10 - Second-order partial derivatives

Abstract

A system having n extensive variables and hence f =n –1 degrees of freedom has a total of f + 1= n first order and a total of 1/2(f + 1)(f + 2) =n (n + 1)/2 distinct second-order partial derivatives of the internal energy. Because the order in which the differentiation is carried out does not affect the value of a “mixed” partial derivative, such derivatives are equal and are not counted separately in the stated total number. The equivalence of the mixed partial derivatives gives rise to a set of important relations known as the Maxwell relations. These important relations often allow the substitution of a derivative which is measured with relative ease in one experiment, for a derivative which is difficult to determine in another. Second-order partial derivatives may be derived from thermodynamic potentials other than the internal energy. This chapter distinguishes several sets of such derivatives.