One mole of an ideal gas at temperature $T_{1}$ expands according to the law $\frac{P}{V^{2}} = a (c o n s t a n t)$ . 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 $\frac{1}{3} R (T_{2} - T_{1})$
Given,
$\frac{P}{V^{2}} = a$
$∴$ the polytropic exponent $η = - 2$
Work done in a polytropic process is
$W = \frac{- n R Δ T}{η - 1}$ ...(1)
Substituting $Δ T = T_{2} - T_{1}$ and $η = - 2$ in (1)
$W = \frac{- R (T_{2} - T_{1})}{- 2 - 1} = \frac{1}{3} R (T_{2} - T_{1})$

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