# Thermodynamic Potentials (Non-Corresponding Bath)

I understand that thermodynamic potentials are another way of stating the 2nd law of thermodynamics (maximization of total entropy) for systems connected to a bath. For example, Helmholtz free energy,$$A(T, V, N)$$, is minimized when a system only exchanges entropy(heat) with a bath. Gibbs free energy $$G(T, P, N)$$ is minimized when a system exchanges both entropy(heat) and volume with a bath. Enthalpy $$H(S, P, N)$$ is minimized when a system exchanges only volume with a bath. However, I have instances where these thermodynamic potentials are applied to cases where the bath under consideration is not that of the thermodynamic potential. For example, if a system is connected to a heat bath where Helmholtz free energy is the relevant thermodynamic potential since we have constant (independent) variables T, V, and N. Does it make sense to talk about the other thermodynamic potentials in this case, are they even defined? would it make sense, for example, to talk about the change in enthalpy or Gibbs free energy for this system?

External conditions (whether the system is isolated or in any kind of equilibrium with an external reservoir) only affect the min-max principles that can be used to find the equilibrium state reached after the relaxation of internal constraints. For example, in an isolated system characterized by fixed pressure, temperature, and the number of particles, Gibbs free energy $$G$$:
2. it has a minimum, at fixed $$T,p,N,$$ with respect to variables describing internal constraints if they are allowed to vary.
These two facts are not in contradiction with the possibility of evaluating, for the same system, the Helmholtz free energy: $$A = G - pV,$$ where $$V = \left.\frac{\partial{G}}{\partial{p}}\right|_{T,N}$$. However, one should bear in mind that in such a case, it is meaningless to minimize $$A(T,V,N;X)$$ with respect to $$X$$, where $$X$$ is a coordinate expressing an internal constraint, to get the equilibrium condition at fixed $$T,p,N$$ because variations of the constraint at fixed $$T,V,N$$ imply a change of the value of $$p$$.