Lattice Energy Formula

Lattice Energy

Lattice energy refers to the energy which is released while two oppositely charged gaseous ions attract to each other and form an ionic solid. The total potential energy of the ionic compounds is also referred as the lattice energy. Lattice energy UL  per mole may be defined as the sum of the electrostatic and repulsive energy. The Born-Lande equation provides lattice energy.

Lattice Energy Formula per mole is symbolized as

\(\begin{array}{l}U_{L} = \left (\frac{N_{A}\alpha Z^{2}e^{2}}{4\pi \varepsilon _{o}r^{2}o} \right )\left ( 1 – \frac{1}{n} \right )\end{array} \)

NA = Avogadro’s constant (6.022 × 1022)

α = Madelung constant

e = Electron charge (1.6022 × 10-19C)

Z+ and Z– = Cation and anion charge

ϵo = Permittivity of free space

n = Born Exponent

r0 = Closest ion distance

UL = equilibrium value of the lattice energy

Solved Examples

Example 1: Compute the Lattice energy of NaCl by using Born-Lande equation.

Given
α = 1.74756
Z– = -1 (the Cl– ions charge)
Z+ = +1 (the charge of the Na+ ion)
NA = 6.022 × 1023 ion pairs mol-1

C = 1.60210 × 10-19C (the charge on the electron)
π = 3.14159
εo  = 8.854185 × 10-12 C2 J-1 m-1
ro = 2.81 × 10-10 m, the sum of radii of  Born-Lande equation.

Na+ and Cl–
n = 8 the average of the values for Na+ and Cl–.

Answer:

Using the Born-Lande equation.

\(\begin{array}{l}U_{L} = \left (\frac{N_{A}\alpha Z^{2}e^{2}}{4\pi \varepsilon _{o}r^{2}o} \right )\left ( 1 – \frac{1}{n} \right )\end{array} \)

Substitute all the values in the equation

\(\begin{array}{l}= \frac{1.74756 \times 6.022\times 10^{23}\times 1\times -1\times \left ( 1.60210\times 10^{-19} \right )^{2}}{4\times 3.14159\times 8.78541\times 10^{-12}\times 2.81\times 10^{-10}}\times \left ( 1-\frac{1}{8} \right )\end{array} \)

UL= – 755 KJmol-1

Example 2: The lattice energy of AgBr is 895 KJ mol-1. Predict the Lattice energy of the isomorphous AgI using Born-Lande equation. The numerics of rc + ra is 321 pm for AgBr and 342 pm for AgI.
Answer:

If the only variance between AgBr and AgI were in the size of the anion, one would expect the lattice energies to be relational to the inverse ratio of rc + ra.Henceforth we expect the Lattice energy of AgI to be

\(\begin{array}{l}= \left ( 895 \right )\left ( \frac{321}{342} \right ) = 840\ Jmole^{-1}\end{array} \)

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