Editing Second virial coefficient
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The second virial coefficient, in three dimensions, is given by | The second virial coefficient, in three dimensions, is given by | ||
:<math>B_{2}(T)= - \frac{1}{2} \int \left( | :<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> | ||
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]]. | ||
==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 | ||
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and the Swiss mathematician Hadwiger in 1950 | 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>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref> | ||
<ref> | <ref>H. Hadwiger "" Experimentia '''7''' pp. 395- (1951)</ref> | ||
<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</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 | ||
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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 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 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> | ||
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where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]]. | where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]]. | ||
==See also== | ==See also== | ||
*[[Virial equation of state]] | *[[Virial equation of state]] | ||
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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)] | *[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/10.1063/1.481106 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)] | ||
[[Category: Virial coefficients]] | [[Category: Virial coefficients]] |