$\ell n\left(\frac{p}{p_{0}}\right)=-a v$

$p=p_{0} e^{-a v}$

For temperature maximum p-v product should be maximum

${T}=\frac{{pv}}{{n} {R}}=\frac{{p}_{0} {ve}^{-{av}}}{{R}}$

$\frac{{dT}}{{d} {v}}=0 \Rightarrow \frac{{p}_{0}}{{R}}\left\{{e}^{-{av}}+{ve}^{-{dv}}(-{a})\right\}$

$\frac{{p}_{0} {e}^{-{dv}}}{{R}}\{1-{av}\}=0$

${v}=\frac{1}{{a}}, \infty$

${T}=\frac{{p}_{0} 1}{{Rae}}=\frac{{p}_{0}}{{Rae}}$

$\text { at } {v}=\infty$

${T}=0$