text: The ideal gas law follows from the van der Waals equation whenever  is sufficiently large (or correspondingly whenever the molar density, , is sufficiently small), Specifically
  * when , then  is numerically indistinguishable from ,
  * and when , then  is numerically indistinguishable from .
  Putting these two approximations into the van der Waals equation when  is large enough that both inequalities are satisfied reduces it to 
  which is the ideal gas law. This is not surprising since the van der Waals equation was constructed from the ideal gas equation in order to obtain an equation valid beyond the limit of ideal gas behavior. 
  What is truly remarkable is the extent to which van der Waals succeeded. Indeed, Epstein in his classic thermodynamics textbook began his discussion of the van der Waals equation by writing: "In spite of its simplicity, it comprehends both the gaseous and the liquid state and brings out, in a most remarkable way, all the phenomena pertaining to the continuity of these two states". Also in Volume 5 of his Lectures on Theoretical Physics Sommerfeld, in addition to noting that "Boltzmann described van der Waals as the Newton of real gases", also wrote "It is very remarkable that the theory due to van der Waals is in a position to predict, at least qualitatively, the unstable [referring to superheated liquid, and subcooled vapor now called metastable] states" that are associated with the phase change process.