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

Answer 1

Answer: (C)

Idea gas equation $PV=nRT$
For one mole of ideal gas $PV=RT$
$\Rightarrow P=\frac{RT}{V}$
Given that $V\propto T^{2/3}$ or $V=k{T}^{2/3}$ ...(i)
Differentiating equation (i),
$dV=k\frac{2}{3}{T}^{-1/3}dT$ ...(ii)
Work done $dW=PdV=\left(\frac{RT}{V}\right)\cdot k\frac{2}{3}{T}^{-1/3}dT$
$dW=\frac{RT}{k{T}^{2/3}}\cdot k\cdot \frac{2}{3}{T}^{-1/3}dT=\frac{2}{3}RdT$ ...(iii)
Integrating equation (iii)
$W=\int dW=\frac{2}{3}\underset{T_1}{\overset{T_2}{\int }}dT=\frac{2}{3}R\left(T_2-T_1\right)=\frac{2}{3}R\Delta T$
Total work done,
$W=\frac{2}{3}\times R\times 30=20R$