The pressure is given by

\[
p=-\left(\frac{\partial A}{\partial V}ight)_{N, T}=n^{2}\left(\frac{\partial A / N}{\partial n}ight)_{T}
\]

and the excess pressure is given by

\(P^{\mathrm{ex}}=P_{c s}-P_{\text {ideal }}=n k T\left(\frac{1+\eta+\eta^{2}-\eta^{3}}{(1-\eta)^{3}}-1ight)=n k T \frac{4 \eta-2 \eta^{2}}{(1-\eta)^{3}}=n^{2}\left(\frac{\partial A^{\mathrm{ex}} / N}{\partial n}ight)_{T}\).

This can be integrated to give

\[
\frac{\beta A^{\mathrm{ex}}}{N}=\int_{0}^{\eta} \frac{4-2 \eta^{\prime}}{\left(1-\eta^{\prime}ight)^{2}} d \eta^{\prime}=\frac{3-2 \eta}{(1-\eta)^{2}}-3=\frac{4 \eta-3 \eta^{2}}{(1-\eta)^{2}}
\]