The virial expansion is a model of thermodynamic equations of state. It expresses the pressure of a gas in local equilibrium as a power series of the density. This equation may be represented in terms of the compressibility factor, , as
<math display="block">Z \equiv \frac{P}{RT\rho} = A + B\rho + C\rho^2 + \cdots</math>
This equation was first proposed by Kamerlingh Onnes. The terms , , and represent the virial coefficients. The leading coefficient is defined as the constant value of 1, which ensures that the equation reduces to the ideal gas expression as the gas density approaches zero.
Second and third virial coefficients
The second, , and third, , virial coefficients have been studied extensively and tabulated for many fluids for more than a century. Two of the most extensive compilations are in the books by Dymond and the National Institute of Standards and Technology's Thermo Data Engine Database and its Web Thermo Tables. Tables of second and third virial coefficients of many fluids are included in these compilations.
thumb|The 2nd and 3rd virial coefficients of argon
Casting equations of the state into virial form
Most equations of state can be reformulated and cast in virial equations to evaluate and compare their implicit second and third virial coefficients. The seminal van der Waals equation of state was proposed in 1873:
<math display="block">P = \frac{RT}{\left(v-b\right)} - \frac{a}{v^2}</math>
where is molar volume. It can be rearranged by expanding into a Taylor series:
<math display="block">Z = 1 + \left(b-\frac{a}{RT}\right)\rho + b^2\rho^2 + b^3\rho^3 + \cdots</math>
In the van der Waals equation, the second virial coefficient has roughly the correct behavior, as it decreases monotonically when the temperature is lowered. The third and higher virial coefficients are independent of temperature, and are not correct, especially at low temperatures.
Almost all subsequent equations of state are derived from the van der Waals equation, like those from Dieterici, Berthelot, Redlich-Kwong, and Peng-Robinson suffer from the singularity introduced by .
Other equations of state, started by Beattie and Bridgeman, are more closely related to virial equations, and show to be more accurate in representing behavior of fluids in both gaseous and liquid phases. The Beattie-Bridgeman equation of state, proposed in 1928,
<math display="block">p=\frac{RT}{v^2}\left(1-\frac{c}{vT^3}\right)(v+B)-\frac{A}{v^2}</math>
where
- <math>A = A_0 \left(1 - \frac{a}{v} \right)</math>
- <math>B = B_0 \left(1 - \frac{b}{v} \right)</math>
can be rearranged as
<math display="block">Z=1 + \left(B_0 -\frac{A_0}{RT} - \frac{c}{T^3}\right) \rho - \left(B_0 b-\frac{A_0 a}{RT} + \frac{B_0 c}{T^3}\right) \rho^2 + \left(\frac{B_0 b c}{T^3}\right) \rho^3 </math>The Benedict-Webb-Rubin equation of state of 1940 represents better isotherms below the critical temperature:
<math display="block">Z = 1 + \left(B_0 -\frac{A_0}{RT} - \frac{C_0}{RT^3}\right) \rho + \left(b-\frac{a}{RT}\right) \rho^2 + \left(\frac{\alpha a}{RT}\right) \rho^5 + \frac{c\rho^2}{RT^3}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)</math>
More improvements were achieved by Starling in 1972:
<math display="block">Z = 1 + \left(B_0 -\frac{A_0}{RT} - \frac{C_0}{RT^3} + \frac{D_0}{RT^4} - \frac{E_0}{RT^5}\right) \rho + \left(b-\frac{a}{RT}-\frac{d}{RT^2}\right) \rho^2 + \alpha\left(\frac{a}{RT}+\frac{d}{RT^2}\right) \rho^5 + \frac{c\rho^2}{RT^3}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)</math>
Following are plots of reduced second and third virial coefficients against reduced temperature according to Starling: