Question

One mole of an ideal gas expands with temperature according to the relation $V=k{T}^{\frac{2}{3}}$ Here V is volume of gas and T is its temperature .k is a constant. What is the work done by the gas when the temperature changes by ${30}^{\circ }C$ is

• A. 10 R
• B. 20 R
• C. 30 R
• D. 40 R

Answer 1

The correct option is B 20 R

$W=\int PdV=\int \frac{RT}{V}dV$
Since $V=k{T}^{\frac{2}{3}}\Rightarrow dV=\frac{2}{3}k{T}^{\frac{-1}{3}}dT$
Eliminating K, we find $\frac{dV}{V}=\frac{2}{3}\frac{dT}{T}$
Hence
$W={\int }_{T_1}^{T_2}\frac{2}{3}\frac{RT}{T}dT=\frac{2}{3}R(T_2-T_1)=\frac{2}{3}R\left(30\right)=20R$