Virial equation of state

The virial equation of state is used to describe the behavior of diluted gases. It is usually written as an expansion of the compresiblity factor, $Z$, in terms of either the density or the pressure. Such an expansion was first introduced by Kammerlingh Onnes. In the first case:

${\frac {pV}{Nk_{B}T}}=Z=1+\sum _{k=2}^{\infty }B_{k}(T)\rho ^{k-1}$.

where

• $p$ is the pressure

• $V$ is the volume

• $N$ is the number of molecules

• $\rho \equiv {\frac {N}{V}}$ is the (number) density

• $B_{k}\left(T\right)$ is called the k-th virial coefficient

Virial coefficients

The second virial coefficient represents the initial departure from ideal-gas behavior

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B_{2}(T)={\frac {N_{0}}{2V}}\int ....\int (1-e^{-u/kT})~d\tau _{1}d\tau _{2}}

where $N_{0}$ is Avogadros number and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\tau_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\tau_2} are volume elements of two different molecules in configuration space. The integration is to be performed over all available phase-space; that is, over the volume of the containing vessel. For the special case where the molecules posses spherical symmetry, so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} depends not on orientation, but only on the separation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} of a pair of molecules, the equation can be simplified to

$B_{2}(T)=-{\frac {1}{2}}\int _{0}^{\infty }\left(\langle \exp \left(-{\frac {u(r)}{k_{B}T}}\right)\rangle -1\right)4\pi r^{2}dr$

Using the Mayer f-function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ij}=f(r_{ij})= \exp\left(-\frac{u(r)}{k_BT}\right) -1 }

one can write the third virial coefficient more compactly as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23} dr_1 dr_2 dr_3 }

References

1. H. Kammerlingh Onnes "", Communications from the Physical Laboratory Leiden 71 (1901)
2. James A Beattie and Walter H Stockmayer "Equations of state",Reports on Progress in Physics 7 pp. 195-229 (1940)