Hard sphere: virial coefficients: Difference between revisions

Revision as of 16:20, 27 May 2010

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 $B_{2}$ by Johannes Diderik van der Waals^[1], $B_{3}$ by Jäger ^[2] and Ludwig Eduard Boltzmann ^[3] ^[4], and $B_{4}$ by Johannis Jacobus van Laar ^[5] as well as Boltzmann ^[6] ^[7]. The calculation of $B_{5}$ had to wait for the Rosenbluths ^[8] in 1954. Thus far no analytical expressions for $B_{5}$ and beyond have been derived. One has:

{\frac {B_{2}}{V(\mathbb {R} ^{3})}}=4

{\frac {B_{3}}{V(\mathbb {R} ^{3})^{2}}}=10

{\frac {B_{4}}{V(\mathbb {R} ^{3})^{3}}}={\frac {2707\pi +[438{\sqrt {2}}-4131\arccos(1/3)]}{70\pi }}=18.3647684

where $V(\mathbb {R} ^{3})$ is the volume of a sphere in three dimensions. For hard disks (ie. 2-dimensional hard spheres) one has^[9]

{\frac {B_{2}}{V(\mathbb {R} ^{2})}}=2

{\frac {B_{3}}{V(\mathbb {R} ^{2})^{2}}}={\frac {16}{3}}-{\frac {4{\sqrt {3}}}{\pi }}

{\frac {B_{4}}{V(\mathbb {R} ^{2})^{3}}}=16-{\frac {36{\sqrt {3}}}{\pi }}+{\frac {80}{\pi ^{2}}}

where $V(\mathbb {R} ^{2})$ is the area of a circle.

Virial / Dimension	2	3	4	5	6	7	8
$B_{3}/B_{2}^{2}$	0.782004...	0.625	0.506340...	0.414063...	0.340941...	0.282227...	0.234614...
$B_{4}/B_{2}^{3}$	0.53223180...	0.2869495...	0.15184606...	0.0759724807...	0.03336314...	0.00986494662...	-0.00255768...
$B_{5}/B_{2}^{4}$	0.33355604(1)	0.110252(1)	0.0357041(17)	0.0129551(13)	0.0075231(11)	0.0070724(10)	0.00743092(93)
$B_{6}/B_{2}^{5}$	0.1988425(42)	0.03888198(91)	0.0077359(16)	0.0009815(14)	-0.0017385(13)	-0.0035121(11)	-0.0045164(11)
$B_{7}/B_{2}^{6}$	0.1148728(43)	0.01302354(91)	0.0014303(19)	0.0004162(19)	0.0013066(18)	0.0025386(16)	0.0034149(15)
$B_{8}/B_{2}^{7}$	0.0649930(34)	0.0041832(11)	0.0002888(18)	-0.0001120(20)	-0.0008950(30)	-0.0019937(28)	-0.0028624(26)
$B_{9}/B_{2}^{8}$	0.0362193(35)	0.0013094(13)	0.0000441(22)	0.0000747(26)	0.0006673(45)	0.0016869(41)	0.0025969(38)
$B_{10}/B_{2}^{9}$	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.^[10]

References

↑ 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)
↑ G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) 105 pp. 15- (1896)
↑ L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) 105 pp. 695- (1896)
↑ L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. 7 pp. 484- (1899)
↑ 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)
↑ John H. Nairn and John E. Kilpatrick "van der Waals, Boltzmann, and the Fourth Virial Coefficient of Hard Spheres", American Journal of Physics 40 pp. 503-515 (1972)
↑ B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review 85 pp. 777-783 (1952)
↑ Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics 22 pp. 881- (1954)
↑ Stanislav Labík, Jirí Kolafa, and Anatol Malijevský, "Virial coefficients of hard spheres and hard disks up to the ninth", Physical Review E 71 pp. 021105 (2005)
↑ 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)

Related reading

[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)

[2] G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) 105 pp. 15- (1896)

[3] L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) 105 pp. 695- (1896)

[4] L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. 7 pp. 484- (1899)

[5] 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)

[6] John H. Nairn and John E. Kilpatrick "van der Waals, Boltzmann, and the Fourth Virial Coefficient of Hard Spheres", American Journal of Physics 40 pp. 503-515 (1972)

[7] B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review 85 pp. 777-783 (1952)

[8] Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics 22 pp. 881- (1954)

[9] Stanislav Labík, Jirí Kolafa, and Anatol Malijevský, "Virial coefficients of hard spheres and hard disks up to the ninth", Physical Review E 71 pp. 021105 (2005)

[10] 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)

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@@ Line 1: / Line 1: @@
 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 (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]]
+In 3-dimensions analytical results were  derived 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 Jäger <ref>G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 15- (1896)</ref>
+and [[Ludwig Eduard Boltzmann]] <ref>L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a)  '''105''' pp. 695- (1896)</ref>
 <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.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
+<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>
+as well as Boltzmann <ref>[http://dx.doi.org/10.1119/1.1986605 John H. Nairn and John E. Kilpatrick "van der Waals, Boltzmann, and the Fourth Virial Coefficient of Hard Spheres", American Journal of Physics '''40''' pp. 503-515 (1972)]</ref>
+<ref>[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review  '''85''' pp. 777-783 (1952)]</ref>.
+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:

Hard sphere: virial coefficients: Difference between revisions

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Hard sphere: virial coefficients: Difference between revisions

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