# 1.2: Virial Equations

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- Contributed by Paul Ellgen
- Retired Teach (Chemistry) at Oklahoma School of Science Mathematics

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.