Question

One mole of an ideal gas at temperature $T_1$ expands according to the law $\frac{P}{V^2}=a\text{(constant)}$. The work done by the gas till temperature of gas becomes $T_2$ is:

• A. $\frac{1}{2}R(T_2-T_1)$
• B. $\frac{1}{3}R(T_2-T_1)$
• C. $\frac{1}{4}R(T_2-T_1)$
• D. $\frac{1}{5}R(T_2-T_1)$

Answer 1

The correct option is B $\frac{1}{3}R(T_2-T_1)$

Work done in polytropic process represented by $P{V}^{\eta }=\text{constant}$ is

$W=-\frac{nR\Delta T}{\eta -1}$

Given that,

$\frac{P}{V^2}=\text{constant}\Rightarrow P{V}^{-2}=\text{constant}$

Here, the polytropic exponent, $\eta =-2$

Number of moles, $n=1$

$\Rightarrow W=-\frac{1R(T_2-T_1)}{-2-1}$

$\Rightarrow W=\frac{1}{3}R(T_2-T_1)$

Hence, option $\left(B\right)$ is correct.