# Principle of Maximum Work for Different Paths

The principle of maximum work states that for any process between two states, the work done by the system is maximised for a reversible process (and heat transfer is minimised), and that the work done by any reversible process between these two states is equal.

I don't see how that can be compatible with going between 2 states with equal entropy by different paths. For example:

-An adiabatic expansion between 2 states with the same entropy.

-An isovolumetric pressurisation followed by an isobaric expansion with the same initial and final states as the adiabatic expansion.

The area below of the second path in a P-V diagram is clearly higher, so we have a reversible process between two states doing less work than another process between those states. How can this apparent contradiction be resolved?

• Your 2nd example is not an adiabatic process, and that the final states are the same as in the 1st example means just that and no more. The states maybe the same but both heat transfer and working are process dependent though their sum is the difference between the corresponding internal energies. Commented Jul 11, 2016 at 19:25
• Does this mean that I should take the principle of maximum work to mean "If a process between two states is not reversible, then a process that delivers more work exists, and that process is reversible"? That seems reasonable, but the statement from Herbert Callen's textbook "the delivery of work is identical for every reversible process" would contradict that. Commented Jul 11, 2016 at 19:42
• I assume you mean chapter 4-5 in Callen. There Callen constrains not only the internal energy change but also the entropy change and in that case the reversible process will always deliver more work than the irreversible one because in the latter the "sub-system" will also have to shed excess heat (due to internal irreversibilities, such as friction) thus the sum can only be the same if the work is less. Commented Jul 11, 2016 at 19:59
• But my second process has the same entropy change as the adiabatic expansion, as they have the same initial and final states. The example I gave is in fact problem 4.5-6 from that section, so if I've phrased it poorly it's also there. Commented Jul 11, 2016 at 20:06
• Ok, so it seems to be a homework problem. Hint: check Eq. 4-13, what is $T_{res}$ in your problem? Commented Jul 11, 2016 at 21:27