text: In 1857 Rudolf Clasius published The Nature of the Motion which We Call Heat. In it he derived the relation  for the pressure, , in a gas with  particles per unit volume (number density), mass , and mean square speed . He then noted that using the classical laws of Boyle and Charles one could write  with  a constant of proportionality. Hence temperature was proportional to the average kinetic energy of the particles. This article inspired further work based on the twin ideas that substances are composed of indivisible particles, and that heat is a consequence of the particle motion as governed by Newton's laws. The work, known as the kinetic theory of gases, was done principally by James Clerk Maxwell, and Ludwig Boltzmann. At about the same time J. Willard Gibbs also contributed, and advanced it by converting it into statistical mechanics.
  This environment influenced Johannes Diderik van der Waals. After initially pursuing a teaching credential, he was accepted for doctoral studies at the University of Leiden under Pieter Rijke. This led, in 1873, to a dissertation that provided a simple, particle based, equation that described the gas–liquid change of state, the origin of a critical temperature, and the concept of corresponding states. The equation is based on two premises, first that fluids are composed of particles with non-zero volumes, and second that at a large enough distance each particle exerts an attractive force on all other particles in its vicinity. These forces were called by Boltzmann van der Waals cohesive forces.
  In 1869 Irish professor of chemistry Thomas Andrews at Queens University Belfast in a paper entitled On the Continuity of the Gaseous and Liquid States of Matter', displayed an experimentally obtained set of isotherms of carbonic acid, HCO, that showed at low temperatures a jump in density at a certain pressure, while at higher temperatures there was no abrupt change; the figure can be seen here. Andrews called the isotherm at which the jump just disappeared the critical point. Given the similarity of the titles of this paper and van der Waals subsequent thesis one might think that van der Waals set out to develop a theoretical explanation of Andrews experiments. However the opposite is true, van der Waals began work by trying to determine a mollecular attraction that appeared in Laplaces theory of capillarity, and only after establishing his equation tested it using Andrews results.
  By 1877 sprays of both liquid oxygen and liquid nitrogen had been produced, and a new field of research, low temperature physics, had been opened. The van der Waals equation played a part in all this especially with respect to the liquefaction of hydrogen and helium which was finally achieved in 1908. From measurements of  and  in two states with the same density, the van der Waals equation produces the values  and . Thus from two such measurements of pressure and temperature one could determine  and , and from these values calculate the expected critical pressure, temperature, and molar volume. Goodstein summarized this contribution of the van der Waals equation as follows:
  Van der Waals was awarded the Nobel Prize in 1910, in recognition of the contribution of his formulation of this "equation of state for gases and liquids".
  As noted previously, modern day studies of first order phase changes make use of the van der Waals equation together with the Gibbs criterion, equal chemical potential of each phase, as a model of the phenomenon. This model has an analytic coexistence (saturation) curve expressed parametrically, , that was first obtained by Plank,  known to Gibbs, and later derived in a beautifully simple and elegant manner by Lekner. A summary of Lekner's solution is presented in a subsequent section, and a more complete discussion in the Maxwell construction.