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.

Question

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.

Accepted Answer

From Eq. 12.4-4 we have

[latex]\begin{aligned}\Delta_{ ext {conf }} \underline{S} &=-R\left[\frac{N}{N+n} \ln \frac{N}{N+n}+\frac{n}{N+n} \ln \frac{n}{N+n} ight] \&=-R\left[x_{ A } \ln x_{ A }+x_{ H } \ln x_{ H } ight]\end{aligned}[/latex] (12.4-4)

[latex]\begin{aligned} ext { Fraction of lattice sites that are vacant } &=\exp \left(-\frac{\Delta_{ vac } \underline{G}}{R T} ight) \&=\exp \left(-\frac{126000 \frac{ J }{ mol }}{8.314\left(\frac{ J }{ mol K } ight) T K } ight)\end{aligned}[/latex]

Using this equation, we find that the fraction of vacant lattice sites is [latex]6.9 imes 10^{-14}[/latex] at 500 K, [latex]2.6 imes 10^{-7} ext { at } 1000 K , ext { and } 4.1 imes 10^{-5}[/latex] at 1500 K

Question 12.4.1: Estimating the Number of Lattice Vacancies It has been estim...

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