Gibbs fundamental equation Legendre transformation

Applying the concepts of Legendre transformation to thermodynamic potentials, we realize that Gibbs fundamental equation [see Eq. (1.22)] can be rewritten as... 

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

There are two ways to derive the Gibbs-Duhem equation for a system (1) Subtract the fundamental equation from the total differential of the thermodynamic potential. (2) Use a complete Legendre transform. As an example of the second method, eonsider fundamental equation 3.3-10 for a single reactant ... 

The last of Eqs. (51) defines a thermodynamic fundamental equation for G = G(N, P, 7) in the Gibbs energy representation. Note that passing from one ensemble to the other amounts to a Legendre transformation in macroscopic thermodynamics [39]. Vq is just an arbitrary volume used to keep the partition function dimensionless. Its choice is not important, as it just adds an arbitrary constant to the free energy. The NPT partition function can also be factorized into the ideal gas and excess contributions. The configurational integral in this case is ... 

In addition to the criteria of criticality in the U-represen-tation, Gibbs presented two alternative sets namely Eqs. (1) and (2) and Eqs. (9) and (10). These are two of many sets which we shall show derive from representations of the Fundamental Equation in alternate potential functions. We shall use the methodology of Legendre transformations to shift from one set of Independent variables to another. ... 

---