1
Lattice energy of ionic solids
Interatomic Forces
Solids are aggregates of atoms, ions or molecules. The bonding between these particles may be
understood in terms of forces that play between them. Attractive and repulsive forces come into play
when these particles approach each other. When atoms are infinitely far apart the forces between them are
zero and the potential energy of interaction is therefore zero. As particles approach each other potential
energy of interaction may be positive or negative depending upon the distance of separation. The potential
energy due to attraction is negative and that of repulsive energy is positive. Thus energies of interaction
are
a) Attractive energy
b) Repulsive energy
The net potential energy is the sum of attractive and repulsive energies which may be written as,
U(r) =
Where r is internuclear distance, a and b are constants representing the forces of attraction and repulsion,
and m and n are small integers characteristic of a given system. The variation of attractive and repulsive
energies as a function of distance of separation is diagrammatically shown in figure 1.
Figure 1: Potential energy as a function of distance of separation.
2
The internuclear distance corresponding to the minimum potential energy is the equilibrium distance and
the net decrease in energy is the binding energy. At equilibrium separation, attractive and repulsive forces
just balance and the potential energy is lowest.
The particles form a stable lattice at equilibrium distance ro at which U(r) is minimum.
Problem 1: The potential energy of a diatomic molecule is given in terms of the internuclear distance of r
by the expression,
U(r) =
(a) Derive an expression for the equilibrium spacing of the atoms and hence obtain the dissociation
energy.
(b) Prove that the attractive force should vary more slowly with r than the repulsive force.
Problem 2: The interaction energy of two particles in the field of each other is given by
U(r) =
(a) Show that the particles form a stable compound for
(b) Show that in stable configuration the energy of attraction is 9 times the energy of repulsion.
(c) Show that the potential energy of the two particles in stable configuration is -
3
Nature of Bonding
The nature of chemical bonding between atoms depends upon the electronegative difference
between bonding atoms. The bonding may be classified into primary and secondary as follow.
Primary bonding:
Ionic:
Covalent:
Metallic:
transfer of valence electrons
sharing of valence electrons
delocalization of valence electrons
Secondary or van der Waals Bonding:
Dipole-dipole
H-bonds
Polar molecule-induced dipole
Fluctuating dipole (weakest)
1. If electronegativity difference = zero. The bonding will be pure covalent. In pure covalent
bond, there will be equal shearing of bonding electrons and such bonding is possible in
homonuclear diatomic molecules such as Cl2, O2, N2.
2. If electonegativity difference = small. The bonding will be polar covalent. In pure covalent
bonding there will be unequal shearing of the bonding electrons between bonding atoms.
3. If electronegativity difference = large. The bonding will be ionic. In ionic bonding the
shearing of electrons will be so unequal that the electron pair is exclusively posses by one of
the bonding atoms. In such situation, it is reasonably correct to say that the electron has
transferred from one atom to another atom forming ions. These ions are not free but
bonded together by columbic forces of attraction and bond is called ionic bond.
Ionic Bonding
A bond will be formed whenever there is a lowering of energy. Let us take an example of formation
of ion-pair molecule.
Na(g)
 Na+(g) + eCl(g) + e Cl-(g)
+
Na (g) + Cl (g) NaCl(s)
Na(g) + Cl(g)  NaCl(g)
Ionization energy = 496 kJ
Electronegativity = -349kJ
Coulomb energy = -584 kJ
∆H = -439kJ
Thus the ion pair molecule is more stable than isolated gaseous atom by 439kJ/mole. In ion pair
molecule, the coulomb attraction between oppositely charged Na+ and Cl- ions is responsible for the
stability of the molecule. Substances which form ion pair molecules in gaseous phase condense to
form ionic crystal. In crystalline solid the alternative presence of oppositely charged ions extends
infinitely in three dimensions that give rise to more lowering of the energy that results in the
formation of crystal which is discussed more elaborately below.
4
Lattice Energy of Ionic Crystal
Lattice energy of a compound is the energy given off when oppositely charged ions in the gas phase
come together to form a solid. The lattice energy of NaCl is the energy given off when one mol of
Na+ and Cl- ions in the gas phase come together to form NaCl crystal
Na+(g) + Cl-(g)
NaCl(s)
Ho = -788 kJ/mol
The lattice energy of NaCl is 788 kJ/mol. The lattice energy of ionic crystal can be determined by BornHaber cycle using Hess law. The Born Haber cycle for the determination of lattice energy is shown in the
figure 2.
