For molecules that are non attracting hard spheres, a = 0 , the vdW virial expansion becomes simply Z = ( 1 − − b ρ ρ ) − − 1 = ∑ ∑ i = 0 ∞ ∞ ( b ρ ρ ) i , which illustrates the effect of the excluded volume alone. It was recognized early on that this was in error beginning with the term ( b ρ ρ ) 2 . Boltzmann calculated its correct value as ( 5 / 8 ) ( b ρ ρ ) 2 , and used the result to propose an enhanced version of the vdW equation ( p + a / v 2 ) ( v − − b / 3 ) = R T [ 1 + 2 b / ( 3 v ) + 7 b 2 / ( 24 v 2 ) ] . On expanding ( v − − b / 3 ) − − 1 , this produced the correct coefficients thru ( b / v ) 2 and also gave infinite pressure at b / 3 , which is approximately the close packing distance for hard spheres. This was one of the first of many equations of state proposed over the years that attempted to make quantitative improvements to the remarkably accurate explanations of real gas behavior produced by the vdW equation.