Real Materials

The thermodynamic equilibrium concentration of vacancies corresponds to the minimum value of the Gibbs function as n is changed. The minimum results from the balance between the enthalpy and entropy contributions.

(dG_n/dn)_T,p = 0 = (d[G₀+nH_V- kT Ln{W}]/dn)_T,p

· For large x, Stirling's approximation gives:

ln x! = x ln x - x.

· Using this to differentiate the above expression for G gives:

H_v - kT[Ln(N - n₀) - Ln(n₀)] = 0

i.e. {n₀/(N - n₀)}={n₀ / N} = exp(- H_V/kT)