Question

Find the amount of work done in joules to increase the temperature of one mole of an ideal gas by ${30}^{\circ }\text{C}$ if it is expanding under the condition $V\propto T^{2/3}$. Answer to the nearest integer.

Answer 1

$PV=RT$ for $1$ mole
$W=\int PdV=\int \frac{RT}{V}dV$
$V=C{T}^{2/3}$
$\therefore dV=\frac{2}{3}C{T}^{-1/3}dT$ or $\frac{dV}{V}=\frac{2}{3}\frac{dT}{T}$
$\therefore W=\underset{T_1}{\overset{T_2}{\int }}RT\left(\frac{2}{3}\right)\frac{dT}{T}=\frac{2}{3}R(T_2-T_1)=166.2\text{J}$