Virial Equations

\[Z= 1+\frac{B}{V} + \frac{C}{V^2} + \frac{D}{V^3} + \cdots \tag*{(6)}\]

  • For wider range of pressures, more terms in the power series (\(1/V\)) can be used.

  • The above equation is often used in the truncated form as \[Z = 1+\frac{B}{V} + \frac{C}{V^2} \tag*{(7)}\] where \(B\) and \(C\) are second third virial coefficients.

  • Another form: \[Z = 1 + B'P+C'P^2+D'P^3+\cdots \tag*{(8)}\] The coefficients of the terms in \(P\) or \(1/V\) are known as virial coefficients (To say strictly, virial coefficients refer to coefficients of \(1/V\) series only).

  • At first sight, these virial equation forms looks like nothing more than a polynomial form of the ideal gas law. However, it turns out to have real physical significance. Statistical thermodynamics shows that the second coefficient (\(B\)) arises from the interaction of two molecules, the third (\(C\)) from the interaction of three molecules at a time, and so on. They can be calculated from known interaction potentials, or used to estimate such potentials from observed \(PVT\) behavior.

  • All virial coefficients (\(B,C,D,\ldots\) and \(B',C',D',\ldots\)) are independent of pressure and density; for pure components, they are functions only of temperature. For mixtures, however, these coefficients depend on composition, as well.