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 for $B_{2}$ by Johannes Diderik van der Waals, $B_{3}$ by Jäger and Ludwig Eduard Boltzmann , and $B_{4}$ by Johannis Jacobus van Laar as well as Boltzmann . The calculation of $B_{5}$ had to wait for the Rosenbluths 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

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

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

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

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.. The values of $B_{11}$ and $B_{12}$ for three dimensional hard spheres are taken from .

See also

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

References

Related reading

• I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics 119 pp. 747-764 (2005)
• 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)
• I. Urrutia "The Fourth Virial Coefficient for Hard Spheres in Even Dimension", Journal of Statistical Physics 187 pp. 29-50 (2022)
• 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)
• 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)
• André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics 129 044509 (2008)
• Miguel Ángel G. Maestre, Andrés Santos, Miguel Robles, and Mariano López de Haro "On the relation between virial coefficients and the close-packing of hard disks and hard spheres", Journal of Chemical Physics 134 084502 (2011)
• Cheng Zhang and B. Montgomery Pettitt "Computation of high-order virial coefficients in high-dimensional hard-sphere fluids by Mayer sampling", Molecular Physics 112 pp. 1427-1447 (2014)