Helmholtz free energy

The Helmholtz free energy $A$ or $F$ is a measure of the energy of a system. It is defined as

where $U$ is the internal energy, $T$ is the temperature and $S$ is the entropy. Helmholtz free energy is particularly useful in systems that a constant temperature.

Mechanically isolated system#

For a mechanically isolated system at constant temperature, $A$ never increases. In fact, from the second law of thermodynamics we get

(to pull $T$ out of the integral it must be constant). From the first we get

For a reversible transformation we get $W=-\Delta A$. But in a mechanically isolated system we have $W=0$, therefore

In other words, for a system of constant temperature, $A$ can only decrease. Thus the most stable state (i.e. lowest energy state) is the one where $A$ is lowest. The equilibrium is reached because the internal energy has a tendency to stabilize the system, whereas entropy tends to make it more disorderly and eventually balance each other out.