Thermodynamics Question 303
Question: A monoatomic ideal gas goes through a process $ p=p _{0}-\alpha V $ where $ p _{0} $ and $ \alpha $ are positive constants and V is its volume. At what volume will the entropy of gas be maximum?
Options:
A) $ \frac{5p _{0}}{6\alpha } $
B) $ \frac{p _{0}}{2\alpha } $
C) $ \frac{p _{0}}{4\alpha } $
D) $ \frac{5p _{0}}{8\alpha } $
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Answer:
Correct Answer: D
Solution:
[d] $ ds=nC _{v}dT+PdV=0 $
$ nR\frac{dT}{dV}+( p _{0}-\alpha V )=0 $
$ pV=nRT $
$ p _{0}V-\alpha V^{2}=nRT $
$ p _{0}-2\alpha V=nR\frac{dT}{dV} $
$ -( p _{0}-\alpha V )( \gamma -1 )=p _{0}-2\alpha V $
$ -p _{0}(\gamma -1)+\alpha (\gamma -1)V=p _{0}-2\alpha V $
$ p _{0}V=\alpha V(\gamma +1) $
$ V=\frac{p _{0}\gamma }{\alpha ( \gamma +1 )} $
$ V=\frac{p _{0}\times \frac{5}{3}}{\alpha ( \frac{5}{3}+1 )}=\frac{5p _{0}}{8\alpha } $
