Second virial coefficient

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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].

Isihara-Hadwiger formula[edit]

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[edit]

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

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[edit]

For the Van der Waals equation of state one has:

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

For the derivation click here.

Excluded volume[edit]

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[edit]

For the second virial coefficient of a mixture [10]

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

See also[edit]

References[edit]

Related reading