Born-Haber Cycle
Figure 2: Born Haber Scheme for determination of lattice energy
∆Hf
= ∆Hsub + I.E + ∆Hdiss + E.A + U
-411
= 108 + 496 + 122 – 349 + U
U
= - 411 - 108 - 496 - 122 + 349 = -788 kJ/mol
Lattice Energy = 788 kJ/mol
The lattice energy is greatest when the ions are small. The lattice energies for the alkali halides is,
therefore largest for LiF and smallest for CsI, as shown in the following table.
5
Table 1: Lattice Energies of Alkali Metals Halides (kJ/mol)
+
Li
Na+
K+
Rb+
Cs+
F1036
923
821
785
740
Cl853
788
715
689
659
Br807
747
682
660
631
I757
704
649
630
604
Problem 3: Calculate the lattice energy of AgCl from the following thermo-chemical data
Ag(s)  Ag(g)
∆H=284 kJ
Cl2(g)  2Cl(g)
∆H= 243 kJ
Ag(g)  Ag+(g) =e∆H=731 kJ
Cl (g) + e-  Cl-(g)
∆H= -349 kJ
Ag(s) + ½ Cl2  AgCl(s)
∆H=-127kJ
Ag+(s) + Cl-(g)  NaCl(s)
∆H= -U
[Ans = -915kJ]
The ionic bond should also become stronger as the charge on the ions becomes larger. The data in the
table below show that the lattice energies for salts of the OH- and O2- ions increase rapidly as the charge
on the ion becomes larger.
Table 2: Lattice Energies of Salts of the OH- and O2- Ions (kJ/mol)
Na+
Mg++
Al+++
OH900
3006
5627
O22481
3791
15916
6
Theoretical Evaluation of lattice energy
Ions when brought closer together attractive and repulsive energy come into play. The attractive energy is
given by coulomb energy. The electrostatic attractive potential energy, Epair, between a pair of ions of
charge z1 and z2 is,
Epair
For a simple lattice of 1:1 crystal like NaCl, the coulomb interactions between one ion and all other lattice
ions need to be summed which is given by
Elattice
where
A
r
= Madelung constant, which is related to the geometry of the crystal
= closest distance between two ions of opposite charge
Born and Lande suggested that a repulsive interaction between the lattice ions would be proportional to
1/rn so that the repulsive energy term, Er, would be expressed:
E
=
Where
B
r
n
= constant
= closest distance between two ions of opposite charge
= Born exponent
The total potential energy of an ion in the lattice can therefore be expressed as the sum of the coulomb
and repulsive potentials:
Etotal =
Figure 3: Variation of Born repulsive energy and Coulombic attractive
energy as a function of internuclear distance
7
For one mole of the ionic crystal,
Differentiating this equation with respect to distance r, we get
At equilibrium, du/dr = 0 and r = ro, so from equation, we have,
Substituting the value of B in terms and simplifying we get the Born–Landé equation for lattice energy as
U(ro) =
Where,
A is Madelung constant
n = Born exponent
E = electrostatic permittivity in vacuum
r= inter-ionic distance
N = Avogadro’s constant.
Born-Lande equation for the ionic crystal of charges z1 and z2 is
U(ro) =
This equation shows that lattice energy of ionic crystal
 Is proportional to the product of charges of the ions and
 Inversely proportional to the inter-ionic distance.
This means that lattice increases with increase in charges on the ions and decreases with increase in size
of the combining ions. This means that the lattice energy is greatest when the ions are small. The lattice
energies for the alkali halides clearly shows that as the ions becomes large, the iner nuclear distance
increases and lattice energy decreases.(Table1) Because of this LiF has the highest lattice energy while
that of CsI is smallest.
The lattice energy become greater as the charge on the ions becomes larger. The lattice energies of some
ionic compounds are given below which clearly shows that lattice energy increase rapidly as the charge
on the ion becomes larger.
NaCl = 788 kJ mol-1 Na2O = 2481 kJ mol-1 MgO = 3791 kJ mol-1 Al2O3 = 15,916 kJ mol-1
8
Evaluation of Madelung constant
The coulomb attractive energy between two oppositely charge ions of unit charge is given by coulomb
law as
=
For one mole of ion pair is,
=
In crystalline state, each ion is surrounded by a definite number of ions of opposite charge and this
extends infinitely in three dimensional. This arrangement makes the system more stable owing to greater
Coulomb force of attraction. The Coulombic force of attraction due to formation of crystalline solid will
be increased by a factor A, where A is Madelung constant. The value of Madelung constant depends on
the geometry of lattice. The value of Madelung constant for different crystal lattice was first evaluated by
Erwing Madelung in 1918. The evaluation of Madelung constant for three dimensional lattice is involving
one. Therefore before considering the evaluation of Madelung constant for three dimensional lattice, let
us consider a hypothetical linear array of ions of alternative sign as shown in the following figure.
