*v* and *T* as state variables

In some cases it is convenient to use molar quantities and mole fractions in the formulation of an EOS. In that case, a slightly different method can be used to calculate . Let us start with a slightly different form of Equation (53)

(64)

and make it molar by dividing by the total number of moles *n**T*

(65)

It is obvious that:

(66)

Here the molar Helmoltz free energy *a**(v,T,x)* is a function of *v, T, *and* x* instead of *V, T, *and* n. *Now instead of differentiating the left side of Equation (66), we use the right hand side and apply the product rule

(67)

Because *x* as well as *v **(**V* is the relevant thermodynamic variable and *v=V/n**T**)* are functions of *n* we have to apply the general chain rule for the differentiation:

(68)