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:

Z = PV/RT

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

Z = PV/RT = 1 + B/V + C/V^{2} + ...

Z = PV/RT = 1 + Bρ + Cρ^{2} + ...

Z = PV/RT = 1 + B'P + C'P^{2} + ...

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.