The relevant results for \(TT_{c}\) follow from eqn. (11.2.10) by neglecting \(n_{0}\) altogether; we get, to the first order in \(a\),

\[
\begin{align*}
\frac{1}{N} A(N, V, T) & =\frac{1}{N} A_{i d}(N, V, T)+\frac{4 \pi a \hbar^{2}}{m \mathrm{v}},  \tag{13a}\\
P & =P_{i d}+\frac{4 \pi a \hbar^{2}}{m \mathrm{v}^{2}}  \tag{14a}\\
\mu & =\mu_{i d}+\frac{8 \pi a \hbar^{2}}{m \mathrm{v}} \tag{15a}
\end{align*}
\]

Remembering that \(\mathrm{v}_{\mathrm{c}} \propto \mathrm{T}^{-3 / 2}\), the various quantities of interest turn out to be

\[
\begin{aligned}
C_{\mathrm{V}} & =-T\left(\frac{\partial^{2} A}{\partial T^{2}}ight)_{N, \mathrm{~V}}=\left(C_{\mathrm{V}}ight)_{i d}+N \frac{2 \pi a \hbar^{2}}{m T}\left\{\begin{array}{cc}
0 & \left(T>T_{c}ight) \\
\left(-\frac{3}{2 \mathrm{v}_{c}}+\frac{6 \mathrm{v}}{\mathrm{v}_{c}^{2}}ight) & \left(T\end{array}ight. \\
K & =-\mathrm{v}\left(\frac{\partial P}{\partial \mathrm{v}}ight)_{T}=K_{i d}+\frac{2 \pi a \hbar^{2}}{m} \begin{cases}4 / \mathrm{v}^{2} & \left(T>T_{c}ight) \\
2 / \mathrm{v}^{2} & \left(T\left(\frac{\partial^{2} P}{\partial T^{2}}ight)_{\mathrm{v}} & =\left(\frac{\partial^{2} P}{\partial T^{2}}ight)_{\mathrm{v}, i d}+\frac{2 \pi a \hbar^{2}}{m T^{2}}\left\{\begin{array}{cl}
0 & \left(T>T_{c}ight) \\
6 / \mathrm{v}_{c}^{2} & \left(T\end{array}ight. \\
\left(\frac{\partial^{2} \mu}{\partial T^{2}}ight)_{\mathrm{v}} & =\left(\frac{\partial^{2} \mu}{\partial T^{2}}ight)_{\mathrm{v}, i d}+\frac{4 \pi a \hbar^{2}}{m T^{2}}\left\{\begin{array}{cl}
0 & \left(T>T_{c}ight) \\
3 / 4 \mathrm{v}_{c} & \left(T\end{array}ight.
\end{aligned}
\]

The thermodynamic relationship quoted in part (b) of the problem is readily verified.

As for the discontinuities at \(T=T_{c}\), we get (setting \(\mathrm{v}=\mathrm{v}_{c}\) )

\[
\begin{aligned}
\Delta C_{\mathrm{V}} & =N \frac{9 \pi a \hbar^{2}}{m T_{c}} \frac{1}{\mathrm{v}_{c}}=N k \frac{9 a \lambda_{c}^{2}}{2} \frac{\zeta(3 / 2)}{\lambda_{c}^{3}}=N k \frac{9 a}{2 \lambda_{c}} \zeta(3 / 2), \\
\Delta K & =-\frac{4 \pi a \hbar^{2}}{m} \frac{1}{\mathrm{v}_{c}^{2}} \\
\Delta\left(\frac{\partial^{2} P}{\partial T^{2}}ight)_{\mathrm{v}} & =\frac{12 \pi a \hbar^{2}}{m T_{c}^{2} \mathrm{v}_{c}^{2}}, \Delta\left(\frac{\partial^{2} \mu}{\partial T^{2}}ight)_{\mathrm{v}}=\frac{3 \pi a \hbar^{2}}{m T_{c}^{2} \mathrm{v}_{c}}
\end{aligned}
\]