A rigid, perfectly insulated cylinder is divided into two equal volumes by a frictionless, thermally conducting, massless piston that is initially locked in place. The left side contains n moles of an ideal monatomic gas at temperature Th and pressure Ph; the right side contains the same gas at temperature Tc<Th and pressure Pc>Ph such that the piston remains stationary while locked. At time t=0 the latch is released, allowing the piston to move and heat to flow through it until the system reaches a new equilibrium.
(a) Identify the conditions (pressure, temperature, and piston position) that define the final equilibrium state.
(b) Derive an expression for the total change in entropy of the gas in terms of n, the universal gas constant R, and the initial and final temperatures of each side.
(c) Show quantitatively that the total entropy change you obtained in part (b) is positive, and explain how this confirms the predicted spontaneous direction of evolution for the isolated cylinder-piston system.