Legendre transformation Maxwell relation

Legendre Transformations, Maxwell Relations, Linking Entropy and Probability, and Derivation of dS/dt... 

In 1982, Nalewajski and Parr took the thermodynamic analogy to its logical conclusion by extending the Legendre-transform structure of classical thermodynamics to DFT [8]. One of their results was the Maxwell relation for Equation 18.6,... 

The number of Maxwell equations for each of the possible thermodynamic potentials is given by D(D — l)/2, and the number of Maxwell equations for the thermodynamic potentials for a system related by Legendre transforms is [ )(D — 1)/2]2D. Examples are given in the following section. 

Standard thermodynamic formalism for the total differential of specific enthalpy in terms of its natural variables (i.e., via Legendre transformation, see equations 29-20 and 29-24b) allows one to calculate the pressure coefQdent of specific enthalpy via a Maxwell relation and the definition of the coefficient of thermal expansion, a. 

Since the definition (5.2.1) is a linear combination of thermodynamic properties, all the usual relations for extensive properties (see Chapter 3) can be expressed in terms of excess properties. Those relations include the Legendre transforms, the four forms of the fundamental equation, the response functions, and the Maxwell relations. Such relations reduce the amount of information needed to compute values for excess properties. 

Legendre transformations and given in Appendix 5. Together with the Maxwell relations, the correlations at the bottom of Fig. 2.19 are derived. 

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