Question

Work done to increase the temperature of one mole of an ideal gas by $30^{\circ} C$, if it is expanding under the condition $V \propto T^{2 / 3}$ is, $(R=8.314 \,J / mol / K )$

• A. 116.2 J
• B. 136.2 J
• C. 166.2 J
• D. 186.2 J

Answer 1

Answer: (C)

We have, $V \propto T^{2 / 3}$
and $p \propto \sqrt{V}$
$\therefore p=k \sqrt{V} $
$\Rightarrow p=k V^{1 / 2}$
Work done, $W=\int p d V$
$W=\int k V^{1 / 2}$
$W=\frac{k V^{3 / 2}}{3 / 2}$
$W=\frac{2}{3} \frac{p}{\sqrt{V}} V^{3 / 2}=\frac{2}{3} p V \left[\because V=\frac{p}{\sqrt{V}}\right]$
$=\frac{2}{3} n R T=\frac{2}{3} \times 1 \times 8.3 \times 30=166.2 \,J$