Consider a (non-ideal) gas with free energy given by F = -a/N(V²T²), where V is the volume, T is the temperature, N is the number of particles in the gas, and a > 0 is a constant (with proper units).
(a) Determine the expression of the pressure P and entropy S of the gas as a function of V and T.
(b) Starting from a given state, with known T and V, the volume of the gas is doubled (i.e., V → V' = 2V) by following two distinct (quasistatic) processes, namely (1) an adiabatic and (2) an isobaric process. Find the ratio of the two final temperatures. At the end of which process is the temperature higher?
(c) Find the maximum work that can be obtained by connecting two containers of the same volume, V, filled with the same amount (particle number N) of the above gas but having different initial temperatures T₁ > T₂.