I am continuing to try to understand maximum work reversible processes (and a subset thereof -- Carnot cycles) better. I am here curious about the following system.
(1) Consider one mole of a gas (say, ideal, or van der Waals) which is expanded isothermally, at temperature $T_h$, from an initial volume $V_i$, to a final volume $V_f$. A thermal reservoir at temperature $T_c$, is available.
My question is about how I can know/prove that there exists a way to take the gas (the primary subsystem) reversibly with respect to a composite system consisting of the gas, the reservoir, a reversible work source (RWS), and some set of auxiliary systems (defined as undergoing no net change) from the initial state to the final state of the primary subsystem in this particular way (i.e. along the given isotherm).
I ask because in the context of the following conundrum given to us by Callen in his classic thermodynamics textbook:
(2) A system can be taken from state A to state B (where $S_B= S_A$) either (a) directly along the adiabat $S$ = constant, or (b) along the isochore AC and the isobar CB. The difference in the work done by the system is the area enclosed between the two paths in a P-V diagram. Does this contravene the statement that the work delivered to a reversible work source is the same for every reversible process?
The answer here -- I think -- is that there is no way to reversibly (with respect to the subsystem+surroundings --i.e. subsystem+whatever composite system you consider it as part of) take the primary subsystem to the final state as in b). Perhaps this answer is wrong though, and there is some separate reason that (2) above does not violate the so-called Maximum Work Theorem.
At any rate, the contrast between the two cases discussed above leads to my question: how can I know whether it is possible to take a (sub)system reversibly (again, with respect to some composite system containing the primary subsystem, a reversible heat source (the reservoir in the first example is a particular case), a reversible work source (RWS), and some set of auxiliary systems)? Is it always possible, or are there some changes of state for which this is not possible? If it is always possible, it would seem that my answer to the paradox in (2) is wrong, and I would appreciate any help in resolving it.