# Difference between revisions of "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 behaviour. The second virial coefficient, in three dimensions, is given by

$B_{2}(T)= - \frac{1}{2} \int \left( \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right) -1 \right) 4 \pi r^2 dr$

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.

In practice the integral is often very hard to integrate analytically for anything other than, say, the hard sphere model, thus one numerically evaluates

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

calculating

$\left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle$

for each $r$ 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

$B_2=RS+V$

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.

## Hard spheres

For the hard sphere model one has [9]

$B_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr$

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

Note that $B_{2}$ 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:

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

## 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.

## Admur and Mason mixing rule

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

$B_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j$

where $x_i$ and $x_j$ are the mole fractions of the $i$th and $j$th component gasses of the mixture.