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)}

Question:

Show that for an ideal Bose gas

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

compare this result with equation (7.1.36). Hence show that

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

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