text: Figure 1 shows four isotherms of the van der Waals equation (abbreviated as vdW) on a pressure, molar volume plane. The essential character of these curves is that: 
  # at some critical temperature,  the slope is negative, , everywhere except at a single point, the critical point, , where both the slope and curvature are zero, ;
  # at higher temperatures the slope of the isotherms is everywhere negative (values of  for which the equation has 1 real root for );
  # at lower temperatures there are two points on each isotherm where the slope is zero (values of ,  for which the equation has 3 real roots for )
  Evaluating the two partial derivatives in 1) using the vdW equation and equating them to zero produces, , and using these in the equation gives .
  This calculation can be done algebraically by noting that the vdW equation can be written as a cubic in , which at the critical point is, 
  . Moreover, at the critical point all three roots coalesce so it can also be written as 
  Then dividing the first by , and noting that these two cubic equations are the same when all their coefficients are equal gives three equations 
   whose solution produces the previous results for .
  Using these critical values to define reduced properties  renders the equation in the dimensionless form used to construct Fig. 1
  This dimensionless form is a similarity relation; it indicates that all vdW fluids at the same  will plot on the same curve. It expresses the law of corresponding states which Boltzmann described as follows:
  This "law" is just a special case of dimensional analysis in which an equation containing 6 dimensional quantities, , and 3 dimensions, [p], [v], [T], must be expressible in terms of 6 − 3 = 3 dimensionless groups. Here  is a characteristic molar volume,  a characteristic pressure, and  a characteristic temperature, and the 3 dimensionless groups are . The reduced properties defined previously are , , and . Recent research has suggested that there is a family of equations of state that depend on an additional dimensionless group, and this provides a more exact correlation of properties. Nevertheless, as Boltzmann observed, the van der Waals equation provides an essentially correct description.
  The vdW equation produces , while for most real fluids . Thus most real fluids do not satisfy this condition, and consequently their behavior is only described qualitatively by the vdW equation. However, the vdW equation of state is a member of a family of state equations based on the Pitzer (acentric) factor, , and the liquid metals, Mercury and Cesium, are well approximated by it.