Question 705039

The partition function $Z = 1 + e^{-\beta E}$. The average energy $\langle E \rangle = \frac{E e^{-\beta E}}{1+e^{-\beta E}}$. The specific heat $C_V = \frac{\partial \langle E \rangle}{\partial T} = \frac{\partial \langle E \rangle}{\partial \beta} \frac{d\beta}{dT}$. Since $\beta = 1/k_BT$, $\frac{d\beta}{dT} = -1/k_BT^2$. Differentiating $\langle E \rangle$ gives $C_V = \frac{E^2}{k_BT^2} \frac{e^{\beta E}}{(1+e^{\beta E})^2}$. At high $T$, $\beta E \to 0$, $e^{\beta E} \approx 1$, thus $C_V \propto 1/T^2$.