Statistical mechanics suggests that  can be expressed by a power series called a virial expansion,
The functions  are the virial coefficients; the th term represents a  particle interaction.
If we note in the expression for   that for , the term  can be expanded in an absolutely convergent series; this yields
This is the virial expansion for the van der Waals fluid. The first virial coefficient is the slope of  at . Notice that it can be positive or negative depending on whether or not , which agrees with the result found previously by differentiation.