Lattice Energy Formula

While studying the bonding between atoms we understood that depending upon the type of bonding the amount of energy that has to be supplied will be varying. Basically, the lattice energy is the amount of energy required to separate a mole of the solid ionic compound into its gaseous form. Sometimes, the overall potential energy of the compound is also referred to as lattice energy. The formula used to estimate the lattice energy is known as the lattice energy formula. 

The lattice energy of an ionic compound is generally represented by the letter Uⳑ and the lattice energy per mole can be represented as the total sum of the electrostatic and repulsive energy. Hence, we can determine the lattice energy by using the fundamental laws of electrostatic i.e., by Coulomb’s law as well as by using the Born-Lande equation. In this article, we will learn the lattice energy formula, born lande equation, and how to calculate lattice energy with suitable solved examples.

Born Lande Equation:

According to the definition of lattice energy, we know that it is the total amount of energy released when the cations and anions in their gaseous state are brought together from infinite separation to form a crystal. Theoretical treatment for the lattice energy was first given by M Born and A Lande. Hence, the lattice energy can be calculated by using Born Lande equations easily.

How to calculate lattice energy? 

The lattice energy can be calculated by using the classic Coulomb's law of electrostatics and by means of the Born Lande equation too. Let us have a look at the born lande equation which is also used as a lattice energy formula. So, there are several factors to take into account, such as the covalent character and electron-electron interactions of ionic solids. This will provide the approach for crystalline lattice energy. The positive ions (Anions) have both attraction and repulsion from ions of different charge and ions of identical charge.

Apart from these, there are few more factors to count for the determination of the lattice energy and they were given by Max Born and Alfred Landé. It led to the formula used for the calculation of the lattice energy for a mole of crystalline solid. It was familiarly known as the Born Lande equation and it is used for calculating the lattice energy of a  particular crystalline ionic compound. The born lande equation is basically derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.

The Born Lande equation is given by the following mathematical expression:

$$U_L=\frac{N_{A}M{Z}^{+}{Z}^{-}{e}^2}{4\pi {ϵ}_{0}r}(1-\frac{1}{n})$$

Where,

$$N_A-$$The Avogadro number

$$M-$$Madelung constant for the lattice

$$Z^{+}-$$The charge of the positive ion

$$Z^{-}-$$The charge of the negative ion

$$e-$$The charge of electrons

$$r-$$The distance between the two ions

$${ϵ}_o-$$The permittivity to free space

$$n-$$Born exponent and its value lies between 5 and 12 and is predetermined experimentally. This number directly related to the electronic configurations of the ions participated

How To Calculate Lattice Energy?

So, now how to find lattice energy?

We can derive the lattice energy of nearly any ionic solid by utilising a modified form of Coulomb’s law. This Lattice Energy Formula derived from the coulomb’s law is as follows:

$$U_L=\frac{K{Q}_1{Q}_2}{r_o}$$

Where,

$$Q_1$$ and $$Q_2-$$The charges on the ions

$$r_o$$The internuclear distance between the ions

We have to note an important point here that is the value of Uⳑ is always a positive number i.e., the value of Uⳑ has to be greater than zero (Uⳑ＞0), and it represents the total amount of energy required to dissociate one mole of an ionic solid compound into the gaseous ions. Here K is the proportionality constant.

As before, $$Q_1$$ and $$Q_2$$are the charges on the ions and $$r_o$$ is the internuclear distance. This lattice energy is directly related to the product of the ion charges, on the other hand inversely related to the internuclear distance.

Example:

1. Calculate the lattice energy of the NaCl crystal.

Sol:

Now, our aim is determining lattice energy of NaCl. The lattice energy can be calculated using the Born Lande Equation given below:

$$U_L=\frac{N_{A}M{Z}^{+}{Z}^{-}{e}^2}{4\pi {ϵ}_{0}r}(1-\frac{1}{n})$$

Where,

$$N_A-$$The Avogadro number whose value is $$6.023\times {10}^{23}$$

$$M-$$Madelung constant for the lattice whose value is 1.7475

$$Z^{+}-$$The charge of the positive ion = +1

$$Z^{-}-$$The charge of the negative ion = -1

   $$e-$$The charge of electrons

$$r-$$The distance between the two ions$$=2.81\times {10}^{-10}m$$

$${ϵ}_o-$$The permittivity to free space $$=8.854185\times {10}^{-12}{C}^2{J}^2{m}^{-1}$$

$$n-$$Born exponent and its value lies between 5 and 12 and is predetermined experimentally. This number directly related to the electronic configurations of the ions participated and for NaCl, it is found to be around 9.1.

Substituting, all the known values in the above equation we get:

$$\Rightarrow U_L=\frac{N_{A}M{Z}^{+}{Z}^{-}{e}^2}{4\pi {ϵ}_{0}r}(1-\frac{1}{n})$$

$$\Rightarrow U_L=\frac{(6.023\ast {10}^{23})(1.7475)(1.69\ast {10}^{-19}{)}^2}{4\pi (8.854\ast {10}^{-12})(2.81\ast {10}^{-10})}(1-\frac{1}{9.1})=756kj$$

$$\Rightarrow U_L=756kj$$

Therefore, the lattice energy of the NaCl is around 756kj.