A cylindrical tube of volume V having adiabatic walls contains an ideal gas. The internal energy of this ideal gas is given by 1.5 nRT . A fixed diathermic wall divides the tube into two equal parts. Initially, the pressure and temperature are p₁, T₁ on the left side and p₂, T₂ on the right side. The system is allowed sufficient time for the temperatures on both sides to become equal. Calculate the amount of heat tha

Step-by-step solution

Let the volume of each part be V₁ = V₂ = V 2 . The number of moles of gas in the left and right parts are respectively: n₁ = p₁ (V/2) R T₁ = p₁ V 2 R T₁ n₂ = p₂ (V/2) R T₂ = p₂ V 2 R T₂ The internal energy of the ideal gas is given as U = 1.5 nRT = 3 2 nRT . This implies the molar heat capacity at constant volume is C_v = 3 2 R . Since the outer walls are adiabatic and the total volume is fixed, the total internal energy of the system remains conserved. U₁ + U₂ = U_ 1f + U_ 2f n₁ C_v T₁ + n₂ C_v T₂ = (n₁ + n₂) C_v