ThermodynamicsHard
Question
A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volume Vo, in which an ideal gas is contained under the same pressure Po and at the same temperature. What work has to be performed in order to increase the volume of one part of gas isothermally $\eta$ times when compared to that of the other by slowly moving the piston?
Options
A.$P_{o}V_{o}\ln\eta$
B.$P_{o}V_{o}\ln\frac{(\eta + 1)^{2}}{4\eta}$
C.$P_{o}V_{o}\ln\frac{(\eta - 1)^{2}}{4\eta}$
D.$2P_{o}V_{o}\ln\eta$
Solution
Work performed on the piston
$= - \left\lbrack \int_{V_{o}}^{\eta V}P_{1}.dV + \int_{V_{0}}^{V}{P_{2}.dV} \right\rbrack\text{ and }(V + \eta.V) = 2V_{0} $$$= P_{0}V_{0}.\ln\frac{(\eta + 1)^{2}}{4\eta}$$
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