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I have seen some material which tries to show that when approaching equilibrium upon different constraints, the changes in some thermodynamic potential is always positive or negative (for example, that Helmoltz free energy decreases for constant temperature and volume).

I cannot make sense of these arguments, however, because the quantities these thermodynamic potentials depend on are only defined at equilibrium. I think, that what one should show is that if there are several equilibria, the one which minimizes/maximizes the appropriate potential is the one which maximizes entropy. Is this the right way to think about thermodynamic potentials?

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The potentials are defined in equilibrium and their values are determined by the constraints of the problem. You remove some of the constraints and a spontaneous process is started that will end in another equilibrium after some time. The direction of the change of the potentials between these two equilibrium states is the concern in these questions. E.g., the Helmholtz energy decreases if the heat exchange occurs only with a fixed temperature reservoir, and since it decreases until it reaches equilibrium, in equilibrium it must be minimum. By the way, it is possible to define the potentials in non-equilibrium, as well.

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