Minimum Principles

The principle that $S$ is a maximum at constant energy allows us to prove the obvious, that $T$ and $P$ are uniform in systems that are in thermal and mechanical contact. This idea is not terribly useful until we consider two systems in mechanical and thermal contact, one so large that it hardly changes at all and the other being the system we are interested in studying. To see if our little system is in equilibrium, we consider all of its internal degrees of freedom. It turns out that if the internal degrees of freedom are the chemical compositions of all the phases in our little system, then what must be minimized is the sum of the partial Gibbs free energies each weighted by the amount of substance present.