Question

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

• A. $32 R$
• B. $15 R$
• C. $20 R$
• D. $25 R$

Answer 1

Answer: (C)

Idea gas equation $P V=n R T$
For one mole of ideal gas $P V=R T$
$\Rightarrow P=\frac{R T}{V}$
Given that $V \propto T^{2 / 3}$ or $V=k T^{2 / 3}$ ...(i)
Differentiating equation (i),
$d V=k \frac{2}{3} T^{-1 / 3} d T$ ...(ii)
Work done $d W=P d V=\left(\frac{R T}{V}\right) \cdot k \frac{2}{3} T^{-1 / 3} d T$
$d W=\frac{R T}{k T^{2 / 3}} \cdot k \cdot \frac{2}{3} T^{-1 / 3} d T=\frac{2}{3} R d T$ ...(iii)
Integrating equation (iii)
$W=\int d W=\frac{2}{3} \int\limits_{T_{1}}^{T_{2}} d T=\frac{2}{3} R\left(T_{2}-T_{1}\right)=\frac{2}{3} R \Delta T$
Total work done,
$W=\frac{2}{3} \times R \times 30=20 \,R$