Consider the PV diagram I've drawn at the end of this post. Consider the processes drawn thereon: (1-2-3) is an isobaric step followed by an isochoric step, while (1-3) is an adiabatic expansion (I couldn't figure out how to flip the arrow, but NB that I am not interested in discussing a cycle here and (1-3) is meant to be interpreted as going from 1 to 3).
My question is as follows. I recently learned (Callen Chapter 4.5) about the maximum work theorem, which gives the maximum work that can be extracted from a given system in taking it between two specified states, while there is access to some fixed "reversible heat source" (RHS) and fixed " reversible work source" (RWS) (on which the work is done). It was proven — and I agree with the proof naturally — that the maximum such work obtains from a reversible $\Delta S_{tot} = 0$ process with respect to the overall, composite system of three subsystems.
Now in this specific case, fix the RHS and RWS. The adiabatic process (1-3) is clearly reversible, since it is quasistatic and since there is no heat transfer. Thus, by the theorem, it must supply the maximum work in going between the two states. But the process (1-2-3) evidently supplies more work (area under the PV diagram), so where am I going wrong? Obviously, one of the hypotheses entering the maximum work theorem must be violated, but I can't imagine which one.
Edit: I believe the correct resolution is to note that, in fact, both lead to precisely the same work done by a RWS (as they must by the MWT); this fact is obscured in this particular case by the fact that the \emph{direct work} done in the given reversible path. That is, the difference in area between the two paths corresponds to different values of $W$ in the process, but we note again that $W \neq -W_{RHS}$ in general; instead, we have $Q + W = -Q_{RHS} – W_{RWS}$ and so $W_a \neq W_b$ does not at all contradict the possibility that $W_b + Q_b + Q_{RHS,b} = -W_{RWS,b} = – W_{RWS,a} = W_a + Q_a + Q_{RHS,a}.$
Best Answer
No, you are just mistaking what the theorem is saying. Along isentropic path 1->3 you have no entropy / heat transfer, and thus the maximum work is whatever you get. Along path 1->2->3 there would be a need for infinitely many heat pads of different temperatures, from which you absorbed some heat. That heat is being converted to work, and that is why the limit of maximum work along path 1->2->3 is a little greater than that of path 1->3 isentropically.