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Lattice Energy Formula

An ionic compound has the overall potential energy, which we refer frequently as the lattice energy. This lattice energy, represented as U per mole might be represented as the total of the electrostatic and repulsive energy terms. We can thus compute this lattice energy by using the fundamental laws of Coulomb as well as by using the Born-Lande equation. This article will explain this concept as well as lattice energy formula with examples. Let us learn it!

Concept of 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 attraction of the two ions releases energy, so this process is of an exothermic kind. Lattice energy can be a very complicated process but it can be easily understood by using Coulomb’s law.

Lattice Energy Formula

Source: en.wikipedia.org

Recall that the reaction of a metal with a non-metal usually produces the ionic compound. It is, electrons transfer from the metal to the non-metal. Ionic compounds are usually rigid, brittle, crystalline substances with flat surfaces in a lattice form.

They do not easily deform, as well as they melt at relatively high temperatures. For example, NaCl melts at \(801 ^{\circ}\) C. These properties form the regular arrangement of the ions in the crystalline lattice. Also from the strong electrostatic attractive forces between ions with opposite charges energy stored in it.

The Formula for Ionic Lattice Energy

We can compute the lattice energy of nearly any ionic solid by using a modified form of Coulomb’s law. This Lattice Energy Formula is as follows:

\(U=−\frac{k′Q_1Q_2}{r_0}\)

U is always a positive number, and it represents the amount of energy required to dissociate 1 mol of an ionic solid into the gaseous ions. Here K is the proportionality constant.

As before, \(Q_1 and Q_2\) are the charges on the ions and \(r_0\) is the inter-nuclear distance. This lattice energy is directly related to the product of the ion charges, on the other hand inversely related to the inter-nuclear distance.

The Formula for Crystalline Lattice Energy

There are factors to consider such as covalent character and electron-electron interactions in ionic solids. This will give the approach for crystalline lattice energy. The positive ions have both attraction and repulsion from ions of opposite charge and ions of the same charge.

There are other factors to consider for the evaluation of the lattice energy given by Max Born and Alfred Landé. It led to the formula for the evaluation of lattice energy for a mole of crystalline solid. This Born–Landé equation is for calculating the lattice energy of a  particular crystalline ionic compound. We derive it from the electrostatic potential of the ionic lattice and a repulsive potential energy term.

Thus its formula is,

\(U= \frac{N_A M Z^2e^2}{4\pi \epsilon_0 r} \left( 1 – \frac{1}{n} \right)\)

Where,

\(N_A\) Avogadro constant
M Madelung constant for the lattice
\(Z^+\) charge number of cation
\(Z^-\) charge number of anion
e elementary charge, \(1.6022 \times 10^{-19} C\)
\(\epsilon_0\) the permittivity of free space
\(r_0\) distance to closest ion
n Born exponent that is typically between 5 and 12 and is determined experimentally. This number related to the electronic configurations of the ions involved

Solved Examples for Lattice Energy Formula

Q.1: Determine the lattice energy for NaCl.

Solution: Using the above formula and known values, we have

\(\begin{align*}U_{NaCl} &= \dfrac{(6.022 \times 10^{23} /mol) (1.74756 ) (1.6022 \times 10 ^{-19})^2 (1.747558)}{ 4\pi \, (8.854 \times 10^{-12} C^2/m ) (282 \times 10^{-12}\; m)} \left( 1 – \dfrac{1}{9.1} \right) \\[4pt] &= – 756 \;kJ \;per\;mol \end{align*}\)

Therefore lattice energy for NaCl is 756 KJ per mol.

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