One mole of an ideal gas at temperature $T_1$ expends according to the law $\frac{P}{{{V^2}}} =a$ (constant). The work done by the gas till temperature of gas becomes $T_2 $ is
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$\frac{P}{V^{2}}=a$
$\therefore$ the polytropic exponent $\eta=-2$
Work done in a polytropic process is
$W=\frac{-n R \Delta T}{\eta-1}$ $...(1)$
Substituting $\Delta T=T_{2}-T_{1}$ and $\eta=-2$ in $( 1)$
$W=\frac{-R\left(T_{2}-T_{1}\right)}{-2-1}=\frac{1}{3} R\left(T_{2}-T_{1}\right)$
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