The virial equation of state is a generalized equation of state that was initially proposed on a purely empirical basis in 1885. Further development of the virial equation was done in 1901 by Kamerlingh-Onnes. In 1927, H. D. Ursell re-developed the virial equation, but this time on a fundamental basis, starting from a statistical-mechanical analysis of intermolecular forces.

The basis of the virial equation is the definition of a compressibility factor Z, defined as:


The compressibility factor can be written in the form of power series:

Z = PV/RT = 1 + B/V + C/V2 + ...

Z = PV/RT = 1 + Bρ + Cρ2 + ...

Z = PV/RT = 1 + B'P + C'P2 + ...

which are respectively called the volume form, the density form and pressure form of the virial equation.

The coefficient B corresponds to interaction between pairs of molecules, C to triplets, and so on. The unprimed terms B, C,... are called the second, third, and so on virial coefficients. In theory, they are functions of temperature only for a given substance. It is clear that higher order molecular interactions usually play a less important role, and thus the virial equation is generally truncated to a low order.

Since the virial equation is a generalized form of an equation of state, it is the fundamental basis for many others. For instance, if we set all the higher order virial coefficients equal to zero (thus assuming no molecular interactions), then Z=1 and we yield the Ideal Gas Law.