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| 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 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, in three dimensions, is given by | | The second virial coefficient, in three dimensions, is given by |
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| :<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> | | :<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> |
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| 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]]. |
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| 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
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| :<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>
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| calculating
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| :<math> \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle</math>
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| 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>.
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| ==Isihara-Hadwiger formula== | | ==Isihara-Hadwiger formula== |
| The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara | | The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara and the Swiss mathematician Hadwiger in 1950. |
| <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>
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| <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>
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| <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>
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| 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>
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| <ref>[http://dx.doi.org/10.1007/BF02168922 H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia '''7''' pp. 395-398 (1951)]</ref>
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| <ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>
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| 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 |
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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. |
| | ====References==== |
| | #[http://dx.doi.org/10.1063/1.1747510 A. Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics '''18''' pp. 1446-1449 (1950)] |
| | #[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)] |
| | #[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)] |
| | #H. Hadwiger "" Mh. Math. '''54''' pp. 345- (1950) |
| | #H. Hadwiger "" Experimentia '''7''' pp. 395- (1951) |
| | #H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955) |
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| ==Hard spheres== | | ==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> | | For hard spheres one has (McQuarrie, 1976, eq. 12-40) |
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| :<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> |
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| Note that <math>B_{2}</math> for the hard sphere is independent of [[temperature]]. See also: [[Hard sphere: virial coefficients]]. | | Note that <math>B_{2}</math> for the [[hard sphere model| hard sphere]] is independent of [[temperature]]. |
| ==Van der Waals equation of state==
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| For the [[Van der Waals equation of state]] one has:
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| :<math>B_{2}(T)= b -\frac{a}{RT} </math>
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| For the derivation [[Van der Waals equation of state#Virial form | click here]].
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| ==Excluded volume==
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| The second virial coefficient can be computed from the expression
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| :<math>B_{2}= \frac{1}{2} \iint v_{\mathrm {excluded}} (\Omega,\Omega') f(\Omega) f(\Omega')~ {\mathrm d}\Omega {\mathrm d}\Omega'</math>
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| where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]].
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| ==Admur and Mason mixing rule==
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| The [[second virial coefficient]] for a mixture of <math>n</math> components is given by (Eq. 11 in
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| <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>)
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| :<math>B_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j</math>
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| 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.
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| ==Unknown==
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| (<ref>I am not sure where this mixing rule was published</ref>)
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| :<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>
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| ==See also== | | ==See also== |
| *[[Virial equation of state]] | | *[[Virial equation of state]] |
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| *[[Boyle temperature]] | | *[[Boyle temperature]] |
| *[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]] | | *[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]] |
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| ==References== | | ==References== |
| <references/>
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| '''Related reading'''
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| *[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)]
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| *[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)]
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| *[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)]
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| *[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)]
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| [[Category: Virial coefficients]] | | [[Category: Virial coefficients]] |