrostatic forces oppose one another: the repulsion between the valence shell electrons and the additional electron, and the attraction between the nucleus and the incoming electron. In contrast, repulsive interactions are dominant when an electron is added to an anion and the process is endothermic. A(g) e- A (g) Electron affinity=EA=-ΔU

An (g) A(n1) (g) e- Ionization energy=ΔU

Ionization energy and Energy Affinity

Ion Protons Electrons Radius(pm)O2- 8 10 126F- 9 10 119

Na+ 11 10 116Mg2+ 12 10 86

Ion Protons Electrons Radius(pm)O2- 8 10 126S2- 16 18 170Se2- 34 36 184Te2- 52 54 207

Ion Protons Electrons Radius(pm)Ti2+ 22 20 100Ti2+ 22 19 81Ti2+ 22 18 75

+++ -

- - -+++

- - -+

Atomic and Ionic Radii Trends

Orbital is like ghost, we cannot see it but some believe that it exists

Atomic units

Conclusions and Appendices

Bonding model

Homonuclear diatomic molecules

The octet rule isoelectronic species

Molecular shape and the VSEPR model

Molecular shape: Stereoisomerism

Electronegativity values

Dipole moments

Chapter 2: Basic Concepts: molecules

A cartoon rendition of Irving Langmuir and G.N. Lewis ; Source: http://www.minerva.unito.it/humor/Langmuir%20&%20Lewis.htm

Lewis, Kossel, and Langmuir made several important proposals on bonding which lead to the development of Lewis Bonding Theory

Elements of the theory: Valence electrons play a fundamental role in chemical bonding. Ionic bonding involves the transfer of one or more electrons from one atom to another. Covalent bonding involves sharing electrons between atoms. Electrons are transferred or shared such that each atom gains an electron configuration of a noble gas

(ns2np6), i.e. having 8 outer shell (valence) electrons. This arrangement is called the octet rule. Exceptions to the octet rule do exist and will be explored later.

Lewis Bonding Theory

Lewis Symbols represent the resulting structures that accommodate the octet rule.

In a Lewis symbol, an element is surrounded by up to 8 dots, where elemental symbol represents the nucleus and the dots represent the valence electrons.

1A(1)ns1

2A(2)ns2

3A(13)ns2np1

4A(14)ns2np2

5A(15)ns2np3

6A(16)ns2np4

7A(17)ns2np5

8A(18)ns2np6

Li Be B

Na Mg Al

C

Si

N O F Ne

P S Cl Ar

2

3

Octet Rule

Ionic bond: e.g. Na ➟ Na- + ➟+Cl Cl

-Cl

-Na

+

metal nonmetal

small ion large ionLoss Gain

Covalent bond: e.g. ClShare

Cl Cl Cl➟nonmetal nonmetal

O

H

H ➟

Lewis formula

O

H

H

Kekulé formula

❙

❙

How can we write Mg3N2???

Homonuclear covalent bondHomonuclear molecule

Homonuclear covalent bondHomonuclear molecule

Homonuclear covalent bondHeteronuclear molecule

Homonuclear covalent bondHeteronuclear molecule

H ● H● H ● H●

F ●●●

●●

●●F●●●

●●

●●F ●●●

●●

●●F●●●

●●

●●

O

●●

●●

●●C●●

●

●2 C●●O

●●

●●

●●●

●

O

●●

●●

●●

●

● ●

●●

● ●C●

●●

●

C●

●

●

●

F ●

● ●

●●

● ● ●

●●

●●

●●

●

●●

●●

●●●

●●

● ●

●● F

F

F

F

+

+

+

+

Covalent Bond

Unequally shared: polar covalent bond

Bond order: Single, Double, Triple, Quadruple, Quintuple, Sextuple

Single e.g. ClShare

Cl Cl Cl➟ O

H

HShare➟ O

H

H

Equally shared: nonpolar covalent bond

❙❙ ❙

Double e.g. O O❙❙

Triple e.g. N N❙❙❙

Quadruple e.g. Cl4Re ReCl4❙❙❙❙

Quintuple e.g. RCr CrR❙❙❙❙❙

;R=large organic group

Sextuple e.g. W W❙❙❙❙❙

❙

Zeruple e.g. The carbon atom in C(PPh3)2 (where Ph stands for a benzene ring) has two lone pair electrons, but no electrons connected to the bonded groups

2005 Philip Power, U of California, Davis

1964 F. Albert Cotton, Texas A&M University

2006 (Roos, Gagliardi, Borin, U of São Paulo)

Xe 3+O 1- Expanded Shells

1. Elements in the third or higher shells can have >8 valence electrons2. Empty d-orbitals are used to hold excess electrons (s2p6d10=18 electrons max)

S

N

F F

F

S

O

Cl O

Cl

Xe

O

O O

S

O

O O

O

S

O

O O

S

N

F F

F

S

O

Cl O

Cl

S

O

O O

O

S

O

O O

I

O

F

FF

FF

I

O

F

FF

FF

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O

O O

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S 2+N 2-

S 2+O 1-

S 2+O 1-

S 1+O 1-

S 0N 0

S 0O 0

Xe 0O 0

S 0O 0,1-

S 0O 0,1-

O 1+I 1-

O 0I 0

Anion Formal Charge Anion Formal ChargeMolecule Expanded to

SNF3

SO2Cl2

XeO3

SO42-

SO32-

IOF5

No octet rule

12

12

14

12

10

12 or 14Or

Decreasing the formal chargeNo octet mean, it is not possible to draw the molecule with 8 electrons around the central atom

Resonance and Formal Charge1. Ability to draw multiple Lewis structures for the same molecule

SO

O OSO

O OSO

O O

a. Taken as a set to represent the true structures

b. Interconverted by movement of e- only

c. Separated by double headed arrows

More resonance structures means a lower energy for the compound; Spreading the electrons overe more atoms lets them occypy more space.

Isoelectronic=resonance structures with identical electronic structuresa) SO3 and CO32-

Resonance structures may be electronically different as well amides

CH3CO

NH2 CH3CO

NH2

distance in pm S-C C-N

SCN- 165 117

HSCN 156 122

Single bond 181 147

double bond 155 ~128

triple bond 116

Data from A. F. Wells, structural Inorganic Chemistry, 5th ed. OXford University Press, New York, 1984, pp. 807, 926, 933-934

distance in pm O-C C-N

OCN- 113 121

HNCO 118 120

Single bond 143 147

double bond 116 ~128

triple bond 113 116

Thiocyanate SCN-

Cyanate OCN-

Fluminate CNO-

S=C=N S−C≡N S≡C−N

O=C=N O−C≡N O≡C−N

C=N=O C−N≡O C≡N−O

1- 1- 2-1+

1- 1- 2-1+2-

1+ 1+ 1+3- 1+ 1+1-

Formal charge=valence electrons in Neutral atom-(unshared valence electrons+half os shared electrons)

Could not explain:

a. Why metals conduct electricity

b. How a semiconductor works

c. Why liquid oxygen is attracted into the magnetic field of a large magnet

d. Be and B compounds1. BeX2 and BX3 compounds have <8 electrons around Be/B if single bonds2. Be/B=X double bonds create large charge separation3. Solids tend to have extended structures relieving charge separation/octet4. Monomers are reactive as Lewis acids (electron pair acceptors), even though the small atom size allow less than an octet around Be/B

Shortcomings of Lewis theory

BeF

F F

F

Be

Be

FF

F

F

F

F

Be FF Be FF

B FF

F

B FF

FB F

F

FB F

F

F

Predicted: The B-F bond length is 131 pm the calculated single-bond length is 152 pm

Predicted

Actual salt

Valence Shell Electron Pair Repulsion Theory

1. All paires of electrons, both bonding pairs and lone pairs, are important in determining the shape of a molecule.

2. Bonding pairs are smaller than lone pairs because there are 2 positively charged nuclei pulling them in.

3. Single bonds are smaller than double bonds and double bonds are smaller than triple bonds.

4. If a central atom (A) is surrounded by different atoms (B and C) in the molecule ABxCy, the relative sizes of B and C can affect the structure of the molecule.

