Hard sphere: virial coefficients

From SklogWiki
Revision as of 11:28, 1 August 2008 by Carl McBride (talk | contribs) (tmp save)
Jump to navigation Jump to search

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 by Johannes Diderik van der Waals (Ref. 1), by Ludwig Eduard Boltzmann (Ref. 2), and by Johannis Jacobus van Laar (Ref. 3). The calculation of had to wait for the Rosenbluths (Refs. 4) in 1954. Thus far no analytical expressions for and beyond have been derived.

One has:


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. 6.

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. L. Boltzmann "", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. 7 pp. 484 (1899)
  3. J. J. Van Laar "", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. 1 pp. 273- (1899)
  4. Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics 22 pp. 881- (1954)
  5. 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)
  6. 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)
  7. Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics 128 034506 (2008)
  8. 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)
This page contains numerical values and/or equations. If you intend to use ANY of the numbers or equations found in SklogWiki in any way, you MUST take them from the original published article or book, and cite the relevant source accordingly.