1
Lattice Energy
Pauling Rules
Bonding Models for Covalent and Ionic
Compounds
Organic (molecular, covalent) vs. Inorganic (nonmolecular, ionic) bonding
G. N. Lewis 1923
Electron pair sharing
Orbital overlap
Chemical bond
Number of bonds = atomic valence
Born, Lande, Magelung, Meyer
1918
Electrostatic attraction
(Coulomb) + Repulsion
1911
2
The lattice enthalpy change, L, is the standard molar enthalpy
change for the process:
M+
(gas) + X
(gas)  MX(solid)
The formation of a solid from ions in the gas phase is always
exothermic
Lattice enthalpies are usually negative
L is the most important energy factor in determining the stability of
an ionic compound
The most stable crystal structure of a given compound is the one
with the highest (most negative) lattice enthalpy L
(entropy considerations neglected)
H L
0
Lattice Enthalpy, L
(L)
3
Lattice Enthalpy, L, kJ/mol
4
d
eZZ
E BA
coul
2
04
1


All compounds adopt
the NaCl structure,
except CsCl, CsBr, CsI
ScN 7547
∆Hform
o = 411 kJ mol1
∆Hsubl
o = 108 kJ mol1
½ D = 122 kJ mol1
EA = 355 kJ mol1
IE = 502 kJ mol1
L = ?Na(s) + 1/2 Cl2 (g)
Na(g) + 1/2 Cl2 (g)
Na(g) + Cl (g)
Na+
(g) + Cl (g)
Na+
(g) + Cl-
(g)
NaCl (s)
0 = ∆Hform
o + ∆Hsubl
o + 1/2 D + IE + EA+ L
0 = 411 + 108 +122 + 502 + (355) + L L = 788 kJ mol1
Born-Haber Cycle
All enthalpies: kJ mol-1 for normal conditions  standard enthalpies
5
Lattice Enthalpy, L
L = ECoul + Erep
One ion pair
(calculated exactly)
(modelled empirically)
n = Born exponent
(experimental measurement of
compressibility or Pauling)
B = a constant
6
d
eZZ
E BA
coul
2
04
1


nrep
d
B
E 
d
ezz
ANE ACoul
0
2
4


Coulombic contributions to lattice enthalpies, ECoul
Coulombic Contribution to L
Coulomb potential
of an ion pair
ECoul : Coulomb potential (electrostatic potential)
A : Madelung constant (depends on structure type)
NA: Avogadro constant
z : charge number
e : elementary charge (1.6022 × 10−19 C)
o: dielectric constant (vacuum permittivity, 8.854 × 10−12 F m−1)
d : shortest distance between cation and anion 7
Lattice Enthalpy, L
1 mol of ions
ECoul = NA A (e2 / 4  e0) (zA zB / d)
A = Madelung constant - a single ion interacts with all other ions
Erep = NA B / dn
L = ECoul + Erep
Find minimum dL/d(d) = 0
nA
BA
A
d
B
N
d
eZZ
ANL 
0
2
4
8
Madelung Constant, A
Count all interactions in the crystal
lattice of one ion with all others
Madelung constant A = 1.3862944……
for an infinite linear chain of ions
= sum of convergent series
The simplest example : 1D lattice
9
Erwin Madelung
(1881 – 1972)
2ln2
4
....
4
1
2
3
1
2
2
1
2
1
1
2
4 0
2
0
2
d
ZZe
d
ZZe
E BABA
coul






Madelung Constant for NaCl
Sum of convergent series
3D ionic solids:
Coulomb
attraction and
repulsion
A single ion
interacts with all
other ions
74756.1....
5
24
2
6
3
8
2
12
6 A
10
M
d
ZZe
d
ZZe
E BABA
coul
0
2
0
2
4
....
5
1
24
4
1
6
3
1
8
2
1
12
1
1
6
4 







Neighbors Distance
6 Cl d
12 Na 2 d
8 Cl 3 d
6 Na 4 d
24 Cl 5 d
Madelung Constant for NaCl
11
Madelung Constants for Some Structural
Types
12
Structure A A/ Coordination
CsCl 1.762675 0.882 (8,8)
NaCl 1.747565 0.874 (6,6)
CaF2 2.51939 0.840 (8,4)
ZnS Wurtzite 1.64132 0.821 (4,4)
ZnS Sphalerite 1.63805 0.819 (4,4)
CdCl2 2.244 0.75 (6,3)
CdI2 2.191 0.73 (6,3)
TiO2 Rutile 2.408 0.803 (6,3)
Al2O3 Corundum 4.172 0.834 (6,4)
Linear Lattice 1.3862944 (2,2)
Ion Pair ? (1,1)
 = the number of ions in a formula unit
Born Repulsion, Erep
Because the electron density of atoms
decreases exponentially towards zero at
large distances from the nucleus the
Born repulsion shows the same
behavior
approximation:
d
E nrep
B

