Wikipedia
The van der Waals equation (or van der Waals equation of state) is an equation relating the density of gases and liquids ( fluids) to the pressure (p), volume (V), and temperature (T) conditions (i.e., it is a thermodynamic equation of state). It can be viewed as an adjustment to the ideal gas law that takes into account the non-zero volume of gas molecules, which are subject to an inter-particle attraction. It was derived in 1873 by Johannes Diderik van der Waals, who received the Nobel Prize in 1910 for "his work on the equation of state for gases and liquids."
It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in statistical thermodynamics, the latter of which provides the partition function of the system and allows thermodynamic functions to be specified. It successfully 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 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 behaviour, in particular that p is a constant function of V at given temperatures. As such, the van der Waals model is not useful only for calculations intended to predict real behavior in regions near the critical point. Empirical corrections to address these predictive deficiencies have been inserted into the van der Waals model, e.g., by James Clerk Maxwell in his equal area rule, and related but distinct theoretical models, e.g., based on the principle of corresponding states, have been developed to achieve better fits to real fluid behaviour in equations of comparable complexity.