Determine the Valence→Determine the valency→Draw Lewis→Count the lone pairs

The true bond angles will usually be distorted from the idealized angles in the pictures above because all bonds and non-bonding electron pairs don't have the same "size".

lone pairs > triple bond > double bond > single bond

Valence Shell Electron Pair Repulsion Theory: Coordination Geometry

AX2

Linear

BeCl2 , CO2 , HCN

AX3

Trigonal planar

SO3, BeCl3

AX2E

Bent

O3, SnCl2

AX4

AX5

Valence Shell Electron Pair Repulsion Theory: Coordination Geometry

AX3E AX2E2

Tetrahedral Trigonal pyramidal Bent

SO4 , POCl3 , SiF4 (H3O)+ , NH3 , PCl3 H2O , Cl2O, (NH2)-

AX4E AX3E2 AX2E3

Trigonal bipyramidal Seesaw T-shaped Linear

SOF4 , PCl5 XeO2 , IF2O2 , SiF4 ClF3 I3-

AX7 AX8

AX6

Valence Shell Electron Pair Repulsion Theory: Coordination Geometry

AX5E AX4E2 AX3E3

Octahedral Square pyramidal Square planar T-shaped

IOF5 , SF6 XeOF4 , BrF5 ICl4-

[Ln(H2O)8]3+

Square antiprismatictrigonal bipyramidal

Valence Shell Electron Pair Repulsion Theory: Second order effectsInfluence of the nature of electron pairs on the angles between bonds It's logical to admit that the non-bonding pairs, less confined in the internuclear space than bonding pairs, occupy a higher volume. And so, we see a diminution of the angles between the bonds when we observe in the order CH4, NH3

and H2O.

109.5º 107º 104.5

[NH4]+ NH3 [NH2]-

Influence of the volume of the multiple bonds The geometric form depends only on bond. So we can class the molecules with bonds in the same groups that these with only bonds. But the volume occupied by the electrons depends on the number of bonds, and then we observe a diminution of the opposite angle of the bond.

109.5º 102º 98º

SiF4 POF3 NSF3

Isomerism

equatorialaxial

Bond Polarity

¥ .

0 0.4 2.0 4.0Electronegativity Difference

covalent ionic

polarnon

polar

3.0-3.0= 0.0

4.0-2.1= 1.9

3.0-0.9= 2.1

Pauling electronegativity values χP

bond enthalpy=D(XY)=½(D(XX)+D(YY))∆χ= χP(Y)-χP(X)=√∆D

Mulliken electronegativity values χM

χM=½(Ionization energy+Electron affinity) eV

Allerd-Rochow electronegativity values χAR

χAR=(3590xZeff /rcov2)+0.744; rcov in pm

Eleectronegativity difference

The symmetrical electron distribution in the bond of a homonuclear diatomic renders the bond non-polar. In a heteronuclear

diatomic, the electron withdrawing powers of the two atoms may be different, and the bonding electrons are drawn closer

towards the more electronegative atom. The bond is polar and possesses an electric dipole moment (µ)

µ=δed e charge of the electron, d bond distance µ electric dipole momentδ point charge

Measure of the pull an atom has on bonding electronsIncreases across period (left to right) highest at FDecreases down group (top to bottom)Large difference in electronegativity, more polar the bondNegative end toward more electronegative atom

System operations and symmetry elements

Successive operations

Point groups

Introduction to character tables

An introduction to Molecular SymmetryChapter 4

Symmetry operations are geometrical operations applied to molecules (or better, molecular models). The application of a symmetry operation to a particular molecular geometry produces a spatial orientation which is indistinguishable from the initial orientation.

The five basic symmetry operations that need to be considered are identity, rotation, reflection (mirroring), inversion, and rotary reflection.

For each symmetry operation there is a corresponding symmetry element, with respect to which the symmetry operation is applied. For example, the rotation (a symmetry operation) is carried out around an axis which is the corresponding symmetry element. Other symmetry elements are planes (for reflections) and points (for inversions).

Different (valid) combinations of symmetry operations that leave at least a single common point (the molecular center) unchanged give rise to the so called point groups (not all arbitrary combinations of symmetry operations and elements are possible, and some combinations of symmetry operations implicitly imply others, see below).

Symmetry

The basic symmetry operations are:

Identity Operation E The identity operation does nothing and leaves any molecule unchanged, the corresponding symmetry element is the entire object (molecule) itself. The reason for including this symmetry operation is that some molecules have only this symmetry element and no other symmetry properties. Another reason is the logical completeness of the mathematical description of group theory.

Inversion i The inversion (the symmetry operation) through a center of inversion (the symmetry element, which must be identical to the center of geometry of the molecule) takes any point in the molecule, moves its to the center, and then moves it out the same distance on the other side again (sometimes called point reflection). The benzene molecule, a cube, and spheres do have a center of inversion, whereas a tetrahedron does not.

Reflection σ The reflection (the symmetry operation) in a plane of symmetry or mirror plane σ (the corresponding symmetry element) produces a mirror image geometry of the molecule. The mirror plane bisects the molecule and must include its center of geometry. If this plane is parallel to the principal axis (and includes it, see below), it is called a vertical mirror plane denoted σv , if it is perpendicular to the principal axis (and bisects it in the molecular center of geometry), it is denoted a horizontal mirror plane σh . Vertical mirror planes bisecting the angle between two Cn axes are called dihedral mirror planes σd (these mirror planes all include principal axis and intersect in it).

n-Fold Rotation Cn The n-fold rotation (the symmetry operation) about a n-fold axis of symmetry (the corresponding symmetry element) produces molecular orientations indistinguishable from the initial for each rotation of 360°/n (clockwise and counter-clockwise). A water molecule has a single C2 axis bisecting the H-O-H bond angle, and benzene has one C6 axis (amongst one C3 axis and seven C2 axes of which the C3 and one C2 axis coincide with the C6 axis). Linear molecules display a C∞ axis (any infinitely small rotation about this axis produces unchanged orientations), and perfect spheres posses an infinite number of symmetry axes along any diameter with all possible integral values of n. If a molecule has one (or more) rotation axes Cn or Sn (see below), the axis with the greatest n is called the principal axis.n-Fold Rotary Reflection Sn The n-fold rotary reflection (or n-fold improper rotation, the symmetry operation) about an n-fold rotary reflection axis (or n-fold axis of improper rotation) is composed of two successive geometry transformations: first, a rotation through 360°/n about the axis of that rotation, and second, reflection through a plane perpendicular (and through the molecular center of geometry) to that axis. Neither of these two operations (rotation or reflection) alone is a valid symmetry operation, but only the outcome of the combination of both transformations.