B and n are constants for a given atom type
n can be derived from compressibility
measurements or Pauling values
(e.g., for NaCl, n = 8)
Repulsion arising from
overlap of electron clouds
13
Max Born
1882 – 1970
The Born-Landé Equation (1918)
 EE repCoulL
 min
0
)
1
1(
4 0
2
0
n
N
d
ezz
A AL
 

set first derivative of the sum to zero
14
Total Lattice Enthalpy from Coulomb interaction and Born repulsion
Lattice Enthalpy Calculation







nd
eZZ
ANL BA
A
1
1
4 0
2

El. config. n Example
He-He 5 LiH, Be2+
Ne-Ne 7 NaF, MgO, Al3+
Ar-Ar 9 KCl, CaS, CuCl, Zn2+,
Ga3+
Kr-Kr 10 RbBr, AgBr, Cd2+, In3+
Xe-Xe 12 CsI, Au+, Tl3+, Ba2+, Hg2+
Born–Mayer
Born–Lande







d
d
d
eZZ
ANL BA
A
*
0
2
1
4
For compounds of mixed ion types, use
the average value (e.g., for NaCl, n = 8).
15
Pauling's approximate values of n
d* = 0.345 Å
The Kapustinskii Equation
Kapustinskii found that if the Madelung constant for a given
structure is divided by the number of ions in one formula unit ()
the resulting values are almost constant:
Structure Madelung constant (A) A/ Coordination
CsCl 1.763 0.882 8:8
NaCl 1.748 0.874 6:6
CaF2 2.519 0.840 8:4
-Al2O3 4.172 0.834 6:4
A general lattice energy equation that can be applied to any crystal
regardless of the crystal structure










