Second virial coefficient

2019-12-03T08:33:29Z

129.241.228.167: /* Admur and Mason mixing rule */

The '''second virial coefficient''' is usually written as ''B'' or as <math>B_2</math>. The second [[Virial equation of state |virial coefficient]] represents the initial departure from [[ideal gas |ideal-gas]] behaviour.
The second virial coefficient, in three dimensions, is given by

:<math>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 </math>

where <math>\Phi_{12}({\mathbf r})</math> is the [[intermolecular pair potential]], ''T'' is the [[temperature]] and <math>k_B</math> 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: virial coefficients | hard sphere model]], thus one numerically evaluates

:<math>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 </math>

calculating

:<math> \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle</math>

for each <math>r</math> using the numerical integration scheme proposed by Harold Conroy <ref>[http://dx.doi.org/10.1063/1.1701795 Harold Conroy "Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals", Journal of Chemical Physics '''47''' pp. 5307 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01597437 I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics '''39''' pp. 65-79 (1989)]</ref>.
==Isihara-Hadwiger formula==
The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara
<ref>[http://dx.doi.org/10.1063/1.1747510 Akira Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics '''18''' pp. 1446-1449 (1950)]</ref>
<ref>[http://dx.doi.org/10.1143/JPSJ.6.40 Akira Isihara and Tsuyoshi Hayashida "Theory of High Polymer Solutions. I. Second Virial Coefficient for Rigid Ovaloids Model", Journal of the Physical Society of Japan '''6''' pp. 40-45 (1951)]</ref>
<ref>[http://dx.doi.org/10.1143/JPSJ.6.46 Akira Isihara and Tsuyoshi Hayashida "Theory of High Polymer Solutions. II. Special Forms of Second Osmotic Coefficient", Journal of the Physical Society of Japan '''6''' pp. 46-50 (1951)]</ref>
and the Swiss mathematician Hadwiger in 1950
<ref>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref>
<ref>[http://dx.doi.org/10.1007/BF02168922 H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia '''7''' pp. 395-398 (1951)]</ref>
<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>
The second virial coefficient for any hard convex body is given by the exact relation

:<math>B_2=RS+V</math>

or

:<math>\frac{B_2}{V}=1+3 \alpha</math>

where

:<math>\alpha = \frac{RS}{3V}</math>

where <math>V</math> is
the volume, <math>S</math>, the surface area, and <math>R</math> the mean radius of curvature.

==Hard spheres==
For the [[hard sphere model]] one has <ref>Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40</ref>

:<math>B_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr
</math>
leading to
:<math>B_{2}= \frac{2\pi\sigma^3}{3}</math>

Note that <math>B_{2}</math> 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:

:<math>B_{2}(T)= b -\frac{a}{RT} </math>

For the derivation [[Van der Waals equation of state#Virial form | click here]].
==Excluded volume==
The second virial coefficient can be computed from the expression

:<math>B_{2}= \frac{1}{2} \iint v_{\mathrm {excluded}} (\Omega,\Omega') f(\Omega) f(\Omega')~ {\mathrm d}\Omega {\mathrm d}\Omega'</math>

where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]].
==Admur and Mason mixing rule==
For the [[second virial coefficient]] of a mixture
<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref> (this reference does not say this)
:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>

==See also==
*[[Virial equation of state]]
*[[Osmotic virial coefficients]]
*[[Boyle temperature]]
*[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398- (1941)]
*[http://dx.doi.org/10.1063/1.481106 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)]
*[http://dx.doi.org/10.1080/00268976.2016.1263763 Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics '''115''' pp. 1191-1199 (2017)]
*[https://doi.org/10.1063/1.5004687 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)]

[[Category: Virial coefficients]]

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Second virial coefficient