A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston s — Thermodynamics and Thermochemistry Chemistry Question
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 $\eta$ times compared to that of the other by slowly moving the piston?
💡 Solution & Explanation
Moving the piston slowly isothermally dictates work evaluated by calculating free energy variance or integration of pressure differential: $W = -\int (P_1 - P_2) dV_1 = -P_o V_o \ln(\frac{V(2V_o - V)}{V_o^2})$. Integrating out to limits yields $P_o V_o \ln \frac{(\eta+1)^2}{4\eta}$.