Work done to increase the temperature of one mole of an ideal gas by $30^{\circ} \mathrm{C}$, if it is…
Work done to increase the temperature of one mole of an ideal gas by $30^{\circ} \mathrm{C}$, if it is expanding under the condition $V \propto T^{2 / 3}$ is, $(R=8.314 \mathrm{~J} / \mathrm{mol} / \mathrm{K})$
$116.2 \mathrm{~J}$
$136.2 \mathrm{~J}$
$166.2 \mathrm{~J}$
$186.2 \mathrm{~J}$
Solution
We have, $V \propto T^{2 / 3}$
and
$p \propto \sqrt{V}$
$\therefore \quad p=k \sqrt{V} \Rightarrow p=k V^{1 / 2}$
Work done,
$W=\int p d V$
$\begin{aligned} W & =\int k V^{1 / 2} \\ W & =\frac{k V^{3 / 2}}{3 / 2} \\ W & =\frac{2}{3} \frac{p}{\sqrt{V}} V^{3 / 2}=\frac{2}{3} p V \quad\left[\because V=\frac{p}{\sqrt{V}}\right] \\ & =\frac{2}{3} n R T=\frac{2}{3} \times 1 \times 8.3 \times 30=166.2 \mathrm{~J}\end{aligned}$