(a) Apply the theory of Section 13.2 to a one-dimensional lattice gas and show that the pressure \(P\) and the volume per particle \(v\) are given by

\[
\frac{P}{k T}=\ln \left[\frac{1}{2}\left\{(y+1)+\sqrt{ }\left[(y-1)^{2}+4 y \eta^{2}\right]\right\}\right]
\]

and

\[
\frac{1}{v}=\frac{1}{2}\left[1+\frac{y-1}{\sqrt{ }\left[(y-1)^{2}+4 y \eta^{2}\right]}\right]
\]

where \(y=z \exp (4 \beta J)\) and \(\eta=\exp (-2 \beta J), z\) being the fugacity of the gas. Examine the high and the low temperature limits of these expressions.

(b) A hard-core lattice gas pertains to the limit \(J \rightarrow-\infty\); this makes \(y \rightarrow 0\) and \(\eta \rightarrow \infty\) such that the quantity \(y \eta^{2}\), which is equal to \(z\), stays finite. Show that this leads to the equation of state

\[
\frac{P}{k T}=\ln \left(\frac{1-ho}{1-2 ho}\right) \quad\left(ho=\frac{1}{v}\right)
\]