# 1.2: Virial Equations

It is often useful to fit accurate pressure-volume-temperature data to polynomial equations. The experimental data can be used to compute a quantity called the compressibility factor, $$Z$$, which is defined as the pressure–volume product for the real gas divided by the pressure–volume product for an ideal gas at the same temperature.

We have

${\left(PV\right)}_{ideal\ gas}=nRT$

Letting P and V represent the pressure and volume of the real gas, and introducing the molar volume, $$\overline{V}={V}/{n}$$, we have

$Z=\frac{\left(PV\right)_{real\ gas}}{\left(PV\right)_{ideal\ gas}}=\frac{PV}{nRT}=\frac{P\overline{V}}{RT}$

Since $$Z=1$$ if the real gas behaves exactly like an ideal gas, experimental values of Z will tend toward unity under conditions in which the density of the real gas becomes low and its behavior approaches that of an ideal gas. At a given temperature, we can conveniently ensure that this condition is met by fitting the Z values to a polynomial in P or a polynomial in $${\overline{V}}^{-1}$$. The coefficients are functions of temperature. If the data are fit to a polynomial in the pressure, the equation is

$Z=1+B^*\left(T\right)P+C^*\left(T\right)P^2+D^*\left(T\right)P^3+\dots$

For a polynomial in $${\overline{V}}^{-1}$$, the equation is

$Z=1+\frac{B\left(T\right)}{\overline{V}}+\frac{C\left(T\right)}{\overline{V}^2}+\frac{D\left(T\right)}{\overline{V}^3}+\dots$

These empirical equations are called virial equations. As indicated, the parameters are functions of temperature. The values of $$B^*\left(T\right)$$, $$C^*\left(T\right)$$, $$D^*\left(T\right)$$, , and $$B\left(T\right)$$, $$C\left(T\right)$$, $$D\left(T\right)$$,, must be determined for each real gas at every temperature. (Note also that $$B^*\left(T\right)\neq B\left(T\right)$$, $$C^*\left(T\right)\neq C\left(T\right)$$, $$D^*\left(T\right)\neq D\left(T\right)$$, etc. However, it is true that $$B^*={B}/{RT}$$.) Values for these parameters are tabulated in various compilations of physical data. In these tabulations, $$B\left(T\right)$$ and $$C\left(T\right)$$ are called the second virial coefficient and third virial coefficient, respectively.