The lattice energy of an ionic crystal is a measure of the strength of the electrostatic forces holding the ions together in the crystal lattice. Born's theory, also known as the Born-Landé equation or Born's rule, provides a theoretical framework to estimate the lattice energy of an ionic crystal based on the charges and sizes of the ions involved. According to this theory interaction energy Ui on the ion i due to other ions is

Uij -> interaction energy between i and j

In case of ionic crystal, there are two types of energies.

1. Coulomb interaction energy

2. The central field repulsive energy which exist only between nearest ions

Repulsive energy

where 𝜆 and 𝜌 are imperical parameters and 𝜌 is a measure of repulsive interaction.

The total lattice energy Utotal of a crystal composed of N molecules.

Utotal = NUi

For state of convenience, we get

rij = PijR

considering repulsive interaction only with rij = R

z - number of nearest neighbour for an ion

At equilibrium separation (R =ro) Utotal is minimum.

dU/dR = 0

From i,

This equation gives the lattice energy for ionic crystal. Here - N𝛼q² /ro is Madelung energy. The cohesive energy per ion pair is

The principal contribution to cohesive energy of ionic crystal is the electrostatic force of coulomb attraction is given by

where 𝛼 = Ƶ1Ƶ2 is a dimensionless constant called Modeling constant.The equation essentially represents the attractive electrostatic interaction between the oppositely charged ions in the crystal lattice. The larger the charges on the ions, the greater the electrostatic attraction, and hence, the higher the lattice energy. Similarly, as the ionic radii decrease (smaller r₀), the ions come closer together, leading to stronger electrostatic forces and a higher lattice energy.

We use the relation for potential energy per ion pair

where Modeling constant is

where Pij is a dimensionless quantity given by

Tij = PijR and R is the nearest neighbour separation

It's important to note that Born's theory provides a simplified approximation and does not capture all factors influencing the lattice energy, such as ionic polarizability or ionic covalency. Additionally, it assumes a regular and highly symmetric crystal lattice, which may not fully reflect the complexities of real crystals. However, Born's theory remains a useful tool for estimating and understanding the lattice energy of many ionic compounds.

ELI5: According to Born's theory, the lattice energy depends on a few things. First, it depends on the charges of the ions. If the ions have bigger charges, they attract each other more strongly, so the lattice energy is higher.

Second, it depends on the distance between the ions. If the ions are closer together, the attraction between them is stronger, and the lattice energy is higher.

So, the lattice energy is like a measure of how much "stickiness" there is between the ions in the crystal. The bigger the charges and the closer the ions are to each other, the stronger they stick together, and the higher the lattice energy.

By using the Born Theory, scientists can estimate this stickiness or lattice energy and understand how strong the attraction is between the ions in an ionic crystal.

This note is taken from Plasma, Msc physics, Nepal.

This note is a part of the Physics Repository.

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