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An ideal gas $(γ = \frac{C_{p}}{C_{v}})$ is taken through a process in which the pressure and the volume vary as P $= α V^{β} .$ If the molar specific heat in this process is $\frac{R}{β},$ then value of $γ$ or the gas will be $(R-$ universal gas constant)

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$1 + α β + α^{2}$

$1 + β + β^{2}$

$α + α^{2} + α β$

$1 + \frac{α}{β} - 2 α β$

answer is B.

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Detailed Solution

$For process {PV}^{x} = constant C = \frac{R}{γ - 1} + \frac{R}{1 - x} \frac{R}{β} = \frac{R}{γ - 1} + \frac{R}{1 + β} \Rightarrow γ = 1 + β + β^{2}$

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