= Solution
Using $dx/dT=-x/T$ in the <internal energy> gives
$$
\boxed{C_V=Nk_{\rm B}\frac{2x^2e^{-x}}{(2+e^{-x})^2},\qquad x=\frac{\delta}{k_{\rm B}T}.}
$$
This is a <Schottky anomaly with unequal degeneracies>, with lower-level <state degeneracy> two and upper-level <state degeneracy> one. Its limiting forms are
$$
\boxed{C_V\sim\frac{Nk_{\rm B}}2x^2e^{-x}\quad(T\to0),\qquad
C_V\sim\frac{2Nk_{\rm B}}9x^2\quad(T\to\infty).}
$$
It vanishes at low <temperature> because exciting the upper orientation is exponentially unlikely. It also vanishes at high <temperature> because the populations have nearly saturated and further heating hardly changes the <internal energy>. Between these limits it has one smooth maximum. The original plot shows its shape; numerical maximization gives $k_{\rm B}T_{\max}/\delta\simeq0.449$ and $C_{\max}/(Nk_{\rm B})\simeq0.241$.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-2-orientation-thermodynamics.png]
{title=Original heat-capacity peak and entropy limits for two lower orientations and one excited orientation}
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