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

Show that for an ideal Bose gas

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

compare this result with equation (7.1.36). Thence, show that

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

as in equation (7.1.48b). Check that, as \(T\) approaches \(T_{c}\) from above, both \(\gamma\) and \(C_{P}\) diverge as \(\left(T-T_{C}\right)^{-1}\).