≡C6

C6 S6

≡

C2

≡

C6σv

σv

C2

C2

σd

σv

C2σ σh

Between two C2Parallel to the principal rotation axes (σv)

Perpendicular to principal rotation axes (σh)Mirror plane (σ)

Identity (E) Inversion (i)

n-fold rotation axis (Cn) n-fold improper axis (Sn)

The basic symmetry operations are:

Symmetry operations examples:

only Ei

σ C2 C3

S4

S1= σS2= i

For this reason, if the molecule has S1 or S2, we use σ and i.

Chiral?All molecules (or molecular conformations) belonging to the C1 , Cn, Dn, or the (very rare) T, O, or I point groups MUST be chiral. All other point groups include symmetry elements that (any Sn axis, including mirror planes σ = S1 or inversion centers i = S2 ) require a molecular geometry to be superimposable to its mirror image geometry, and therefore these molecules MUST be achiral

Polar molecules must have a permanent electric dipole moment. Only molecules belonging to the point groups Cn (including C1 ), Cnv , and Cs , may have a permanent dipole moment, which in the case of Cn and Cnv must lie along the rotation axis. All other point groups posses symmetry operations that include Cn axes (n > 1, and thus no dipole moment perpendicular to these axes), and symmetry operations which take one end of the molecule into the other (therefore no dipole moment along the principal axis can be observed).

Permanent electrical dipole moment?

cont.

EXAMPLES

cont.

EXAMPLES

cont.

C∞v

EXAMPLES

cont.

D∞h

EXAMPLES

cont.

EXAMPLES

EXAMPLES

Chemists and physicists use a special convention for representing character tables which is applied especially to the so-called point groups, which are the 32 finite symmetry groups possible in a lattice. In the following example:

Character Table

1. The symbol used to represent the group in question (in this case C3v).

2. The conjugacy classes, indicated by number and symbol, where the sum of the coefficients gives the group order of the group.

3. Mulliken symbols, one for each irreducible representation.

4. An array of the group characters of the irreducible representation of the group, with one column for each conjugacy class, and one row for each irreducible representation.

5. Combinations of the symbols x, y, z, Rx, Ry and Rz the first three of which represent the coordinates x,y and z and the last three of which stand for rotations about these axes. These are related to transformation properties and basis representations of the group.

6. All square and binary products of coordinates according to their transformation properties.

3. Mulliken Symbols The character tables contain various information, for instance the number and kind of irreducible representations for a specific point group. Usually these irreducible representations are denoted using a scheme suggested by Robert S. Mulliken (1896-1986, awarded with the Nobel prize in 1966) in the early 1930s. Here, an overview on the meaning of Mulliken's symbols is given.

1. The dimension of characters are denoted with one of the following capital letters:

dimension Mulliken symbol1 A and B2 E3 T threefold degeneracy

twofold degeneracy

not degenerated

one electron system

a and b

e

t

2. If Cn represents rotation about the principal axis, the one-dimensional characters are denoted dependent of the value obtained for χ(Cn ).

χ(Cn ) denoted as+1 A-1 B

cont.

symmetric

antisymmetric

3. Mulliken Symbols

3. Indices reflect an additional classification of symmetry. If the molecule possesses an axis C2 or a plane of reflection σ or σd perpendicular to the principal axis Cn, the values for function ψ change or keep their sign and are therefore regarded as symmetric and antisymmetric respectively.

function ψ IndexSign unaffected 1

Change sign 2

4. Dependent on the effect of inversion i, Mulliken's symbols take the indices g for gerade and u for ungerade.

5. The way reflection on a horizontal plane affects a function is denoted by primed or doubly primed symbols.

χ(i) IndexSign unaffected g

Change sign u

χ(σh ) IndexSign unaffected ‘

Change sign “

4. Orbital symmetry (C2v as an example) χ(E) χ(C2(z)) χ(σv(xz)) χ(σv(yz))z

x

y

pz

z

x

y

py

+1 +1 +1 +1

+1 -1 -1 +1

x

y

px

+1 -1 +1 -1

z

4. Translational motion

z

y

x

OH H χ(E) χ(C2(z)) χ(σv(xz)) χ(σv(yz))

+1 -1 -1 +1

doesn’t translate translate translate doesn’t translateH translation:

+1 +1 -1 -1

doesn’t rotate rotate rotateH rotation along z : doesn’t translate

Infrared (IR) active vibrations

IR : one dimensional

Raman: two dimensional

There are two types of spectroscopy that involve vibrational transitions. infrared spectroscopy. During infrared spectroscopy experiments we observe transitions between vibrational energy levels of a molecule induced by the absorption of infrared (IR) radiation. The second type of vibrational spectroscopy is Raman spectroscopy. In Raman spectroscopy, vibrational transitions occur during the scattering of light by molecules.

At room temperature almost all molecules are in their lowest vibrational energy levels with quantum number n = 0. For each normal mode, the most probable vibrational transition is from this level to the next highest level (n = 0 -> 1). The strong IR or Raman bands resulting from these transitiions are called fundamental bands. Other transitions to higher excited states (n = 0 -> 2, for instance) result in overtone bands. Overtone bands are much weaker than fundamental bands

Valence bond (VB) theory: hybridization of atomic orbitals

Valence bond (VB) theory: multiple bonding in polyatomic molecules

Chapter 5 - Part 1

Valence Bond Theory and Hybrid Orbitals

The Linear Combination of Atomic Orbitals (LCAO) is a mathematical method to construct a new set of orbitals - the hybrid orbitals - by mixing appropriate atomic orbitals (AOs)

Valence bond theory describes a chemical bond as the overlap of atomic orbitals.repulsion occurs

some overlap attraction begins

maximum overlap attraction begins

Potential Energy

r74 pm

-436 kJ/mol

0H•+•H→H:H

i ci,11 ci,22 ... ci,nn

Orthogonalityci, j2

j 1 and ci, j ck , j

jk 0

2sp-orbitals

e.g. BeH2 Be

1s 2s 2p

1s 2s 2p

1s 2sp 2p

Be

Be

BeH2

1s 2sp 2pH H

px px

+ =

+ = - = 1 1 22s 1 22 px

2 1 22s 1 22 px

2sp2-orbitalse.g. BH3 B

1s 2s 2p

1s 2s 2p

1s 2sp2 2p

B

B

H H

px px

1s 2sp2 2p

BH3

H

BH3

+ + =

+ +

+ +

=

=

1 1 32s 1 62 px1 22 py

2 1 32s 2(1 32 px)

3 1 32s 1 62 px (1 22 py

)

