Materials Science - Derivation of Kapustinskii equation

[10-D King]

1. The equilibrium bond length r₀ in ion components (or salts) can always be obtained from the tabulated values of cationic, r₊, and anionic radii, r_-, respectively as
r₀=r₊ + r_-

Show how on this basis, and that above (refering to the first part of the problem where i was to verify that the magdelung constants divided by the number of ions in one formula unit is constant(A/v)), the Born-Lande and Borne-Mayer equations become the Kapustinskii equations.

2. Borne-Lande equation

Borne-Mayer Equation

3. When carrying out the first part of the equation i obtained an average (A/v) value of 0.839 and i can easily derive the actual Borne equations but no matter what i try i can't get a decent answer for the derivation of the Kapustinskii equation

Any assistance would be of great appreciation.

 

[Nate810]

The Kapustinskii equation can be derived from the Borne-Lande and Borne-Mayer equations in the following way. From the Borne-Lande equation, we have A/V = 0.839 = r0^3/[(1/r+ + 1/r-)^2] where r0 is the equilibrium bond length and r+ and r− are the cationic and anionic radii, respectively. By substituting this value into the Borne-Mayer equation, we get A/V = 0.839 = (r0/r+)^2 + (r0/r−)^2 which simplifies to A/V = 0.839 = (r0/r+) + (r0/r−). This is the Kapustinskii equation.