Second virial coefficient

The second virial coefficient is usually written as B, or $B_{2}$. The second virial coefficient is given by

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

where $\Phi _{12}({\mathbf {r} })$ is the intermolecular pair potential. Notice that the expression within the parenthesis of the integral is the Mayer f-function.

For any hard convex body

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

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

where

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

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

Hard spheres

For hard spheres one has

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

leading to

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

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

References

McQuarrie, 1976, eq. 12-40