Consider an ideal Fermi gas, with energy spectrum \(\varepsilon \propto p^{s}\), contained in a box of "volume" \(V\) in a space of \(n\) dimensions. Show that, for this system,

(a) \(P V=\frac{s}{n} U\);

(b) \(\frac{C_{V}}{N k}=\frac{n}{s}\left(\frac{n}{s}+1\right) \frac{f_{(n / s)+1}(z)}{f_{n / s}(z)}-\left(\frac{n}{s}\right)^{2} \frac{f_{n / s}(z)}{f_{(n / s)-1}(z)}\);

(c) \(\frac{C_{P}-C_{V}}{N k}=\left(\frac{s C_{V}}{n N k}\right)^{2} \frac{f_{(n / s)-1}(z)}{f_{(n / s)}(z)}\);

(d) the equation of an adiabat is \(P V^{1+(s / n)}=\) const.; and

(e) the index \((1+(s / n))\) in the foregoing equation agrees with the ratio \(\left(C_{P} / C_{V}\right)\) of the gas only when \(T \gg T_{F}\). On the other hand, when \(T \ll T_{F}\), the ratio \(\left(C_{P} / C_{V}\right) \simeq 1+\) \(\left(\pi^{2} / 3\right)\left(k T / \varepsilon_{F}\right)^{2}\), irrespective of the values of \(s\) and \(n\).