2sp3-orbitalse.g. CH4 C

1s 2s 2p

1s 2s 2p

1s 2sp3 2p

C

C

H H

px px

1s 2sp3

CH4

H H

CH4

+ +

++

- -

--

x

y

+-

+z-z

1 1 4 2s 1 4 2 px1 4 2 py

1 4 2 pz

2 1 4 2s 1 4 2 px1 4 2 py

1 4 2 pz

3 1 4 2 s 1 4 2 px1 4 2 py

1 4 2 pz

4 1 4 2s 1 4 2 px1 4 2 py

1 4 2 pz

CH3−CH3

CH2=CH2

CH≡CH

sp3-sp3

sp2-sp2

sp-sp

1s 2sp3

CH3

1s 2sp2

CH2

1s 2sp

CH

2p

2p

4σ

3σ and π

2σ and 2π

Multiple bonding in polyatomic molecules

H−C≡N

C [He]2s22p2

N [He]2s22p3

H 1s2

H

1s 2s 2p

H NC

2π

σσ

H

N [He]2s22p3

1s 2s 2p

sp3 lone pair1s

O [He]2s22p2

1s 2s 2p

1s sp2 two lone pairs

Examples: lone pairs in ammonia and water

Examples: Benzene, Conjugated polymers and resonance

px px px px px px px px

π

σ σ σ σσ σ σ

π π π

px px px px px px px px

π

σ σ σ σσ σ σ

π π

px px px px px px px px

σ σ σ σσ σ σ C ...C ...C ...C ...C ...C ...C ...C

C=C-C=C-C=C-C=C

C-C=C-C=C-C=C-C

π π π π π π π

Hybridization vs. Shape

sp s, px 2 linear sp2 s, px,y 3 trigonal-planarsp3 s, px,y,z 4 tetrahedralsd3 s, dxy, xz, yz 4 tetrahedral

sp2d s, px,y, dxy, xz, yz 4 square-planarsp3d s, px,y,z, dxy, xz, yz 5 square-pyramidal

sp3d s, px,y, z, dx2-y2 5 trigonal-bipyramidalsp3d2 s, px,y, z, dz2 6 octahedralsp3d3 7 pentagonal-bipyramidalsp3d4 8 square-antiprismatic

CO2

CO32-

BF4-

CrO42-

[PtIICl4]2-

Ni(PR3)2Br3

PF5

SF5

IF7

TaF83-

Hybrid Orbital used n Orientation Example

Group theory and Hybridization

Determine the shape of the molecule by VSEPR techniques and consider each sigma bond to the central atom and each lone pair on the central atom to be a vector pointing out from the center.

Find the reducible representation for the vectors, using the appropriate group and character table, and find the irreducible representations that combine to form the reducible representation.

The atomic orbitals that fit the irreducible representations are those used in the hybrid orbitals.

e.g. BH3

1. VSEPR x

ΓBH

B

H

H

H

y

How many H-B bonds have been changed after applying the operation?

3 0 1 3 0 1

Summation of =s s=12

Now we reduce the reducible presentation to irreducible ones:

ΓBH 3 0 1 3 0 11 2 3 1 2 31 1 1 1 1 1

ΓBH 3 0 1 3 0 11 2 3 1 2 3

1 1 -1 1 1 -1

×

×

3 0 3 3 0 3

3 0 -3 3 0 -3

(∑)/s = 12/s=12/12=1

0 A2`

1 A1`

(∑)/s = 0/s=0/12=0

Finally after testing all the irreducible representation we obtain= A1`+E`

Related orbitals s from A1` px py or dxy dx2-y2 from E` Hybridization sp2 or sd2

According to VSEPR of trigonal planar p is less energetic than d so the hybridization is sp2

Molecular orbital (MO) theory : Homonucelar diatomic molecules: (chapter 2)

Molecular orbital (MO) theory : Heteronuclear diatomic molecules (chapter 2)

Molecular orbital (MO) theory: the ligand group orbital approach and application to triatomic molecules

Molecular orbital (MO) theory applied to the polyatomic molecules BH3, NH3 and CH4

Chapter 5 - Part 2

In molecular orbital (MO) theory, we begin by placing the nuclei of a given molecule in their equilibrium positionsand then calculate the molecular orbitals (i.e. regions of space spread over the entire molecule) that a single electron might occupy. Each MO arises from interactions between orbitals of atomic centers in the molecule, and such interactions are:

. Allowed if the symmetries of the atomic orbitals are compatible with one another;

. Efficient if the region of overlap between the two atomic orbitals is significant;

. Efficient if the atomic orbitals are relatively close in energy.

An important ground-rule of MO theory is that the number of MOs that can be formed must equal the number of atomic orbitals of the constituent atoms.

You have to apply aufbau principles on MO’ as we did for atomic ones

MOLECULAR ORBITAL THEORY

Hydrogen

1s*

1s

1s*

1s

Antibonding (LUMO)

bonding (HOMO)

Bond order = ½(bonding-antibonding)The symbol (*) instead of antibonding

Bond order= ½(2-0)=1 Bond order= ½(1-0)= ½

H2 H2+

Bond energy=270 kJ/mol Bond energy=452 kJ/mol

2H+

H2+

H2 -452

-270

0 kJ

1s 1s1s1s

H : 1s1

Bond length=74 pm Bond length=105 pm

1s*

1s

1s*

1s

Antibonding

bonding

Bond order= ½(2-2)=0 Bond order= ½(2-1)= ½

He2 He2+

Bond energy=0.05 kJ/mol Bond energy=301 kJ/mol

1s 1s1s1s

Helium

Molecule not observed

THE MORE BOND ENERGY, THE MORE BOND STABILITY

px py pz

Other Homonuclear Molecules

1s1s

2 p*

2 s*

2 s

2 p

2 p*

2 p

px py pz

1s

2 s*

2 s

px py pz

B2 , C2 , N2 O2 , F2

+

➾

➾

Linear Combination of Atomic Orbitals LCAOAO AO

σp

+

➾

+

+

+

+

➾

➾

➾

➾+

+ ➾

2px

2py

2py

2px

1s

1s

σs*

σs

2πp*

σp*

2πp

Orbital Potential Energy

+-

The s...s and p...p Overlaps

px py pz

1s1s

2 p*

2 s*

2 s

2 p

2 p*

2 p

px py pz

O2

PARAMAGNETIC: Unpaired electronsDiradical

Magnetism in Oxygen

http://sol.sci.uop.edu/%7Ejfalward/magneticforcesfields/magneticforcesfields.html

The variation in orbital energies for Period 2 homonuclear diatomic molecules as far as F2

O2 F2Li2 Be2 B2 C2 N2

1u

1 g

2 u

2 g

1 u

1 g

g , u crossing

Bond order 1 220 11 3Bond distance (pm)Bond energy (kJ/mol)

267 121124- 141159 110110 498607- 159297 945

MO s of Heteronuclear MoleculesMO’s of Heteronuclear

MoleculesThe 2s orbital of the F atom has an energy about 27eV lower than that of the hydrogen 1s, so there is very little interaction between them. The F orbital retains a pair of electrons. The F 2pz, orbital and the H 1s, on the other hand, have similar energies and matching A, symmetries, allowing them to combine into bonding σ and antibonding σ* orbitals. The F 2px and 2py orbitals have B1 and B2 symmetries and remain nonbonding, each with a pair of electrons. Overall, there is one bonding pair of electrons and three lone pairs.

π

σ

σ

π

σ*

1s

2p

2s

a1

b1 b2

a1

-13.6eV

-14.05eV

-40.17eV

H HF FH F

H

H

σ

σ

σ*

π

HF

Molecular Orbitals and Photoelectron Spectrum of CO. Molecular orbitals lσ and lσ* are from the 1s orbitals and are not shown, The el and e2 labels in the left- hand column are for the C∞ symmetry labels; the bl and b2 labels are for C2v

symmetry. (Photoelectron spectrum reproduced with permission from J. L. Gardner and J. A. R. Samson, J. Chem. Phys., 1975,62,1447).

