= Solution
Differentiate the electronic energy at fixed volume:
$$
\boxed{C_{V,\mathrm{el}}=Nk_{\mathrm B}\left(\frac\theta T\right)^2
\frac{e^{\theta/T}}{(1+e^{\theta/T})^2}.}
$$
At $T\to0$, $C_{V,\mathrm{el}}\sim Nk_{\mathrm B}(\theta/T)^2e^{-\theta/T}\to0$: thermal energy is insufficient to populate the upper manifold. At $T\to\infty$, $C_{V,\mathrm{el}}\sim Nk_{\mathrm B}\theta^2/(4T^2)\to0$: the two populations are already approaching equality and the electronic energy has a finite ceiling.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2024/paper-2-electronic-thermodynamics.png]
{title=Original electronic internal-energy and Schottky heat-capacity curves for the two-manifold model from 0 to 600 K}
Between these limits the heat capacity has a <Schottky anomaly>, peaking near $T=\theta/2.399\simeq72.0$ K at $C_{V,\mathrm{el}}\simeq0.439Nk_{\mathrm B}$. 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.
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