Hard sphere: virial coefficients

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_2} by Johannes Diderik van der Waals (Ref. 1), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_3} by Ludwig Eduard Boltzmann (Ref. 2), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_4} by Johannis Jacobus van Laar (Ref. 3). The calculation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_5} had to wait for the Rosenbluths (Refs. 4) in 1954. Thus far no analytical expressions for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_5} and beyond have been derived. One has:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B_2}{V(\mathbb{R}^3)}=4}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B_3}{V(\mathbb{R}^3)^2}=10}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B_4}{V(\mathbb{R}^3)^3}= \frac{2707\pi+[438\sqrt{2}-4131 \arccos(1/3)]}{70\pi}= 18.3647684}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\mathbb{R}^3)} is the volume of a sphere in three dimensions. For hard disks (ie. 2-dimensional hard spheres) one has (Ref. 6)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B_2}{V(\mathbb{R}^2)}=2}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B_4}{V(\mathbb{R}^2)^3}= 16-\frac{36\sqrt{3}}{\pi}+\frac{80}{\pi^2}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\mathbb{R}^2)} is the area of a circle.

Virial / Dimension	2	3	4	5	6	7	8
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_3/B_2^2}	0.782004...	0.625	0.506340...	0.414063...	0.340941...	0.282227...	0.234614...
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. 7.

See also

• Equations of state for hard disks
• Equations of state for hard spheres

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 "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. 7 pp. 484- (1899)
3. 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)
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. 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)
6. 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)
7. 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)
8. 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)
9. 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)
10. André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics 129 044509 (2008)