(a) Show that the isothermal compressibility \(\kappa_{T}\) and the adiabatic compressibility \(\kappa_{S}\) of an ideal Fermi gas are given by

\[
\kappa_{T}=\frac{1}{n k T} \frac{f_{1 / 2}(z)}{f_{3 / 2}(z)}, \quad \kappa_{S}=\frac{3}{5 n k T} \frac{f_{3 / 2}(z)}{f_{5 / 2}(z)}
\]

where \(n(=N / V)\) is the particle density in the gas. Check that at low temperatures

\[
\kappa_{T} \simeq \frac{3}{2 n \varepsilon_{F}}\left[1-\frac{\pi^{2}}{12}\left(\frac{k T}{\varepsilon_{F}}ight)^{2}ight], \quad \kappa_{S} \simeq \frac{3}{2 n \varepsilon_{F}}\left[1-\frac{5 \pi^{2}}{12}\left(\frac{k T}{\varepsilon_{F}}ight)^{2}ight] .
\]

(b) Making use of the thermodynamic relation

\[
C_{P}-C_{V}=T\left(\frac{\partial P}{\partial T}ight)_{V}\left(\frac{\partial V}{\partial T}ight)_{P}=T V \kappa_{T}\left(\frac{\partial P}{\partial T}ight)_{V}^{2}
\]

show that

\[
\begin{aligned}
\frac{C_{P}-C_{V}}{C_{V}} & =\frac{4}{9} \frac{C_{V}}{N k} \frac{f_{1 / 2}(z)}{f_{3 / 2}(z)} \\
& \simeq \frac{\pi^{2}}{3}\left(\frac{k T}{\varepsilon_{F}}ight)^{2} \quad\left(k T \ll \varepsilon_{F}ight) .
\end{aligned}
\]

(c) Finally, making use of the thermodynamic relation \(\gamma=\kappa_{T} / \kappa_{S}\).