The Kapustinskii Equation

The Born-Mayer equation gives the total electrostatic interaction energy for a given structure in terms of the Madelung constant, A, for that structure. Something else that varies from structure to structure is the number of ions within the formula unit for a given structure.

Structure Coordination Number Madelung constantA A/ν
ν is the number of ions in the formula unit
Sodium Chloride (NaCl) 6:6 1.74756 0.88
Cesium Chloride (CsCl) 8:8 1.76267 0.87
Zinc Blende (ZnS) 4:4 1.638 0.82
Wurtzite (ZnS) 4:4 1.64132 0.82
Fluorite (CaF2) 8:4 2.51939 0.84
Rutile (TiO2) 6:3 2.408 0.80
Corundum (Al2O3) 6:4 4.1719 0.83

Kapustinskii noted that the ratio of the Madelung constant, A, to the total number of ions per formula unit for a series of compounds with different structures deviate by less than 10% from 0.87, the value found for the NaCl structure.

By replacing A with 0.87ν, taking the average value of n, and making the approximation that the internuclear separation is the sum of the ionic radii, he derived a new expression for the total electrostatic interaction from the Born-Lande equation. This is known as the Kapustinskii equation.

The Kapustinskii Equation:(note that r+ and r are measured in pm)

The Kapustinskii equation predicts lattice enthalpies to within 10% of the experimental values for most compounds.