Question

Find the amount of work done when the temperature of one mole of an ideal gas increases by ${30}^{\circ }C$, if it is expanding under the condition $V\propto T^{2/3}$.
(Take $R=8.314\text{J/K mol}$)

• A. $166.2\text{J}$
• B. $136.2\text{J}$
• C. $126.2\text{J}$
• D. None of these

Answer 1

The correct option is A $166.2\text{J}$

$PV=RT$ for $1\text{mole}$ of ideal gas.

Work done $W=\int PdV=\int \frac{RT}{V}dV$
Given, $V=C{T}^{2/3}$ where $C$ is a constant.

$\therefore dV=\frac{2}{3}C{T}^{-1/3}dT$
or $\frac{dV}{V}=\frac{2}{3}\frac{dT}{T}$

$\Rightarrow W={\int }_{T_1}^{T_2}RT\left(\frac{2}{3}\right)\frac{dT}{T}$
$=\frac{2}{3}R(T_2-T_1)$
$=\frac{2}{3}\times 8.314\times 30=166.2\text{J}$