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 is given by

$B_2(T)=-\frac{1}{2}\int (⟨\exp (-\frac{{\mathrm{\Phi }}_{12}(\mathbf{r})}{k_{B}T})⟩-1)4\pi r^{2}dr$

where ${\mathrm{\Phi }}_{12}(\mathbf{r})$ 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.

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 (McQuarrie, 1976, eq. 12-40)

$B_2(T)=-\frac{1}{2}\underset{0}{\overset{\sigma }{\int }}(⟨0⟩-1)4\pi r^{2}dr$

leading to

$B_2=\frac{2\pi {\sigma }^3}{3}$

Note that $B_2$ for the hard sphere is independent of temperature.

See also

• Boyle temperature