One mole of an ideal gas at temperature $T_{1}$ expands according to the law $\frac{P}{V^{2}} = a (constant)$ . The work done by the gas till temperature of gas becomes $T_{2}$ is:

\frac{1}{2} R (T_{2} - T_{1})

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\frac{1}{3} R (T_{2} - T_{1})

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\frac{1}{4} R (T_{2} - T_{1})

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\frac{1}{5} R (T_{2} - T_{1})

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Solution

The correct option is B $\frac{1}{3} R (T_{2} - T_{1})$
Work done in polytropic process represented by $P V^{η} = constant$ is

$W = - \frac{n R Δ T}{η - 1}$

Given that,

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

Here, the polytropic exponent, $η = - 2$

Number of moles, $n = 1$

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

$\Rightarrow W = \frac{1}{3} R (T_{2} - T_{1})$

Hence, option $(B)$ is correct.

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