# The maximum work theorem for different reversible processes

Citing the Callen thermodynamics book (page 105): "for all processes leading from the specified initial state to the specified final state of the primary system, the delivery of work is maximum for a reversible process. Furthermore the delivery of work is the same for all reversible processes"

Similarly the Hardy thermodynamics book state (page 152): "All reversible process between two specific state delivery the same amount of work"

I don't see how it can be true, cause I can think a number of two different reversible processes which have the same initial and final state but clearly delivery different work.

As an example consider an isothermal reversible process from state P1,V1,T1 to state P2,V2,T1 and another process composed of the reversible constant-volume process from state P1,V1,T1 to state P2,V1,T2 followed by the constant-pressure process from state P2,V1,T2 to state P2,V2,T1. The work delivered is the area underlying the curve, which is clearly different in the two processes above.

Looking at the proof of the theorem my thought to reconcile the example above with the text books is that the maximum work theorem applies to non-infinitesimal process only if the heat source is a reservoir, cause in this case the temperature is constant and it can be brought outside of the entropy integral. In the constant-volume + constant-pressure reversible process above the heat source can't be a reservoir cause the temperature of the system continuously changes during the process, and if two reservoirs (T1 and T2) were used then the process wouldn't be reversible cause it wouldn't be an equilibrium process.

• I think they are trying to compare same types of processes that is reversible isothermal process vs an irreversible isothermal process. The work done is greatest for the reversible case. – harshit54 Jan 3 at 11:51
• it doesn't seem it refers only to isothermal process, cause isothermal is not mentioned and in the proof it mentions heat capacity depending on temperature (page 106). – Andrea Jan 3 at 11:59
• Sorry, i don't have the book. And I was just giving an example. Two processes of the same kind are being compared. – harshit54 Jan 3 at 12:02
• I totally agree with your assessment. What they said can only be correct if a single constant temperature reservoir is being used. – Chet Miller Jan 3 at 13:34