Hard sphere: virial coefficients: Difference between revisions

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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 (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>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:
<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:



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

where is the volume of a sphere in three dimensions. For hard disks (ie. 2-dimensional hard spheres) one has[9]

where is the area of a circle.

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

See also

References

  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)

Related reading

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