Hard sphere: virial coefficients

From SklogWiki
Jump to: navigation, 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 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)
\(B_{11}/B_2^{10}\) 0.000122 (4)
\(B_{12}/B_2^{11}\) 0.000027 (7)

This table is taken directly from Table 1 in Ref.[10]. The values of \(B_{11}\) and \(B_{12}\) for three dimensional hard spheres are taken from [11].

See also[edit]

References[edit]

  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)
  11. Richard J. Wheatley "Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres", Physical review Letters, 110 200601 (2013)]

Related reading

40px-Stop hand nuvola.svg.png 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.