An incompressible liquid of specific volume \(v_{l}\), is in equilibrium with its own vapour and an inert gas in a closed vessel. The vapour obeys the law

\[p(v-b)=\Re T\]

Show that

\[\ln \left(\frac{p_{v}}{p_{0}}\right)=\frac{1}{\Re T}\left\{\left(p-p_{0}\right) v_{\ell}-\left(p_{v}-p_{0}\right) b\right\}\]

where \(p_{0}\) is the vapour pressure when no inert gas is present, and \(p\) is the total pressure.