Gibbs Free Energy is Minimized at constant Pressure and Temperature

The Gibbs free energy is partitioned into a potential for each chemical species $i$, $\mu_i$ and the number of moles of $i$, $N_i$. Then, the Gibbs free energy, $G = \ensuremath{\sum_{i=1}^{C}} \mu_i N_i$ must be a minimum. Knowing that it is a minimum means its derivative is zero--and this gives us a bunch of useful equations that apply to states of equilibrium. The fact that it is a minimum also means that its second derivative is positive definite--this implies restrictions on the properties of stable materials (such as the bulk modulus and the heat capacity must be positive).