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 V o , in which an idea

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 V o , in which an ideal gas is contained under the same pressure P o and at the same temperature. What work has to be performed in order to increase isothermally the volume of one part of gas η times compared to that of the other by slowly moving the piston?

TestHub · Accepted Answer

Final volumes: V a = η V b and V a + V b = 2V o , so V b = (2V o )/(1+η), V a = (2η V o )/(1+η). Work done on the gases (both isothermal, nRT = P o V o for each part) = -P o V o [ln(V a )/(V o ) + ln(V b )/(V o )] = -P o V o ln(4η)/((1+η) 2 ) = P o V o ln((η+1) 2 )/(4η). Matches (b).

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