An adiabatic vessel of total volume V is partitioned into two equal parts by a fixed, conducting separator. The left part holds one mole of an ideal gas ( U = 1.5 nRT ) and the right part contains two moles of the same gas. Initially, the pressure on each side is p . The system is allowed to rest for sufficient time until a steady state is achieved. Determine the heat given to the gas in the right part.

Step-by-step solution

Let the initial temperatures of the left and right parts be T₁ and T₂ respectively. Using the ideal gas equation pV = nRT : For the left part: p ( V 2 ) = 1 R T₁ T₁ = pV 2R For the right part: p ( V 2 ) = 2 R T₂ T₂ = pV 4R Since the vessel is adiabatic, the total internal energy of the system remains constant. Let the final equilibrium temperature be T_f . n₁ C_v T₁ + n₂ C_v T₂ = (n₁ + n₂) C_v T_f 1 pV 2R + 2 pV 4R = (1 + 2) T_f pV R = 3 T_f T_f = pV 3R The separator is fixed, so the volume of the right part is con