 rr
G
rr
ZZ
vKL BA
1
K, G = constants
16
The most important advantage of the Kapustinskii equation
• it is possible to apply the equation for lattice calculations of
crystals with polyatomic ions (e.g., KNO3, (NH4)2SO4 ...)
• a set of „thermochemical radii“ was derived for further
calculations of lattice enthalpies
The Kapustinskii Equation
17
Experimental and Calculated Lattice
Enthalpies
NaCl
Born–Lande calculation L =  765 kJ mol1
Only ionic contribution considered
Experimental Born–Haber cycle L =  788 kJ mol1
Lattice Enthalpy consists of ionic and covalent contributions
18
Applications of Lattice Enthalpy calculations
• thermal stabilities of ionic solids
• stabilities of oxidation states of cations
• solubility of salts in water
• calculations of electron affinity data
• lattice enthalpies and stabilities of „non existent“ compounds
Five principles which could be used to determine the
structures of complex ionic/covalent crystals
Pauling’s Rule no. 1 Coordination Polyhedra
• A coordinated polyhedron of anions is formed about each cation
• Cation-Anion distance is determined by sums of ionic radii
• Cation coordination environment is determined by radius ratio
Pauling’s Rules
19
Coordination Polyhedra
20
Different Types of Radii
21
The radius of one ion was fixed to a
reasonable value
(r(O2-) = 140 pm) (Linus Pauling)
That value is then used to compile a
set of self consistent values for all
other ions
P – Pauling radius
G – Goldschmidt radius
S – Shannon radius
Variation of the Electron Density
along the Li – F Axis in LiF
22
Variation of Ionic Radii with Coordination Number
23
As the coordination
number (CN) increases,
the Ionic Radius increases
Variation of Atomic Radii
through the Periodic Table
24
R.D. Shannon and C.T. Prewitt, Acta Cryst. B25, 925-945 (1969)
R.D. Shannon, Acta Cryst. A32, 751-767 (1976)
Ionic Radii
As the coordination number (CN) increases, the Ionic Radius
increases
Sr 2+
CN Radius, Å
6 1.32
8 1.40
9 1.45
10 1.50
12 1.58
As the oxidation state increases,
cations get smaller
(6-fold coordination, in Å)
Mn2+ 0.810
Mn3+ 0.785
Mn4+ 0.670
Ti2+ 1.000
Ti3+ 0.810
Ti4+ 0.745 25
Ionic Radii
The radius increases down a group in the periodic table
The exception - 4d/5d series in the transition metals - the lanthanide
contraction
(6-fold coordination, in Å)
Al3+ 0.675
Ga3+ 0.760
In3+ 0.940
Tl3+ 1.025
Ti4+ 0.745
Zr4+ 0.86
Hf4+ 0.85
Right to left across the periodic table
the radius decreases
(6 coordinate radii, in Å)
La3+ 1.172
Nd3+ 1.123
Gd3+ 1.078
Lu 3+ 1.001
26
1. Ionic radii increase down a group
(Lanthanide contraction restricts the increase of heavy ions)
2. Radii of equal charge ions decrease across a period
3. Ionic radii increase with increasing coordination number
the higher the CN the bigger the ion
4. The ionic radius of a given atom decreases with increasing
charge (r(Fe2+) > r(Fe3+))
5. Cations are usually the smaller ions in a cation/anion
combination (exceptions: r(Cs+) > r(F-))
6. Frequently used for rationalization of structures:
„radius ratio“ r(cation)/r(anion) (< 1)
General Trends for Ionic Radii
27
Cation/Anion Radius Ratio
CN r/R
12 – hcp/ccp 1.00 (substitution)
8 – cubic 0.732 – 1.00
6 – octahedral 0.414 – 0.732
4 – tetrahedral 0.225 – 0.414
Optimal radius ratio
for given CN
ions are in touch
28
29
Structure Map
Structural map as function of
radius ratios for AB
compounds
Structural map as function of
radius ratios for A2BO4
compounds
Dependence of the structure type on parameters, such as ionic
radii, ionicity, electronegativity etc.
30
Structure Map
Dependence of the structure type (coordination number) on
the electronegativity difference and the average principal
quantum number (size and polarizability)
AB compounds
31
the electronegativity difference
Pauling’s Rule no. 2 Bond Strength
The strength of an electrostatic bond sij= valence / CN
The bond valence sum (BVS) of each ion equals its oxidation state Vi
The valence of an ion (Vi, equal to the oxidation state of the ion) is equal to a
sum of the valences of its bonds (sij): Vi =  sij
In a stable ionic structure the charge on an ion is balanced by the sum of
electrostatic bond strengths (sij) to the ions in its coordination polyhedron
TiO2 (Rutile) Ti - oxidation state of +4, coordinated to 6 oxygens
VTi = +4 = 6 (sij) sij = +2/3
The bond valence of oxygen, coordinated by 3 Ti atoms
Vo = 3 (sij) = 3 (-2/3) = 2
Each bond has a valence of sij with respect to the cation and sij with respect to the
anion
Pauling’s Rules
32
Bond Strength
Brown, Shannon, Donnay, Allmann:
Correlation of the valence of a bond sij with the (experimental)
bond distance dij
Rij = standard single bond length - determined empirically from (many)
structures where bond distances and ideal valences are accurately known
Tables of Rij values for given bonding pairs (i.e., Nb-O, Cr-N, Mg-F, etc.)
have been calculated, just as tables of ionic radii are available
A constant b = 0.37
R = d s = e0 = 1
R  d s = e1 < 1 a bond longer than R is weaker than 1
R  d s = e1 > 1 a bond shorter than R is stronger than 1
b
dR
s
ijij
ij

 exp
33
Bond Valence Sum (BVS)
Correlation of the valence of a bond sij with the (experimental) bond
distance dij
Use of the bond valence sum (BVS) concept:
A) To check experimentally determined structures for correctness, or
bonding instabilities
B) To predict new structures
C) To locate light atoms such as hydrogen or Li ion, which are hard to find
experimentally
D) To determine ordering of ions which are hard to differentiate
experimentally, such as Al3+ and Si4+, or O2- and FE)
To check/confirm oxidation states of atoms (Co2+ /Co3+ , Fe2+ / Fe3+ )
b
dR
s
ijij
ij

