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 compressibility factor, \( Z \), in terms of either the
density or the pressure. Such an expansion was first introduced in 1885 by Thiesen
\[ \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}\].
where
- \( p \) is the pressure
- \( V \) is the volume
- \( N \) is the number of molecules
- \(T\) is the temperature
- \(k_B\) is the Boltzmann constant
- \( \rho \equiv \frac{N}{V} \) is the (number) density
- \( B_k\left( T \right) \) is called the k-th virial coefficient
Contents |
[edit] Virial coefficients
The second virial coefficient represents the initial departure from ideal-gas behaviour
\[B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2\]
where \(N_A\) is Avogadros number and \(d\tau_1\) and \(d\tau_2\) are volume elements of two different molecules in configuration space.
One can write the third virial coefficient as
\[B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23} dr_1 dr_2 dr_3\]
where f is the Mayer f-function (see also: Cluster integrals).
See also
[edit] Convergence
For a commentary on the convergence of the virial equation of state see
[edit] Quantum virial coefficients
Using the path integral formulation one can also calculate the virial coefficients of quantum systems
[edit] References
- ↑ M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik 24 pp. 467-492 (1885)
- ↑ H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden 71 pp. 3-25 (1901)
- ↑ H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 4 pp. 125-147 (1902)
- ↑ H. D. Ursell "The evaluation of Gibbs' phase-integral for imperfect gases", Mathematical Proceedings of the Cambridge Philosophical Society 23 pp. 685-697 (1927)
- ↑ M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics 89 pp. 1005-1017 (1996)
- ↑ J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics 5 pp. 841-847 (1964)
- ↑ A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter 20 283102 (2008)
- ↑ Giovanni Garberoglio and Allan H. Harvey "Path-integral calculation of the third virial coefficient of quantum gases at low temperatures", Journal of Chemical Physics 134, 134106 (2011)
Related reading
- James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics 7 pp. 195-229 (1940)
- Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928
