# Understanding the Gibbs Free energy principle

So, I have just derived that $$dG\leq0$$ for a closed system at constant temperature and volume. But, I am finding it hard to understand how this corresponds to the equilibrium state which has minimum Gibbs free energy. I mean, mathematically, it is a second derivative, so I am thinking like the change in Gibbs free energy can be either negative or zero, but how it can be minimum? It is simply the first derivative, for the function like $$G$$, for it to be minimum, both its first derivative should be zero and second derivative be greater than zero. How can I comprehend this relation regarding Gibbs free energy be negative?

1. The condition for $$dG \le 0$$ should be $$T = const$$ and $$P=const$$, instead of $$V=const$$.
2. $$dG\le0$$ describes the free energy change of a process, not a state. For a reversible process, $$dG=0$$, while for a irreversible process $$dG<0$$.
3. The second law of thermodynamics says: Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. Minimum $$G_{system}$$ corresponds to maximum entropy $$S_{system + environment}$$. In this sense, minimum $$G$$ at equilibrium is true by definition, instead of being a consequence of $$dG\le0$$.
4. As far as we know, there are no perpetual motion machines. So every system we know has a minimum energy $$G_{min}$$. We say states of system that has $$G_{min}$$ is in thermodynamic equilibrium. Since $$dG\le0$$, we know every system is heading towards equilibrium, however slowly it may go.