# Second virial coefficient

The **second virial coefficient** is usually written as *B* or as . The second virial coefficient represents the initial departure from ideal-gas behaviour.
The second virial coefficient, in three dimensions, is given by

where is the intermolecular pair potential, *T* is the temperature and is the Boltzmann constant. Notice that the expression within the parenthesis
of the integral is the Mayer f-function.

In practice the integral is often *very hard* to integrate analytically for anything other than, say, the hard sphere model, thus one numerically evaluates

calculating

for each using the numerical integration scheme proposed by Harold Conroy ^{[1]}^{[2]}.

## Isihara-Hadwiger formula[edit]

The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara
^{[3]}
^{[4]}
^{[5]}
and the Swiss mathematician Hadwiger in 1950
^{[6]}
^{[7]}
^{[8]}
The second virial coefficient for any hard convex body is given by the exact relation

or

where

where is the volume, , the surface area, and the mean radius of curvature.

## Hard spheres[edit]

For the hard sphere model one has ^{[9]}

leading to

Note that for the hard sphere is independent of temperature. See also: Hard sphere: virial coefficients.

## Van der Waals equation of state[edit]

For the Van der Waals equation of state one has:

For the derivation click here.

## Excluded volume[edit]

The second virial coefficient can be computed from the expression

where is the excluded volume.

## Admur and Mason mixing rule[edit]

The second virial coefficient for a mixture of components is given by (Eq. 11 in
^{[10]})

where and are the mole fractions of the th and th component gasses of the mixture.

## Unknown[edit]

(^{[11]})

## See also[edit]

## References[edit]

- ↑ Harold Conroy "Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals", Journal of Chemical Physics
**47**pp. 5307 (1967) - ↑ I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics
**39**pp. 65-79 (1989) - ↑ Akira Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics
**18**pp. 1446-1449 (1950) - ↑ Akira Isihara and Tsuyoshi Hayashida "Theory of High Polymer Solutions. I. Second Virial Coefficient for Rigid Ovaloids Model", Journal of the Physical Society of Japan
**6**pp. 40-45 (1951) - ↑ Akira Isihara and Tsuyoshi Hayashida "Theory of High Polymer Solutions. II. Special Forms of Second Osmotic Coefficient", Journal of the Physical Society of Japan
**6**pp. 46-50 (1951) - ↑ H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math.
**54**pp. 345- (1950) - ↑ H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia
**7**pp. 395-398 (1951) - ↑ H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)
- ↑ Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40
- ↑ I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids
**1**pp. 370-383 (1958) - ↑ I am not sure where this mixing rule was published

**Related reading**

- W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics
**9**pp. 398- (1941) - G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics
**112**pp. 5364-5369 (2000) - Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics
**115**pp. 1191-1199 (2017) - Elisabeth Herold, Robert Hellmann, and Joachim Wagner "Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry", Journal of Chemical Physics
**147**204102 (2017)