Second virial coefficient

The second virial coefficient is usually written as B or as $B_{2}$. The second virial coefficient represents the initial departure from ideal-gas behavior. The second virial coefficient, in three dimensions, is given by

$B_{2}(T)=-{\frac {1}{2}}\int \left(\left\langle \exp \left(-{\frac {\Phi _{12}({\mathbf {r} })}{k_{B}T}}\right)\right\rangle -1\right)4\pi r^{2}dr$

where $\Phi _{12}({\mathbf {r} })$ is the intermolecular pair potential, T is the temperature and $k_{B}$ is the Boltzmann constant. Notice that the expression within the parenthesis of the integral is the Mayer f-function.

Isihara-Hadwiger formula

The second virial coefficient for any hard convex body is given by the exact relation

$B_{2}=RS+V$

or

${\frac {B_{2}}{V}}=1+3\alpha$

where

$\alpha ={\frac {RS}{3V}}$

where $V$ is the volume, $S$, the surface area, and $R$ the mean radius of curvature.

References

1. A. Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics 18 pp. 1446-1449 (1950)
2. 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)
3. 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)

Hard spheres

For hard spheres one has (McQuarrie, 1976, eq. 12-40)

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

leading to

$B_{2}={\frac {2\pi \sigma ^{3}}{3}}$

Note that $B_{2}$ for the hard sphere is independent of temperature.

See also

• Virial equation of state
• Osmotic virial coefficients
• Boyle temperature
• Joule-Thomson coefficient