In thermodynamics, the van der Waals equation is an equation of state for a fluid in the liquid-vapor transition state in which particles begin to experience measurable attraction between each other, versus no noticeable attraction (Boltzmann chaos assumption), as in the ideal gas model equation of state (ideal gas law). The van der Waals equation is defined as follows:

$\left(p + \frac{a}{v^2}\right)\left(v-b\right) = RT$

where a is the 'degree of association factor', b the particle volume factor, p the pressure, v the volume, R the ideal gas constant, and T the the temperature of the body. The equation was proposed in 1873 by Dutch physicist Johannes van der Waals, who would go on to win the 1910 Nobel Prize for this work. [1]

Derivation
In 1864, German physicist Rudolf Clausius began to write the ideal gas law as:

$~ pv = RT ~$

where p is the pressure, v the volume, and T the absolute temperature of the body of non-interactive gas particles. In systems where either the pressure is increased, volume decreased, or temperature decreased, the gas can be turned into a liquid state, and the above equation becomes inapplicable. The system will be become a non-dilute aggregate of moving particles, interacting through inter-particle forces, and will fail to comply with Boyle’s law. These two effects, according to Van der Waals, can attributed to attraction between the particles, signified by a new constant “a”, and particle volume effects, signified by a new constant “b”, and thus the new vapor-approaching-liquid state of the body of particles can thus be quantified by the following equation: [2]

$p = \frac{RT}{v-b} - \frac{a}{v^2} \,$

in its original form, or as it came to be written in later years:

$\left(p + \frac{a}{v^2}\right)\left(v-b\right) = RT$

With the introduction of the Avogadro constant N, being the number of particles in a mol of substance, a term coined in 1909 by French chemist Jean Perrin, the van der Waals equation became reformulated as:

$\left(p + \frac{n^2 a}{V^2}\right)\left(V-nb\right) = nRT$

where n²a/V² is the "internal pressure" representing the attraction exerted between the molecules at short distances by the so-called "Van der Waals forces", and nb is the "covolume" representing the in compressible volume occupied by the gaseous molecules. [1]