The Born–Landé equation is a means of calculating the lattice energy of a crystalline ionic compound. In 1918 Max Born and Alfred Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term. Where:

Derivation

The ionic lattice is modeled as an assembly of hard elastic spheres which are compressed together by the mutual attraction of the electrostatic charges on the ions. They achieve the observed equilibrium distance apart due to a balancing short range repulsion.

Electrostatic potential

The electrostatic potential energy, Epair, between a pair of ions of equal and opposite charge is: where For a simple lattice consisting ions with equal and opposite charge in a 1:1 ratio, interactions between one ion and all other lattice ions need to be summed to calculate EM, sometimes called the Madelung or lattice energy: where

Repulsive term

Born and Lande suggested that a repulsive interaction between the lattice ions would be proportional to 1⁄r so that the repulsive energy term, ER, would be expressed: where

Total energy

The total intensive potential energy of an ion in the lattice can therefore be expressed as the sum of the Madelung and repulsive potentials: Minimizing this energy with respect to r yields the equilibrium separation r0 in terms of the unknown constant B: Evaluating the minimum intensive potential energy and substituting the expression for B in terms of r0 yields the Born–Landé equation:

Calculated lattice energies

The Born–Landé equation gives an idea to the lattice energy of a system. ! Compound ! Calculated ! Experimental !NaCl !LiF !CaCl2

Born exponent

The Born exponent is typically between 5 and 12. Approximate experimental values are listed below: !Ion configuration !He !Ne !Ar, Cu+ !Kr, Ag+ !Xe, Au+ !n