Second virial coefficient: Difference between revisions

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


:<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>  
:<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  
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]].
of the integral is the [[Mayer f-function]].
==For any hard convex body==
 
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
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>
:<math>\frac{B_2}{V}=1+3 \alpha</math>
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where <math>V</math> is
where <math>V</math> is
the volume, <math>S</math>, the surface area,  and <math>R</math> the mean radius of curvature.
the volume, <math>S</math>, the surface area,  and <math>R</math> the mean radius of curvature.
==Hard spheres==
==Hard spheres==
For hard spheres one has (McQuarrie, 1976, eq. 12-40)
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>B_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr  
</math>
</math>
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:<math>B_{2}=  \frac{2\pi\sigma^3}{3}</math>
:<math>B_{2}=  \frac{2\pi\sigma^3}{3}</math>


Note that <math>B_{2}</math> for the [[hard sphere model| hard sphere]] is independent of [[temperature]].
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==
The [[second virial coefficient]] for a mixture of <math>n</math> components is given by (Eq. 11 in
<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>)
:<math>B_{ {\mathrm {mix}} } =  \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j</math>
where <math>x_i</math> and <math>x_j</math> are the mole fractions of the <math>i</math>th and <math>j</math>th component gasses of the mixture.
==Unknown==
(<ref>I am not sure where this mixing rule was published</ref>)
:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>
==See also==
==See also==
*[[Virial equation of state]]
*[[Virial equation of state]]
*[[Osmotic virial coefficients]]
*[[Boyle temperature]]
*[[Boyle temperature]]
*[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]]
*[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]]
==References==
==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]]
[[Category: Virial coefficients]]

Latest revision as of 14:36, 10 December 2019

The second virial coefficient is usually written as B or as . The second virial coefficient represents the initial departure from ideal-gas behaviour. The second virial coefficient, in three dimensions, is given by

where is the intermolecular pair potential, T is the temperature and 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

calculating

for each 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

or

where

where is the volume, , the surface area, and the mean radius of curvature.

Hard spheres[edit]

For the hard sphere model one has [9]

leading to

Note that 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:

For the derivation click here.

Excluded volume[edit]

The second virial coefficient can be computed from the expression

where is the excluded volume.

Admur and Mason mixing rule[edit]

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

where and are the mole fractions of the th and th component gasses of the mixture.

Unknown[edit]

([11])

See also[edit]

References[edit]

Related reading