If we note in the expression for Z ( ρ ρ r , T r ) that for ρ ρ r < 3 , the term ( 3 − − ρ ρ r ) − − 1 can be expanded in an absolutely convergent series; this yields Z ( ρ ρ r , T r ) = 1 + [ 1 − − 27 / ( 8 T r ) ] − − 1 ] ( ρ ρ r / 3 ) + ∑ ∑ k = 2 ∞ ∞ ( ρ ρ r / 3 ) k This is the virial expansion for the van der Waals fluid. The first virial coefficient is the slope of Z ( ρ ρ r , T r ) at ρ ρ r = 0 . Notice that it can be positive or negative depending on whether or not T r > or < 27 / 8 , which agrees with the result found previously by differentiation.