In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules.
The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions (i.e. all collisions are perfectly elastic). The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant:
To account for the volume occupied by real gas molecules, the Van der Waals equation replaces in the ideal gas law with , where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole, given by Avogadro constant times the volume of a single molecule:
The second modification made to the ideal gas law accounts for interaction between molecules of the gas. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form , where a is a constant whose value depends on the gas.
The complete Van der Waals equation is therefore:
For n moles of gas, it can also be written as:
When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT.
This equation approximates the behavior of real fluids above their critical temperatures and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. In particular, p is a constant function of V at given temperatures in these regions. As such, the Van der Waals model is not useful for calculations intended to predict real behavior in regions near critical points. Corrections to address these predictive deficiencies include the equal area rule and the principle of corresponding states.
The equation was named for its developer, the Dutch physicist Johannes Diderik van der Waals.
Overview and history
The Van der Waals equation is a thermodynamic equation of state based on the theory that fluids are composed of particles with non-zero volumes, and subject to a (not necessarily pairwise) inter-particle attractive force. It was based on work in theoretical physical chemistry performed in the late 19th century by Johannes Diderik van der Waals, who did related work on the attractive force that also bears his name. The equation is known to be based on a traditional set of derivations deriving from Van der Waals' and related efforts, as well as a set of derivation based in statistical thermodynamics, see below.
Van der Waals' early interests were primarily in the field of thermodynamics, where a first influence was Rudolf Clausius's published work on heat in 1857; other significant influences were the writings by James Clerk Maxwell, Ludwig Boltzmann, and Willard Gibbs. After initial pursuit of teaching credentials, Van der Waals' undergraduate coursework in mathematics and physics at the University of Leiden in the Netherlands led (with significant hurdles) to his acceptance for doctoral studies at Leiden under Pieter Rijke. While his dissertation helps to explain the experimental observation in 1869 by Irish professor of chemistry Thomas Andrews (Queen's University Belfast) of the existence of a critical point in fluids, science historian Martin J. Klein states that it is not clear whether Van der Waals was aware of Andrews' results when he began his doctorate work.
Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi-quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). It was in this dissertation that the first derivations of what we now refer to as the Van der Waals equation appeared. James Clerk Maxwell reviewed and lauded its published content in the British science journal Nature, and Van der Waals began independent work that would result in his receipt of the Nobel Prize in 1910, which emphasized the contribution of his formulation of this "equation of state for gases and liquids".
