Editing Hard sphere: virial coefficients
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The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest. | The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest. | ||
In 3-dimensions analytical results were derived for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www. | In 3-dimensions analytical results were derived (all in 1899) for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=220&view=image&startrow=1 J. D. van der Waals "Simple deduction of the characteristic equation for substances with extended and composite molecules", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 138-143 (1899)]</ref>, <math>B_3</math> by [[Ludwig Eduard Boltzmann]] | ||
<ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]] | <ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]] | ||
<ref>[http://www. | <ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=203&view=image&startrow=1 J. J. Van Laar "Calculation of the second correction to the quantity b of the equation of condition of Van der Waals", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 273-287 (1899)]</ref>. The calculation of <math>B_5</math> had to wait for the Rosenbluths | ||
The calculation of <math>B_5</math> had to wait for the Rosenbluths | |||
<ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881- (1954)]</ref> in 1954. Thus far no analytical expressions for <math>B_5</math> and beyond have been derived. One has: | <ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881- (1954)]</ref> in 1954. Thus far no analytical expressions for <math>B_5</math> and beyond have been derived. One has: | ||
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| <math>B_{10}/B_2^9</math> || 0.0199537(80)|| 0.0004035(15)|| 0.0000113(31)|| -0.0000492(48) || -0.000525(16) || -0.001514(14) || -0.002511(13) | | <math>B_{10}/B_2^9</math> || 0.0199537(80)|| 0.0004035(15)|| 0.0000113(31)|| -0.0000492(48) || -0.000525(16) || -0.001514(14) || -0.002511(13) | ||
|} | |} | ||
This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref> | This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref> | ||
==See also== | ==See also== | ||
*[[Equations of state for hard disks]] | *[[Equations of state for hard disks]] | ||
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<references/> | <references/> | ||
'''Related reading''' | '''Related reading''' | ||
*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)] | *[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)] | ||
*[https://doi.org/10.1007/s10955-022-02913-7 I. Urrutia "The fourth virial coefficient for hard spheres in even dimension", Journal of Statistical Physics '''187''' pp. 29-50 (2022)] | *[https://doi.org/10.1007/s10955-022-02913-7 I. Urrutia "The fourth virial coefficient for hard spheres in even dimension", Journal of Statistical Physics '''187''' pp. 29-50 (2022)] | ||
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*[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)] | *[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)] | ||
*[http://dx.doi.org/10.1063/1.2958914 André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics '''129''' 044509 (2008)] | *[http://dx.doi.org/10.1063/1.2958914 André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics '''129''' 044509 (2008)] | ||
[[category:virial coefficients]] | [[category:virial coefficients]] | ||
[[category: hard sphere]] | [[category: hard sphere]] | ||
{{numeric}} | {{numeric}} |