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An ideal gas undergoes a process in which $P V^{- a} =$ constant, where $V$ is the volume occupied by the gas initially at pressure $P$ . At the end of the process, rms speed of gas molecules has become $a^{1 / 2}$ times of its initial value. What will be the value of $C_{V}$ so that energy transferred by the heat to the gas is $' a'$ times of the initial energy $(Δ Q = a P V)$

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$\frac{(a^{2} + 1) R}{a^{2} - 1}$

$\frac{(a^{2} + 1) R}{(a^{2} + 1)}$

$\frac{(a + 1) R}{(a - 1)}$

$\frac{(a - 1) R}{(a + 1)}$

answer is A.

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Detailed Solution

$Δ Q = a P V; n c d T = a n R T$

$c d T = a R T \to (1)$

$∴$ polytropic process

$C = C_{V} + \frac{R}{1 - x}; C = C_{V} + \frac{R}{1 + a} \to (2)$

Sub (2) in (1)

Given $V_{r m s f} = a^{\frac{1}{2}} V_{r m s i}$ $∴ T_{f} = a T_{i}; d T = a T - T$

By solving above equations $C_{V} = \frac{(a^{2} + 1) R}{a^{2} - 1}$

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