Show that for an ideal Fermi gas [frac{1}{z}left(frac{partial z}{partial T} ight)_{P}=-frac{5}{2 T} frac{f_{5 / 2}(z)}{f_{3 / 2}(z)}]

Show that for an ideal Fermi gas

\[\frac{1}{z}\left(\frac{\partial z}{\partial T}\right)_{P}=-\frac{5}{2 T} \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)}\]

compare with equation (8.1.9). Hence show that

\[\gamma \equiv \frac{C_{P}}{C_{V}}=\frac{(\partial z / \partial T)_{P}}{(\partial z / \partial T)_{v}}=\frac{5}{3} \frac{f_{5 / 2}(z) f_{1 / 2}(z)}{\left\{f_{3 / 2}(z)\right\}^{2}} .\]

Check that at low temperatures

\[\gamma \simeq 1+\frac{\pi^{2}}{3}\left(\frac{k T}{\varepsilon_{F}}\right)^{2}\]