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
\[{\left(PV\right)}_{ideal\ gas}=nRT \nonumber \]
Letting P and V represent the pressure and volume of the real gas, and introducing the molar volume, \(\overline{V}={V}/{n}\), we have
\[Z=\frac{\left(PV\right)_{real\ gas}}{\left(PV\right)_{ideal\ gas}}=\frac{PV}{nRT}=\frac{P\overline{V}}{RT} \nonumber \]
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
\[Z=1+B^*\left(T\right)P+C^*\left(T\right)P^2+D^*\left(T\right)P^3+\dots \nonumber \]
For a polynomial in \({\overline{V}}^{-1}\), the equation is
\[Z=1+\frac{B\left(T\right)}{\overline{V}}+\frac{C\left(T\right)}{\overline{V}^2}+\frac{D\left(T\right)}{\overline{V}^3}+\dots \nonumber \]
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.
