Utilizing the result obtained in Problem 8.13, we have for a Fermi gas at low temperatures

\[
\begin{equation*}
\frac{C_{\mathrm{V}}}{N k}=\frac{\pi^{2}}{3} \frac{a\left(\varepsilon_{F}\right)}{N} k T \tag{1}
\end{equation*}
\]

Now, the density of states for the relativistic gas is given by, see eqn. (8.5.7),

\[
a(\varepsilon)=\frac{8 \pi V}{h^{3}} p^{2} \frac{d p}{d \varepsilon}=\frac{8 \pi m V}{h^{3}} p\left\{1+\left(\frac{p}{m c}\right)^{2} \right\}^{1 / 2},
\]

where \(p=p(\varepsilon)\). Substituting this result into (1) and making use of eqn. (8.5.4), we get

\[
\frac{C_{\mathrm{V}}}{N k}=\frac{\pi^{2} m}{p_{F}^{2}}\left\{1+\left(\frac{p_{F}}{m c}\right)\right\}^{1 / 2} k T
\]

which leads to the desired result.

In the non-relativistic case \(\left(p_{F} \ll m c \right.\) and \(\left.\varepsilon_{F}=p_{F}^{2} / 2 m \right)\), we obtain the familiar expression (8.1.39); in the extreme relativistic case ( \(p_{F} \gg m c\) and \(\varepsilon=\) \(p c)\), we obtain

\[
\frac{C_{\mathrm{V}}}{N k}=\pi^{2}\left(\frac{k T}{\varepsilon_{F}}\right)
\]

consistent with expression (7) of the solution to Problem 8.13.