## Maxwell Relations and other Useful Formulas

The fundamental differentials described in Section 1.10 are of the form of Eq (6.1). They provide the starting point for obtaining many useful thermodynamic relations. The fundamental differentials are represented as exact differentials of u(s,v), h(s,p), f(T,v), and g(T,p):

Note that each of the energy-like properties has a pair of "natural" variables associated with it. The variables in each pair are those that appear as differentials in the above equations. However, these associations are not immutable; it is possible, for example, to assume u to be a function of p and T and write the total differential du based on these variables instead of the "natural" pair s and v.

Equating the coefficients of the differentials in the second equalities of Eqs (6.9) -(6.12) produces eight relations:

A most useful foursome of thermodynamic relations is obtained by noting that the form of the fundamental differentials of Eqs (6.9) - (6.12) is the same as the total differential of Eq (6.1). Consisting solely of thermodynamic properties, the fundamental differentials are exact (in the mathematical sense defined earlier), and Eq (6.4) applies to them. For example, for the fundamental differential du = Tds -pdv, the correspondence with the general formula sets M = T, N = -p, x = s, and y = v. With these variables, Eq (6.4) gives:

The remaining fundamental differentials in Eqs (6.10) - (6.12) yield:

Equations (6.17) - (6.20) are collectively known as the Maxwell relations. In the following sections, these relations, together with the fundamental differentials and Eqs (6.13) - (6.16), will be utilized to derive a number of useful relationships between the thermodynamic properties of simple substances.

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