Legendre transforms and free energies

The natural variables of the internal energy U are the entropy S, volume V, and total number of moles N of the system. In many situations, however, these variables are not convenient. 

We can easily arrive at a new function that has different natural variables by performing a Legendre transform. For example, to arrive at a new state properly that posesses the independent variables T and V, we define the Helmholtz free energy A as  

Inserting this relation into Eq. (1.3), generalized to open systems, we find 

From this equation, we see that for a system with the temperature, volume, and total number of moles held fixed (i.e., dT = 0, dY = 0, and dN = 0), a process is spontaneous if it decreases the Helmholtz free energy. 

In addition, at equilibrium where the equality holds, we find 

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