# Equilibrium, Thermodynamic Potentials and Phase Rule

In thermodynamics there are four thermodynamic potentials, for simplicity in the following I'll consider only the Gibbs free energy. In order to explain my doubt let me write two observations:

• It is possible to show that the Gibbs free energy, for a process that occurs at constant temperature $$T$$ and pressure $$P$$, is minimum at equilibrium.

• The phase rule tells that for a single component and phase system, the number of degrees of freedom is 2 ($$F=C-P+2$$, where $$F$$ is the number of degrees of freedom, $$C$$ is the number of components and $$P$$ is the number of phases).

Now, it seems to me that the condition that Gibbs free energy is minimum at constant $$P$$ and $$T$$ does not apply for single component and phase systems, because if $$T$$ and $$P$$ are given, then every thermodynamic quantity in the system is given, and there is no process at all. In other words, $$G=G(T,P)$$, so there is no point in saying that $$G$$ is minimum at equilibrium, because in this condition $$G$$ is a fixed quantity.

What is wrong with this reasoning? Please, if possible use an example to show how $$G$$ is minimized in these cases.

• There's nothing wrong with this reasoning. If you cut off all ways for a system to evolve, it makes no sense to talk about how the thermodynamic potentials change during evolution. This system remains static, and $G$ doesn't increase over time, which is consistent with minimization. (Of course, in this idealized example, $G$ doesn't decrease, either.) Jul 26, 2022 at 21:24
• Helmholtz Potential Minimum Principle: The equilibrium value of any unconstrained internal parameter in a system in diathermal contact with a heat reservoir minimizes the Helmholtz potential over the manifold of states for which $T=T^r$. Jul 27, 2022 at 10:05
• In this case it seems that only one degree of freedom need to be fixed, however i don't understand how to use this principle to find the equilibrium state. Can you give an example? Jul 27, 2022 at 10:07

The vapor/liquid transition is the classical example of a process at constant $$T$$ and $$P$$. The Gibbs energy is at a minimum whether we are dealing with one component or $$N$$ components.

In fact we use the minimization of $$G$$ to determine the saturation pressure: Of all possible tie lines that we can draw between the liquid branch of the isotherm and the vapor branch, the saturation pressure corresponds to the one that minimizes the Gibbs energy. The equivalent mathematical condition is $$\mu^L=\mu^V$$.

• Thanks for the answer, i think this example is useful. However, as i asked in the question i'd like, if possible, a single-phase, single-component example of free energy minimization. Aug 10, 2022 at 10:11