Estimating the Number of Lattice Vacancies
It has been estimated that the Gibbs energy change on forming vacancies in a crystal of copper is approximately 126 kJ/mol and is independent of temperature. Estimate the fraction of the lattice sites that are vacant at 500, 1000, and 1500 K.
From Eq. 12.4-4 we have
\begin{aligned}\Delta_{\text {conf }} \underline{S} &=-R\left[\frac{N}{N+n} \ln \frac{N}{N+n}+\frac{n}{N+n} \ln \frac{n}{N+n}\right] \\&=-R\left[x_{ A } \ln x_{ A }+x_{ H } \ln x_{ H }\right]\end{aligned} (12.4-4)
\begin{aligned}\text { Fraction of lattice sites that are vacant } &=\exp \left(-\frac{\Delta_{ vac } \underline{G}}{R T}\right) \\&=\exp \left(-\frac{126000 \frac{ J }{ mol }}{8.314\left(\frac{ J }{ mol K }\right) T K }\right)\end{aligned}
Using this equation, we find that the fraction of vacant lattice sites is 6.9 \times 10^{-14} at 500 K, 2.6 \times 10^{-7} \text { at } 1000 K , \text { and } 4.1 \times 10^{-5} at 1500 K