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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left\langle \exp \left(-{\frac {\Phi _{12}({\mathbf {r} })}{k_{B}T}}\right)\right\rangle }

for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_2=RS+V}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B_2}{V}=1+3 \alpha}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{RS}{3V}}

where $V$ is the volume, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , the surface area, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} the mean radius of curvature.

Hard spheres

For the hard sphere model one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr }

leading to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{2}= \frac{2\pi\sigma^3}{3}}

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{2}(T)= b -\frac{a}{RT} }

For the derivation click here.

Excluded volume

The second virial coefficient can be computed from the expression

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{2}= \frac{1}{2} \iint v_{\mathrm {excluded}} (\Omega,\Omega') f(\Omega) f(\Omega')~ {\mathrm d}\Omega {\mathrm d}\Omega'}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{\mathrm {excluded}}} is the excluded volume.

Admur and Mason mixing rule

For the second virial coefficient of a mixture

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)