Reference sodium ion
-
Na+
Cl
-
Na+
Cl
4ro
3 ro
2 ro
ro
-
Na+
Cl
0
ro
-
+
Na
Cl
+
Na
2 ro
3 ro
4 ro
Figure 5: Hypothetical array of cations and anions in one dimensional lattice.
Let us pick up a positively charged ion for reference, and let ro be the distance between adjacent ions. The
total coulomb energy due to all ions per mole is then given by,
+
+
+
+
2 ln(1+1)]
9
2 ln 2]
Where, A = 2 ln 2 = 1.386 is the Madelung constant for a linear array of oppositely charge ions.
Thus even in 1 dimensional array of oppositely charge ions the attractive energy is 38.6% more
than that of ion pair molecule.
In real crystal like NaCl, the attractive energy is still more but it is not easy to sum the series
conveniently. Let us consider the lattice of NaCl, Rock salt structure. The unit cell of NaCl is
given in figure 6.
Let us consider ththe central Na+ ion as a reference. This central ion is surrounded by
6
12
8
6
Cl- ion at ro
Na+ at
Cl- at
Na+ at
Figure 6: Unit cell of NaCl
The total coulomb energy of interaction per mole is given by,
10
Where A
=
= 6 8.485 + 4.620
The series is far from convergence and not easy to evaluate. Various models have been developed to
evaluate the value of Madelung constant for different types of lattices which have been well discussed by
M.Toshi. The Madelung constant for some typical ionic crystal lattices are given below.
Rock salt structure
1.74756
CsCl structure
1.76267
Zinc Blende
1.63806
The difference in coulombic energy between cation anion ion pair in gaseous state and that in crystalline
state is due to the cumulative attraction of given ion in ionic crystal by surrounding charge ions at various
distance as determined by the geometry of the crystal . As a result the electrostatic energy is increased by
the factor A, and this factor is called Madelung constant. Madelung constant is a factor which indicates by
what factor the binding energy increases compare to ion pair when the ions of the crystal arranged in a
crystal lattice.
M.P.Toshi, Solid State Physics 16, 1, 1964.
Evaluation of Born repulsive potential exponent
The lattice energy of ionic crystal, like NaCl, is given by Born-Mayer equation,
(1)
Where n is Born repulsive potential exponent. The value of this can be evaluated from compressibility
of the crystal. Compressibility, β, is the reciprocal of bulk modulus, κ,
(2)
But bulk modulus, κ, is given by
Κ=
(3)
Where dP is the change is pressure and dV is the corresponding change in volume produced by the
application of pressure.
(4)
According to I law of thermodynamics,
11
dq = dU + PdV
In isothermal process, dq = 0
du = -PdV
Differentiating this with respect to V,
(5)
Substituting the value of –dP/dV from equation 5 to equation 4, we get
(6)
But we have,
Differentiating with respect to V
At equilibrium, du/dr = 0, so
(7)
Substituting the value of d2U/dV2 from equation 7 to equation 6, we get
(8)
Le us take a unit cell of NaCl,
Edge length, a = 2ro
Volume of unit cell, a3 = (2ro)3 = 8 ro3
In unit cell there are 4 units of NaCl,
12
volume per unit cell = a3/4 = 8ro3/4 = 2ro3
volume per mole of NaCl,
V=2 Nro3
(9)
Where N is Avogadro's constant.
Differentiating equation with respect to r
(10)
Substituting the value V and dr/dV from equation 9 and 10 to equation 8
(11)
To evaluate d2U/dr2, we have to take the expression for Born-Mayer equation of lattice energy, as
Differentiating with respect to r
(12)
Differentiating again with respect to r
(13)
At equilibrium, du/dr = 0 and r = ro, so from equation 12 we have,
(14)
Substituting the value of B from equation 14 at equilibrium condition, to equation 13, we get
13
(15)
Substituting the value of d2U/dr2 from equation 15 to the equation 11, we get
On simplification, we get
(16)
+1
(17)
This is the final equation to evaluate Born repulsive potential exponent for an ionic crystal of rock salt
structure. Here,
εo is compressibility of the crystal
ro equilibrium distance between cation and anion
A is Madelung constant.
e is electronic charge
When proper values of different constant s are substituted for NaCl in the equation, the value of n is found
to be 9.47.