Real gases can be characterized by their difference from ideal. This is done by writing the mechanical equation of state in the form p v = Z R T where Z , called the compressibility factor, is usually expressed either as a function of pressure and temperature, or density and temperature, and in each case in the limit p or ρ ρ = 1 / v → → 0 , Z = 1 , the ideal gas value. In the second case p = Z ( ρ ρ , T ) ρ ρ R T . Thus for a van der Waals fluid the compressibility factor is Z = 1 / ( 1 − − b ρ ρ ) − − a ρ ρ / ( R T ) , or in terms of reduced variables Z = 3 3 − − ρ ρ r − − 9 ρ ρ r 8 T r where 0 ≤ ≤ ρ ρ r = 1 / v r ≤ ≤ 3 . At the critical point, T r = ρ ρ r = 1 , Z = Z c = 3 / 2 − − 9 / 8 = 3 / 8 .