Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 2 / A8 / b / Solution

Differentiate the electronic energy at fixed volume:At $T\to 0$, $C_{V,\mathrm{el}}\sim N{k}_{\mathrm{B}}(\theta /T{)}^2{e}^{-\theta /T}\to 0$: thermal energy is insufficient to populate the upper manifold. At $T\to \infty$, $C_{V,\mathrm{el}}\sim N{k}_{\mathrm{B}}{\theta }^2/(4T^2)\to 0$: the two populations are already approaching equality and the electronic energy has a finite ceiling.

Between these limits the heat capacity has a Schottky anomaly, peaking near $T=\theta /2.399\simeq 72.0$ K at $C_{V,\mathrm{el}}\simeq 0.439N{k}_{\mathrm{B}}$. The energy rises monotonically toward $N\Delta /2$; its slope is largest at the heat-capacity peak. At 600 K it has not reached its infinite-temperature limit exactly.