An isolated system consists of two subsystems labelled 1 and 2 analysed in exercise 3.6.1. Using the second law (2.2), show that in a stationary state when T₁ > T₂ the entropy production rate ΠS of the whole system is positive when heat flows across these two subsystems despite the fact that, Π_S1 = Π_S2 = 0.

Data from exercise 3.6.1

An isolated system consisting of two closed subsystems A and B is separated by a diathermal wall. Initially, they are held at temperatures Tⁱ_A
and Tⁱ_B. Subsystem A contains N_A moles of gas. The internal energy of the gas is given by UA = c_AN_ART_A, where T_A is the temperature of the gas, R is a positive constant and c_A is a dimensionless coefficient. Likewise, there are NB moles of gas in subsystem B and the internal energy of the gas is given by U_B = c_BN_BRT_B.
a) Determine the change of the internal energy U_A due to the thermalisation process.
b) Compare the initial temperature Tⁱ_B and the final temperature T_f of the system if the size of subsystem B is much larger than that of subsystem A.