Using the correspondence established in Section 12.4, apply the results of the preceding problem to the case of a lattice gas. Show, in particular, that the pressure, \(P\), and the volume per particle, \(v\), are given by

\[P=\mu \mu_{0} H-\frac{1}{8} q \varepsilon_{0}\left(1+\bar{L}^{2}\right)-\frac{1}{2} k T \ln \left(\frac{1-\bar{L}^{2}}{4}\right)\]

and

\[v^{-1}=\frac{1}{2}(1 \pm \bar{L})\]

Check that the critical constants of this system are \(T_{c}=q \varepsilon_{0} / 4 k, P_{c}=k T_{c}\left(\ln 2-\frac{1}{2}\right)\), and \(v_{c}=2\), so that the quantity \(k T_{c} / P_{c} v_{c}=1 /(\ln 4-1) \simeq 2.589\).