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 Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara and the Swiss mathematician Hadwiger in 1950. 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)
4. H. Hadwiger "" Mh. Math. 54 pp. 345- (1950)
5. H. Hadwiger "" Experimentia 7 pp. 395- (1951)
6. H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)

Hard spheres

For the hard sphere model 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.

Excluded volume

The second virial coefficient can be computed from the expression

$B_{2}={\frac {1}{2}}\iint v_{\mathrm {excluded} }(\Omega ,\Omega ')f(\Omega )f(\Omega ')~{\mathrm {d} }\Omega {\mathrm {d} }\Omega '$

where $v_{\mathrm {excluded} }$ is the excluded volume.

See also

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