A gas expands with temperature according to the relation $V=k{T}^{\frac{2}{3}}$ . What is the work done when the temperature changes by ${30}^{0}C$

$W={\displaystyle \int PdV={\displaystyle \int \frac{RT}{V}dV}}$

Since  $V=k{T}^{2/3}\Rightarrow dV=\frac{2}{3}K{T}^{-1/3}dT$

Eliminating $K$ , we find $\frac{dV}{V}=\frac{2}{3}\frac{dT}{T}$

Hence $W={\displaystyle {\int }_{T_1}^{T_2}\frac{2}{3}\frac{RT}{T}dT}=\frac{2}{3}R(T_2-T_1)=\frac{2}{3}R\left(30\right)=20R$