Bond order=½(bonding-antibondig)=½(8-2)=3

C≡O- +

CO

CO2

MO’s of Polyatomic Molecules

2nodes

3nodes

1node

4nodes

2nodes

5nodes

3nodes

7nodes

3nodes

7nodes

3nodes

nonbonding

nonbonding

bonding

bonding

bonding

bonding

bonding

bonding

antibonding

antibonding

antibonding

antibonding

antibonding

antibonding

0 nodes

CO2

O-●-O O=C=O

CO2

CO2

π - Bonding in CO2

CO2

Molecular orbitals of nonlinear molecules can be determined by the same procedures. Water will be used as an example, and the steps of the previous section will be used.

1. Water is a simple triatomic bent molecule with a C2 axis through the oxygen and two mirror planes that intersect in this axis. The point group is therefore C2v.

2. The C2 axis is chosen as the z axis and the xz plane as the plane of the molecule. Because the hydrogen 1s orbitals have no directionality, it is not necessary to assign axes to the hydrogens.

3. Because the hydrogen atoms determine the symmetry of the molecule, we will use their orbitals as a starting point. The characters for each operation for the Is orbitals of the hydrogen atoms can be obtained easily. The sum of the contributions to the character (I, 0, or - 1) for each symmetry operation is the character for that operation, and the complete list for all operations of the group is the reducible representation for the atomic orbitals. The identity operation leaves both hydrogen orbitals unchanged, with a character of 2. Twofold rotation interchanges the orbitals, so each contributes 0, for a total character of 0. Reflection in the plane of the molecule (σv) leaves both hydrogens unchanged, for a character of 2; reflection perpendicular to the plane of the molecule (σv') switches the two orbitals, for a character of 0.

H2O

4. The representation Γ can be reduced to the irreducible representations A1 +B1, representing the symmetries of the group orbitals. These group orbitals can now be matched with orbitals of matching symmetries on oxygen. Both 2s and 2pz orbitals have A1 symmetry, and the 2px orbital has B1 symmetry. In finding molecular orbitals, the first step is to combine the two hydrogen 1s orbitals. The sum of the two, 1/√2[Ψ(Ha)+Ψ(Hb)], has symmetry A1 and the difference, 1/√2[Ψ(Ha)-Ψ(Hb)], has symmetry B1. These group orbitals, or symmetry-adapted linear combinations, are each treated as if they were atomic orbitals, so they contribute equally to the group orbitals. The normalizing factor is 1/√2. In general, the normalizing factor for a group orbitals is N=1/√(Σci2), where ci=the coefficients on the atomic orbitals. Each group orbital is treated as a single orbital in combining with the oxygen orbitals.

5. The same type of analysis can be applied character when a p orbital changes sign. Each orbital can be treated independently.The s orbital is unchanged by all the operations, so it has A1 symmetry.The px orbital has the B1 symmetry of the x axisThe py orbital has the B2 symmetry of the y axisThe pz orbital has the A1 symmetry of the z axisThe x, y, and z variables and the more complex functions in the character table assists in assigning representations to the atomic orbitals.6. The atomic group orbital combines with the same symmetry are combined into molecular orbitals. They are numbered Ψ1 through Ψ6 in order of their energy, with 1 the lowest and 6 the highest

H2O

Hydrogen orbitals

E C2 σv σv’

+1 +1 +1 +1

+1 -1 +1 -1

A1

B1

H+H

H-H

H2O

Oxygen orbitals

E C2 σv σv’

+1 -1 +1 -1

+1 +1 +1 +1

+1 -1 -1 +1

+1 +1 +1 +1A1s

B2px

A1pz

B1py

H2O

C2v E C2 σv σv’

A1 1 1 1 1

A2 1 1 -1 -1

B1 1 -1 1 -1

B2 1 -1 -1 1

Unchanged bonds E C2 σv σv’

ΓO-H 2 0 2 0

ΓO-H=A1+B1 means bonds (O of symmetry a1) and (H of symmetry a1)

H2O

H2O

D3h E 2C3 3C2 σh 2S3 3σv

A1` 1 1 1 1 1 1

A2 ` 1 1 -1 1 1 -1

E ` 2 -1 0 2 -1 0

A1`` 1 1 1 -1 -1 -1

A2 `` 1 1 -1 -1 -1 1

E `` 2 -1 0 -2 1 0

ΓB-H=A1`+E`Unchanged bonds E 2C3 3C2 σh 2S3 3σv

ΓN-H 3 0 1 3 0 1

x

y

BH3

E C3 C32 C2(1) C2(2) C2(3) σh S3 S32 σv

(1)σv

(2)σv

(3)

ψ1 ψ1 ψ2 ψ3 ψ1 ψ3 ψ2 ψ1 ψ2 ψ3 ψ1 ψ3 ψ2

ψ1

ψ2 ψ3

LGO(1)=ψ1+ψ2+ψ3+ψ1+ψ3+ψ2+ψ1+ψ2+ψ3+ψ1+ψ3+ψ2=4ψ1+4ψ2+4ψ3=1/√3(ψ1+ψ2+ψ3)

LGO(2)=1/√6(2ψ1-ψ2-ψ3)

LGO(3)=1/√2(ψ2-ψ3)

2s orbital has a1` symmetryE 2C3 3C2 σh 2S3 3σv

1 1 1 1 1 1

2pz orbital has a2` symmetry1 1 -1 1 1 -1

2 -1 0 2 -1 0 2px and 2py orbital has e` symmetry

B

BH3

LGO(1) orbital has a1` symmetryE 2C3 3C2 σh 2S3 3σv

1 1 1 1 1 1

2 -1 0 2 -1 0LGO(2) and LGO(3) orbital has e` symmetry

nonbonding

doesn’t belong to a1`+e`

C3v E 2C3 3C2

A1 1 1 1

A2 1 1 -1

E 2 -1 0

ΓN-H=A1+EUnchanged bonds E 2C3 3C2

ΓN-H 3 0 1

NH3

NH32pz and 2s orbital has a1 symmetry

2 -1 0 2px and 2py orbital has e symmetry

LGO(1) orbital has a1 symmetry

LGO(2) and LGO(3) orbital has e symmetry

doesn’t belong to a1+e

x

y

E 2C3 3C2

1 1 1

1 1 1

C3v E 2C3 3C2

1 1 1

1 1 1

NH3

Packing of spheres

The packing-of-spheres model applied to the structures of elements

Polymorphism in metals

Metallic radii

Melting points and standard enthalpies of atomization of metals

Alloys and intermetallic compounds

Bonding in metals and semiconductors

Semiconductors

Sizes of ions

Ionic lattices

Crystal structures of semiconductors

Lattice energy: The Born-Haber cycle

Lattice energy: ‘calculated’ versus experimental values

Applications of lattice energy

Defects in solid state lattices

Structures and energetics of metallic and ionic solidsChapter 6

Structures of the 14 possible crystal structures (Bravais lattices)

Bravais Lattices

x

y

x

z

A AB ABA

A AB ABC

x

z

x

y

Packing of Spheres

Hexagonal close packing (hcp)

Cubic close (ccp) or face-centered (fcc) packing

Cube

Octahedral hole

Tetrahedralhole

Simple Cubic non-close Packing Body-Centered Cubic non-close Packing

The Packing-of-Spheres Model Applied to the Structures of Elements

Group 18 Elements ccp (except He)

H2 and F2hcp,packing-of-spheres model to the crystalline structures of such molecules is only valid because they contain freely rotating molecules

Metallic elements

h hexagonal closet-packing❒cubic closet-packing❍ body-centered cubic packing✺ structures type of its own* slightly distorted

If a substance exists in more than one crystalline form, it is polymorphic.

