The relevant results of the preceding problem are

\[
\begin{align*}
& \frac{A}{N}=k T\left\{\frac{1+L^{*}}{2} \ln \frac{1+L^{*}}{2}+\frac{1-L^{*}}{2} \ln \frac{1-L^{*}}{2}\right\}-\frac{1}{2} q J L^{* 2}-\mu B L^{*}  \tag{1}\\
& \bar{N}_{+}=\frac{1}{2} N\left(1+L^{*}\right), \bar{N}_{-}=\frac{1}{2} N\left(1-L^{*}\right) \tag{2}
\end{align*}
\]

where \(L^{*}\) satisfies the maximization condition

\[
\begin{equation*}
\frac{1}{2}\left\{\ln \left(1+L^{*}\right)-\ln \left(1-L^{*}\right)\right\}=\beta\left(q J L^{*}+\mu B\right) \tag{3}
\end{equation*}
\]

Combining (1) and (3), we get

\[
\begin{equation*}
\frac{A}{N}=\frac{1}{2} k T \ln \frac{1-L^{* 2}}{4}+\frac{1}{2} q J L^{* 2} \tag{4}
\end{equation*}
\]

Now, using the correspondence given in Section 11.4 and remembering that \(L^{*}\) is identical with \(\bar{L}\), we obtain from eqns. (2) and (4) the desired results for the quantities \(P\) and \(\mathrm{v}\) pertaining to a lattice gas.

For the critical constants of the gas, we first note from eqn. (12.5.13) that \(T_{c}=q J / k\), i.e. \(q \varepsilon_{0} / 4 k\); the other constants then follow from the stated results for \(P\) and \(\mathrm{v}\), with \(B=0\) and \(\bar{L}=0\).