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

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

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

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

calculating

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

for each $r$ using the numerical integration scheme proposed by Harold Conroy .

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.

Hard spheres

For the hard sphere model 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. See also: Hard sphere: virial coefficients.

Van der Waals equation of state

For the Van der Waals equation of state one has:

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

For the derivation click here.

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

For the second virial coefficient of a mixture

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

See also

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

References

Related reading

• W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics 9 pp. 398- (1941)
• G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics 112 pp. 5364-5369 (2000)
• Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics 115 pp. 1191-1199 (2017)
• Elisabeth Herold, Robert Hellmann, and Joachim Wagner "Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry", Journal of Chemical Physics 147 204102 (2017)