***

*

*

ra

ra

c

b

ra

b

scca=2r

bccb2 =a2 +a2

c2 =a2 +b2 3a2

c= 3a=4r

a= 4r3

fccb2 =a2 +a2

16r2 =2a2

a= 8r

Relationship between Atomic Radius and the Edge Length in Three Different Unit Cells

Gold (Au) crystallizes in a cubic close-packed structure (the face-centered cubic unit cell) and has a density of

19.3 g/cm3. Calculate the atomic radius of gold in picometers and its percentage occupancy.

density of density of unit cellunit cell

Volume of Volume of unit cellunit cell

Edge of unit Edge of unit cellcell

Radius of Radius of Au atomAu atom

•

Edge

•• Face

Corner

Body (Center)

Corner=1⁄8 atomEdge=1⁄4 atomFace=1⁄2 atomCenter=1 atom

Atoms per unit cell

Atoms per unit cells=how many atoms can be situated inside a

unit cell (fractions are permitted)

8*(1/8)+6*(1/2)=4 atoms per unit cell

Mass of the unit cell=4 atoms

1 unit cell

1 mol6.022 1023 atoms

197.0g Au1 mol Au

1.311031g/unit cell

Volume of the unit cell=md

1.311031

19.3 g/cm3 6.79 1023cm3

a V3 6.79 1023cm33 4.08 108 cm

r a8

4.08 108 cm8

1.44 108 cm=144pm

8r a

( 8 )3r3 a3 22.63r3

(Atoms per unit cell) * 43r3 16

3r3 16.76r3

Unit cell edge length=

Unit cell volume=

The unit cell contains 4 spheres, and the volume of a sphere=4/3πr3

Volume of the unit cell occupied=

Percentage occupancy=Volume of the unit cell occupied/Unit cell volume*100%=16.76/22.63*100%=74%

Atomic Radii and Percentage occupancy

Holes occupancy=1-74%=26%

Pauling considered a series of alkali metal halides, each member of which contained isoelectronic ions (NaF, KCl, RbBr, CsI). In order to partition the ionic radii, he assumed that the radius of each ion was inversely proportional to its actual nuclear charge less an amount due to screening effects (can be calculated using Slater's rules).

For example, the difference in anion-cation distance between the halides NaX and KX is close to 36 pm irrespective of the anion X, and it is natural to attribute this to the difference in radii between Na+ and K+. To separate the observed distances into the sum of two ionic radii is, however, difficult to do in an entirely satisfactory way. One procedure is to look for the minimum value in the electron density distribution between neighboring ions, but apart from the experimental difficulties involved such measurements do not really support the assumption of constant radius. Sets of ionic radii are therefore all based ultimately on somewhat arbitrary assumptions. Several different sets have been derived, the most widely used being those of Shannon and Prewitt, based on the assumed radius of 140 pm for O2 in six-coordination.

Goldschmidt and, more recently, Shannon and Prewitt, concentrated on the analysis of experimental data (mostly fluorides and oxides) with the aim of obtaining a set of onic radii which, when combined in pairs, reproduced the observed internuclear distances, they found that rion for a given ion increases slightly with an increase in coordination number.

Cation Li+ Na+ K+ Rb+ Cs+

r+/pm 76 102 138 149 170Anion F- Cl - Br - I -

r-/pm 133 181 196 220

Ionic Radii

2rx-

d=2rx-+2rx+

d

Li+

Cl-

Ionic Radii Ratio

For LiF, this corresponds to an NaCl type lattice, in

agreement with the observed structure. In fact each of

the group 1 halides (except CsCl, CsBr and CsI) at

298K and 1 bar pressure adopts the NaCl type

structure; CsCl, CsBr and CsI adopt the CsCl type

lattice. Radius ratio rules predict the correct structures

in only some cases; they predict tetrahedral

coordination for the cations in LiBr and LiI, and cubic

coordination in NaF, KF, KCl, RbF, RbCl, RbBr and

CsF (in addition to CsCl, CsBr and CsI).

Metallic Radii

CN=how many atoms can make bond with the cation/anion* Inorganic Chemistry, Houescroft and Sharpe page 153

2rmetal

Coordination number

Relative Radius

12 8 6 4

1.00 0.97 0.96 0.88

For example in the case of K (potassium) the lattice system is cubic body centered=K with 8 other K around.

r12coordiante

r8coordinate

1

0.97

From Table 6.2 * r8coordinate 0.97 * r12coordinate 0.97 * 235 228pm

Binary compounds (compounds consisting of two elements) may have very simple crystal structures and can be described in several different ways. If the larger ions (usually the anions) are in close-packed structures, ions of the opposite charge occupy these holes, depending primarily on two factors:

1. The relative sizes of the atoms or ions. The radius ratio (usually r+/r- but sometimes r-/r+, where r+ is the radius of the cation and r- is the radius of the anion) is generally used to measure this. Small cations will fit in the tetrahedral or octahedral holes of a close-packed anion lattice. Somewhat larger cations will fit in the octahedral holes, but not in tetrahedral holes, of the same lattice. Still larger cations force a change in structure.

2. The relative numbers of cations and anions. For example, a formula of M2X will not allow a close- packed anion lattice and occupancy of all of the octahedral holes by the cations because there are too many cations. The structure must either have the cations in tetrahedral holes, have many vacancies in the anion lattice, or have an anion lattice that is not close-packed

Binary compounds

NaCl is made up of face-centered cubes (fcc) of sodium ions and face-centered cubes of chloride ions combined, but offset by

half a unit cell length in one direction so that the sodium ions are centered in the edges of the chloride lattice (and vice versa). If

all the ions were identical, the NaCl unit cell would be made up of eight simple cubic unit cells. Many alkali halides and other

simple compounds share this same structure. For these crystals, the ions tend to have quite different sizes, usually with the anions

larger than the cations. Each sodium ion is surrounded by six nearest-neighbor chloride ions, and each chloride ion is surrounded

by six nearest-neighbor sodium ions.

Na+

Cl-

Corner

Edge

Face

Center

Total

Na+ Cl-

8 0

0

6

0

12

0

1

8*1/8=1 Na+

12*1/4=3 Cl-

6*1/2=3 Na+

1*1=1 Cl-

4 Na+ and 4 Cl-

CN 6 6

Cation is smallAnion is large

Corner

Edge

Face

Center

Total

Cs+ Cl-

8 0

0

0

0

0

0

1

8*1/8=1 Cs+

0*1/4=3 Cl-

0*1/2=3 Cs+

1*1=1 Cl-

1 Cs+ and 1 Cl-

CN 8 8

As mentioned previously, a sphere of radius 0.73r will fit exactly in the center of a cubic

structure. Although the fit is not perfect, this is what happens in CsCl where the chloride

ions form simple cubes with cesium ions in the centers. In the same way, the cesium ions

form simple cubes with chloride ions in the centers. The average chloride ion radius is 0.83

times as large as the cesium ion (167 pm and 202 pm, respectively), but the inter-ionic

distance in CsCl is 356 pm, about 3.5% smaller than the sum of the average ionic radii.

