A cylindrical vessel fitted with a light piston contains an amount n (in moles) of a monatomic gas at an initial temperature T₀ . The surrounding air is at a temperature T_s ( > T₀ ) and the atmospheric pressure is p_a . Heat can be conducted between the gas and the surroundings through the bottom of the cylinder, which has a surface area A , thickness x , and thermal conductivity K . Assuming the process is slow, de

Step-by-step solution

Since the piston is light and the process is slow, the pressure of the gas remains constant at atmospheric pressure p_a . Thus, the process is isobaric. For a monatomic gas, the molar heat capacity at constant pressure is C_P = 5 2 R . The rate of heat conduction into the gas is given by: dQ dt = KA(T_s - T) x In an isobaric process, the heat absorbed is dQ = nC_P dT . Therefore: n ( 5 2 R ) dT dt = KA(T_s - T) x Rearranging the variables to integrate: dT T_s - T = 2KA 5nRx dt Integrating both sides from t = 0 (whe