# Second virial coefficient

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

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

where ${\displaystyle \Phi _{12}({\mathbf {r} })}$ is the intermolecular pair potential, T is the temperature and ${\displaystyle k_{B}}$ 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

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

calculating

${\displaystyle \left\langle \exp \left(-{\frac {\Phi _{12}({\mathbf {r} })}{k_{B}T}}\right)\right\rangle }$

for each ${\displaystyle r}$ using the numerical integration scheme proposed by Harold Conroy [1][2].

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

${\displaystyle B_{2}=RS+V}$

or

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

where

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

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

## Hard spheres

For the hard sphere model one has [9]

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

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

Note that ${\displaystyle B_{2}}$ for the hard sphere is independent of temperature. See also: Hard sphere: virial coefficients.

## Van der Waals equation of state

For the Van der Waals equation of state one has:

${\displaystyle B_{2}(T)=b-{\frac {a}{RT}}}$

## Excluded volume

The second virial coefficient can be computed from the expression

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

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

## Admur and Mason mixing rule

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

${\displaystyle B_{\mathrm {mix} }=\sum _{i=1}^{n}\sum _{j=1}^{n}B_{ij}x_{i}x_{j}}$

where ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$ are the mole fractions of the ${\displaystyle i}$th and ${\displaystyle j}$th component gasses of the mixture.

## Unknown

([11])

${\displaystyle B_{ij}={\frac {\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^{3}}{8}}}$