 exp
CN
z
sv i
iji 
34
Bond Valence Sum (BVS)
b
dR
s
ijij
ij

 exp
CN
z
sv i
iji 
35
FeTiO3 (mineral Ilmenite) possesses the corundum structure – an hcp array of
oxides with cations filling 2/3 of octahedral holes.
Decide which oxidation states (valences) are present:
Fe(II) Ti(IV) or Fe(III) Ti(III)
Bond Distances (dexp, Å) Tabulated Rij values Constants
FeO = 3×2.07 and 3×2.20 R0(FeO) = 1.795 Å b = 0.30
TiO = 3×1.88 and 3×2.09 R0(TiO) = 1.815 Å b = 0.37
Oxygen valence and coordination number?
Each oxygen is bound to Fe and Ti with both bond distances
Pauling’s Rule no. 3 Polyhedral Linking
The presence of shared edges, and particularly shared faces decreases
the stability of a structure
This is particularly true for cations with large valences and small
coordination number
Avoid shared polyhedral edges and/or faces
Pauling’s Rules
36
Polyhedral Linking
37
The Coulombic interactions in stable structures
- Maximize the cation-anion interactions (attractive)
- Minimize the anion-anion and cation-cation interactions (repulsive)
- Increase the coordination number
- Decrease the cation-anion distance
- If ions too close - electron-electron repulsions
The cation-cation distance decreases and the Coulomb repulsion
increases as
• the degree of sharing increases (corner < edge < face)
• CN decreases (cubic < octahedral < tetrahedral)
• cation oxidation state increases (a stronger Coulomb repulsion)
Polyhedral Linking
38
Pauling’s Rule no. 4 Cation Evasion
In a crystal containing different cations those with large valence and
small coord. number tend not to share polyhedral elements (anions)
Pauling’s Rules
Perovskite, CaTiO3
CaII 12-coordinate CaO12 cuboctahedra
share FACES
TiIV 6-coordinate TiO6 octahedra
share only VERTICES
39
Pauling’s Rule no. 5 Environmental Homogeneity
The rule of parsimony
The number of chemically different coordination environments
for a given ion tends to be small
Once the optimal chemical environment for an ion is found, if
possible, all ions of that type should have the same
environment
Pauling’s Rules
40
Strukturbericht Symbols
A partly systematic method for specifying the structure of a crystal
A - monatomic (elements), B - diatomic with equal numbers of atoms of each
type (AB), C - a 2-1 abundance ratio (AB2), D0 - 3-1, etc.
Structure type Struktur
bericht
Space group
(S.G. No.)
Lattice
Cu A1 Fm-3m (225) fcc
W, Fe A2 Im-3m (229) bcc
Mg A3 P63/mmc (194) hcp
C - diamond A4 Fd-3m (227) diamond
NaCl B1 Fm-3m (225)
CsCl B2 Pm-3m (221)
ZnS B3 F43m (216) Zincblende
ZnS B4 P63/mc (186) Wurtzite
CaF2 C1 Fm-3m (225) Fluorite
41
Pearson Symbols
Indicate the crystal symmetry and the number of atoms in the unit cell
e.g., NaCl - a face-centered (F) cubic (c) structure with 8 atoms in the unit cell = cF8
monoclinic (m), hexagonal (h), orthorhombic (o), asymmetric (a), primitive (P)
the Pearson symbol does not necessarily specify a unique structure (see cF8)
Structure type Pearson
Symbol
Struktur
bericht
Space group
(S.G. No.)
Cu cF4 A1 Fm-3m (225)
W, Fe cI2 A2 Im-3m (229)
Mg hP2 A3 P63/mmc (194)
C - diamond cF8 A4 Fd-3m (227)
NaCl cF8 B1 Fm-3m (225)
CsCl cP2 B2 Pm-3m (221)
ZnS (zb) cF8 B3 F43m (216)
ZnS (w) hP4 B4 P63/mc (186)
CaF2 cF12 C1 Fm-3m (225)
42
Space Group Symbols (230)
primitive (P), face-centered (F), body-centered (I), base-centered (A, B, C),
rhombohedral (R)
S. G. Class Centering Symbol syntax (examples)
Triclinic P P1, P-1
Monoclinic P, C, B Paxis, Pplane, Paxis/plane (P21, Cm, P21/c)
Orthorhombic P, F, I, C, A Paxisaxisaxis, Pplaneplaneplane (Pmmm, Cmc21)
Tetragonal P, I P4, P4axisaxisaxis, P4planeplaneplane (I4/m, P4mm)
Trigonal P, R P3axis, P3plane (R-3m)
Hexagonal P P6, P6axisplane (P63/mmc)
Cubic P, F, I Paxis3plane, Pplane3plane (Pm-3m, Fm-3m)
43