The relation between the energy minimum and entropy maximum

[Gabriel Maia]

Hi. This is the problem 5.1-1 from the second edition of Callen's Thermodynamics. It says

Formulate a proof that the energy minimum principle implies the entropy maximum principle. That is, show that if the entropy were not maximum at constant energy then the energy could not be minimum at constant entropy.
HINT: First show that the permissible increase in entropy in the system can be exploited to extract heat from a reversible heat source (initially at the same temperature as the system) and to deposit it in a reversible work source. The reversible heat source is thereby cooled. Continue the argument.

Let's suppose that the system is at equilibrium and the energy is minimum but the entropy is not maximum. I increase the entropy of the system by extracting heat from a reversible heat source. Since the system and the heat source are at the same temperature there is no variation in the system's energy.

This heat is then deposited in a reversible work source. So the system does work on this source. How come the source is cooled and not the system?

I would think that work deposited on the reversible work source would keep the entropy the same but lower the energy of the system, and therefore, the energy would decrease even though it was supposed to be a minimum, concluding the argumentation that minimum energy goes hand-in-hand with maximum entropy. How come the source is cooled? Where is the flaw in my reasoning?

Thank you very much.

 

[member 553083]

Your reasoning is not flawed, but the conclusion is incorrect. The source is cooled because the heat extracted from the reversible heat source is used to do work on the reversible work source. This means that the energy in the system has been converted into useful work, and the entropy of the system has increased. Since the energy in the system is now lower (due to the work done on the work source), the system is at a lower energy state than before, thus proving that the energy minimum principle implies the entropy maximum principle.