An ideal gas with the adiabatic exponent $\gamma$goes through a process P = Po – αV, where Po and α

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An ideal gas with the adiabatic exponent $\gamma$goes through a process P = Po – αV, where Po and α are positive constants and V is the volume. At what volume will the gas entropy have the maximum value?

Accepted Answer

$dS = \frac{n.C_{V,m}.dT + P.dV}{T}$For maximum entropy, $\frac{dS}{dV} = 0$$	ext{Or, }n.C_{V,m}\frac{dT}{dV} + P = 0(1) $$$	ext{Now, }P = \frac{RT}{V} = P_{0} - \alpha V \Rightarrow \frac{dT}{dV} = \frac{1}{R}\left( P_{0} - 2\alpha V ight)(2)$$From (1) and (2),$1 	imes \frac{R}{\gamma - 1} 	imes \frac{1}{R}\left( P_{0} - 2\alpha V ight) + \left( P_{0} - \alpha V ight) = 0 $$$	herefore V = \frac{\gamma.P_{0}}{\alpha(\gamma + 1)}$$

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