### Virial Equations of State

The virial equations of state are power series expansions for
the compressibility factor in either specific volume:

$z =
\frac{Pv}{RT}=1+\frac{B}{v}+\frac{C}{v^2}+\frac{D}{v^3}+...$

or pressure:

$z = \frac{Pv}{RT}=1+B'P+C'P^2+D'P^3+...$

An interesting consequence of this form for an equation of state
is that the coefficients can be directly related to a first
principles (statistical mechanics and/or physical chemistry)
analysis of intermolecular potentials.

- $B$ is related to "two body" interactions
- $C$ is related to "three body" interactions
- ...

##### Note:

As pressure increases, multi-body interactions become more
important and more terms in the virial equation should be used.
Conversely, as pressure decreases, even two-body interactions
become negligible and the ideal gas law is recovered.

While these coefficients can be derived from first principles,
there are a number of correlations that are in common use. One
example includes the following (for the virial equation truncated
at two-body interaction):

$B=\frac{RT_c}{P_c}\left (B_0+\omega B_1\right)$

$B_0 = 0.083 - \frac{0.422}{T_r^{1.6}}$

$B_1 = 0.139 - \frac{0.172}{T_r^{4.2}}$

##### Outcome

Calculate P, v, or T from non-ideal equations of state (cubic
equations, the virial equation, compressibility charts, and
ThermoSolver)