Only CsCl, CsBr, Csl, TlCl, TlBr, TII, and CsSH have this structure at ordinary

temperatures and pressures, although some other alkali halides have this structure at high

pressure and high temperature. The cesium salts can also be made to crystallize in the NaCl

lattice on NaCl or KBr substrates, and CsCl converts to the NaCl lattice at about 469ºC.

Cation is largeAnion is large

The wurtzite form of ZnS is much rarer than zinc blende, and is formed at higher temperatures

than zinc blende. It also has zinc and sulfide each in a tetrahedral hole of the other lattice, but each

type of ion forms a hexagonal close-packed lattice. As in zinc blende, half of the tetrahedral holes

in each lattice are occupied. S

Zn

One sulfide layer and one zinc layer of wurtzite. The third layer contains sulfide ions, directly above the zinc ions. The fourth layer is zinc ions, directly above the sulfides of the first layer.

ZnS has two common crystalline forms, both with coordination number

4. Zinc blende is the most common zinc ore and has essentially the

same geometry as diamond, with alternating layers of Zn and S. It can

also be described as having zinc ions and sulfide ions, each in face-

centered lattices, so that each ion is in a tetrahedral hole of the other

lattice. The stoichiometry requires half of these tetrahedral holes to be

occupied, with alternating occupied and vacant sites.diamondZnS blend

ZnS wurtzite

The fluorite structure can be described as having the calcium ions in a cubic close-packed lattice, with

eight fluoride ions surrounding each one and occupying all of the tetrahedral holes. An alternative

description of the same structure has the fluoride ions in a simple cubic array, with calcium ions in

alternate body centers. The ionic radii are nearly perfect fits for this geometry. There is also an

antifluorite structure in which the cation-anion stoichiometry is reversed. This structure is found in all

the oxides and sulfides of Li, Na, K, and Rb, and in Li2 Te and Be2 C. In the antifluorite structure,

every tetrahedral hole in the anion lattice is occupied by a cation, in contrast to the ZnS structures, in

which half the tetrahedral holes of the sulfide ion lattice are occupied by zinc ions.

Fluorite shown as Ca2+ in a cubic close packed lattice,

each surrounded by eight F- in the tetrahedral holes.

Ni

As

TiO2 in the rutile structure has distorted TiO6 octahedra that form columns by sharing edges, resulting

in coordination numbers of 6 and 3 for titanium and oxygen, respectively. Adjacent columns are

connected by sharing corners of the octahedra. The oxide ions have three nearest-neighbor titanium

ions in a planar configuration, one at a slightly greater distance than the other two. The unit cell has

titanium ions at the corners and in the body center, two oxygens in opposite quadrants of the bottom

face, two oxygens directly above the first two in the top face, and two oxygens in the plane with the

body-centered titanium forming the final two positions of the oxide octahedron. The same geometry is

found for MgF2 , ZnF2 , and some transition metal fluorides. Compounds that contain larger metal ions

adopt the fluorite structure with coordination numbers of 8 and 4.

The nickel arsenide structure has arsenic atoms in identical close-packed layers stacked directly over

each other, with nickel atoms in all the octahedral holes. The larger arsenic atoms are in the center of

trigonal prisms of nickel atoms. Both types of atoms have coordination number 6, with layers of

nickel atoms close enough that each nickel can also be considered as bonded to two others. An

alternate description is that the nickel atoms occupy all the octahedral holes of a hexagonal close-

packed arsenic lattice. This structure is also adopted by many MX compounds, where M is a

transition metal and X is from Groups 14, 15, or 16 (Sn, As, Sb, Bi, S, Se, or Te). This structure is

easily changed to allow for larger amounts of the nonmetal to be incorporated into nonstoichiometric

materials.

Ti

O

NiAs - a semiconductor

Rutile

Perovskite is an example of a double oxide; it does not, as the formula might imply, contain[TiO3 ]2- ions, but is a mixed Ca(II) and

Ti(IV) oxide.

Many double oxides or fluorides such as BaTiO3 , SrFeO3 , NaNbO3 , KMgF3 and KZnF3 crystallize with a perovskite lattice.

Deformations of the lattice may be caused as a consequence of the relative sizes of the ions, e.g. in BaTiO3 , the Ba2+ ion is relatively

large (rBa2+=142pm compared with rCa2+=100 pm) and causes a displacement of each Ti(IV) centre such that there is one short TiO

contact. This leads to BaTiO3 possessing ferroelectric properties

The structures of some high-temperature superconductors are also related to that of perovskite. Another mixed oxide lattice is that of

spinel, (complicated structure and its structure will not be drawn here) MgAl2 O4 .

Corner

Edge

Face

Center

Ba2+ O2-

0 0

0

0

1

0

6

0

1*1=1 Ba2+

6*1/2=3 O2-

8*1/8=1 Ti+

CN 6 2

Perovskite Ti+

8

0

0

0

8

ccp (fcc)

BaTiO3 a ferroelectric complex

CdI2 Graphite

SiO32-

Si2O52-

SiO32-

Si4O116-

Si6O1812-Si3O96-

Common Silicate Structures

Quartz

=

Consider the first tetrahedron in the chains of SiO32- to have four oxygen atoms, or SiO4. Extending the chain by adding SiO3 units with the fourth position sharing an oxygen atom of the previous tetrahedron results in an infinite chain with the formula SiO3 . The charge can be calculated based on Si4+ and O2-. Si2 O52- can be described similarly. Beginning with one Si2 O7 unit can start the chain. Adding Si2 O5 units (two tetrahedra sharing one corner and each with a vacant comer for sharing with the previous unit) can continue the chain indefinitely. Again, the charge can be calculated from the formula.

ß-cristoballite

Alloys and Intermetallic Compounds

Substitutional alloys:

In a substitutional alloy, atoms of the solute occupy sites in the lattice of solvent metal, e.g. Sterling Silver (92.5% Ag and 7.5% Cu)

1. Both metals must have the same lattice (ccp, hcp...etc)

2. radius of the first metal is approximately equal to the other.

Interstitial alloys:

In a interstitial alloy, atoms with small radius occupy the

octahedral holes, e.g. Stainless steel Cr in Fe and Carbon

steel (see Box 6.2 page 159, of Inorganic chemistry

Housecroft and Sharpe)

Intermetallic compounds:

When melts of some metal mixtures solidify, the alloy formed may possess a definite structure type that is different from those of the

pure metals. e.g. CuZn (ß-Brass)

Replace (an) atom(s) from

the lattice by different one

Insert (an) atom(s) in the

lattice.

Metals

Explain why metals are usually lustrous (mirror-like)

In the band model there is a continuum of empty energy levels rather than discrete energy levels. Thus light quanta of all energies

within a wide range of wavelengths will be absorbed equally, and then the energized electrons will re-emit the light when they

fall back into their ground-state orbitals. This is the mechanism for reflection of light of all frequencies, which we call "luster".

Predict how the Group II metals differ from group I in density, melting point, and mechanical strength.

In any given period the Group II ions are smaller and may thus approach each other more closely. At the same time twice as

many electrons are present sea. The closer approach and the much greater electrostatic interactions between the 2+ ions and the

sea of high negative charge density lead to greater density and much greater bonding energy, which in turn leads to much higher

melting point and hardness. Actually going from Group I to Group II increases the density by a factor of about 2 and raises the

melting point by hundreds of degrees.

