The virial equation of state
is a generalized equation of
that was initially proposed on a purely empirical
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
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
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
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.