The idea of corresponding states originated when van der Waals cast his equation in the dimensionless form, . However, as Boltzmann noted, such a simple representation could not correctly describe all substances. Indeed, the saturation analysis of this form produces , namely all substances have the same dimensionless coexistence curve. In order to avoid this paradox an extended principle of corresponding states has been suggested in which  where  is a substance dependent dimensionless parameter related to the only physical feature associated with an individual substance, its critical point.
The first candidate for  was the critical compressibility factor , but because that quantity is difficult to measure accurately, the acentric factor developed by Kenneth Pitzer, , is more useful. The saturation pressure in this situation is represented by a one parameter family of curves, . Several investigators have produced correlations of saturation data for a number of substances, the best is that of Dong and Lienhard,
which has an rms error of  over the range  
Figure 3 is a plot of  vs . for various values of  as given by this equation. The ordinate is logarithmic in order to show the behavior at pressures far below the critical where differences among the various substances (indicated by varying values of ) are more pronounced.
Figure 4 is another plot of the same equation showing  as a function of  for various values of . It includes data from 51 substances including the vdW fluid over the range . This plot shows clearly that the vdW fluid () is a member of the class of real fluids; indeed it quantitatively  describes the behavior of the liquid metals cesium () and mercury () whose values of  are close to the vdW value. However, it describes the behavior of other fluids only qualitatively, because specific numerical values are modified by differing values of their Pitzer factor, .