Metals usually feel cool to the touch compared with other materials because they are such good conductors of heat. How

can we explain this unusual thermal conductivity?

In most materials heat is conducted by atom-to-atom transfer of vibrational motion from the hot end to the cold end. In metals the

thermal energy is transferred primarily by the motion of the electron sea's free electrons, which are very mobile.

Conductors

LiLi LiLi

LiLi LiLi LiLi

LiLi LiLi LiLi LiLi

LiLi LiLi..........

6.0226.022××10102323 Li (metallic solid)Li (metallic solid)

Li2

Li3

Li4

LiAvogadro

empty orbitalsempty orbitals

filled orbitalsfilled orbitals

In any solid, metallic, ionic or covalent the valence orbitals of atoms and molecules combine to form molecular orbitals. The number of molecular orbitals is equal to the number of atomic orbitals, and for any material, this is a very large number but the energy difference between highest and lowest doesn't change by much. Many orbitals separated by very small energy gaps give rise to bands of orbitals.

For lithium metal, the band is made up of molecular orbitals formed from the combination of 2s atomic orbitals and about half of the orbitals are filled. There is only a very small energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). When an electron goes to an unoccupied orbital with available thermal energy, it is free to travel throughout the metal. The "hole" left in the HOMO can travel too.

Band gap

Bandgap Fermi Level➵

Ele

ctro

nic

(orb

ital e

nerg

y)

nonmetallic band picture metallic solid band picture

electron sea are in blue color

The band gap in a nonmetallic solid is important for electrical

and optical properties. A solid with a small band gap is a

semiconductor with a conductivity that (unlike the case with a

metal) increases as temperature is raised. The band gap also

determines the minimum photon energy required to excite an

electron from the valence band (VB) to the conduction band

(CB), and hence the threshold for optical absorption by a solid.

VB

CB

In silica (SiO2) there are covalent bonds between each silicon and 4 oxygens around it that are

combinations of Si 3s and 3p atomic orbitals and O 2s and 2p atomic orbitals. We can think of the

molecular orbitals forming similar bands but there is a large energy separation or band gap between the

occupied and unoccupied levels. The electrons in the orbitals of silica are not free to travel and the

material is an insulator.

In pure, crystalline silicon, there is also a band gap between the occupied and unoccupied molecular orbitals but it is smaller. Electrons in the filled valence band can be promoted to the conduction band by thermal energy or when the material absorbs light. The electrons would then be free to travel through the material and holes (electron vacancies) would travel through the valence band.

Semiconductors

Intrinsic semiconductors: pure semiconductors e.g. C, Si, Ge and a-Sn

Extrinsic (n- and p-type) semiconductors: extrinsic semiconductors contain dopants; a dopant is an impurity introduced into a semiconductor in minute amounts to enhance its electrical conductivity, e.g. Ge-doped Si. Doping involves the introduction of only a minutely small proportion of dopant atoms, less than 1 in 106, and extremely pure Si or Ge must first be produced.

Acceptor level

Small band gap}

p-type

Small band gap

Donor level

}

n-type

In a p-type semiconductor (e.g. Ga-doped Si), electrical conductivity arises from thermal population of an acceptor level which leaves vacancies (positive holes) in the lower band.

In an n-type semiconductor (e.g. As-doped Si), a donor level is close in energy to the conduction band.

Carbon in diamond form is an insulator with extremely high resistivity. But in graphite form its interatomic spacing is larger, making the band gap small enough to support some electrical conduction

Used as p-dopants to produce p-type semiconductors

Tin can be considered to be a semiconductor with a very small band gap, but at room temperature it supports metallic conduction

Silicon and germanium are the intrinsic semiconductors employed in solid state electronics.

Used as n-dopants to produce n-type semiconductors.

The bands overlap in lead, making it metallic conductor.

B C2p2 N

Al Si3p2 P

Ga Ge4p2 As

In Sn5p2 Sb

Tl Pb6p2 Bi

The electrical conductivity of a metal decreases with temperature

A metal is characterized by the fact that its electrical resistivity increases as the temperature increases, i.e. its electrical conductivity decreases as the temperature increases.

The electrical conductivity of a semiconductor increases with temperature.

Resistivity: Conductors vs. Semiconductors

The degree to which a substance or device opposes the passage of an electric current, causing energy dissipation. Ohm's law resistance (measured in ohms) is equal to the voltage divided by the current.

ohms

PerovskitesMgB2

Cu

Y

Ba

Mg

B

YBa2 Cu3 O7

The conductivity of metals and of semiconductors changes with temperature. Why? It's because the electrons flowing through a material will always pass close by positively charged nuclei. This impedes flow. As temperature increases, there is more thermal motion of the nuclei and this resistance increases. Resistivity is a property of the metal or semiconductor. In superconductors, the resistivity goes to zero at some temperature.

Some of the recently discovered high temperature superconductors are based on perovskites. You drew the lattice of CaTiO3 last week and most of you drew a cubic lattice of Ca with Ti in the cubic hole and oxygen atoms on the edges. Another perovskite with the same structure is BaCuO3 You could also draw the same lattice as a cubic array of copper atoms. Now imagine that some of the oxygen atoms have been removed and 1/3 of the barium atoms have been exchanged for yttrium. You would have the basic structure of a 1-2- 3 material or YBa2 Cu3 O7 .

Superconductors

Lattice Energy

The energy of the crystal lattice of an ionic compound is the energy released when ions come together from infinite separation to form a crystal:

M(g) ++X(g)-→MX(s)

In the crystal lattice there are more interactions. The summation of all of these geometrical interactions is known as the Madelung constant, A. The energy then can be expressed as following:

From Coulomb’s law, the attraction is obtained as following:

Ec ZZ

4 0 rThe energy increases as the interionic distance decreases.

It is common to express Z+ and Z- as multiples of electronic charge:

Ec e2ZZ

4 0 r

Ec Ae2ZZ

4 0 r

Madelung Constant

a distance r

a distance √2 r

a distance √3 r

a distance √4 r=2r

6X z

12M z

8X z

6M z

U e2

40

[(6r

z z ) ( 122r

z z ) ( 83r

z z ) ( 64r

z z )...]

U e2

40

[(6r

z z ) ( 122r

z2 ) ( 8

3rz z ) ( 6

4rz

2 )...]

U z z e2

40r[6 (

12 z2 z

) ( 83

) (3zz

)...]... ...

Madelung Constant

U z z e2

40r[6 (

12 z2 z

) ( 83

) (3zz

)...]zz

1 for NaCl

U z z e2

40r[6 12

2 8

3 3...]

U N

AvogadroA z z e2

40r

A Madelung constant.NAvogadro Avogadro’s number=6.0223x1023 atom/mol|z+| e.g. NaCl; Na has a charge of +1 then |+1|=1|z-| e.g. NaCl; Cl has a charge of -1 then |-1|=1e 1.602x10-18 C ε0 permittivity of vacuum=8.854x10-12 Fm-1

r the ionic radius

Lattice type ASodium chloride (NaCl) 1.7476Caesium chloride (CsCl) 1.7627

Wurtzite (α-ZnS) 1.6413Zinc blend (ß-ZnS) 1.6381

Fluorite (CaF2) 2.5194Rutile (Ti2O) 2.408

Cadmium iodide (CdI2) 2.355