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

${\displaystyle {\frac {B_{2}}{V(\mathbb {R} ^{3})}}=4}$
${\displaystyle {\frac {B_{3}}{V(\mathbb {R} ^{3})^{2}}}=10}$
${\displaystyle {\frac {B_{4}}{V(\mathbb {R} ^{3})^{3}}}={\frac {2707\pi +[438{\sqrt {2}}-4131\arccos(1/3)]}{70\pi }}=18.3647684}$

where ${\displaystyle V(\mathbb {R} ^{3})}$ is the volume of a sphere in three dimensions. For hard disks (ie. 2-dimensional hard spheres) one has[9]

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

where ${\displaystyle V(\mathbb {R} ^{2})}$ is the area of a circle.

 Virial / Dimension 2 3 4 5 6 7 8 ${\displaystyle B_{3}/B_{2}^{2}}$ 0.782004... 0.625 0.506340... 0.414063... 0.340941... 0.282227... 0.234614... ${\displaystyle B_{4}/B_{2}^{3}}$ 0.53223180... 0.2869495... 0.15184606... 0.0759724807... 0.03336314... 0.00986494662... -0.00255768... ${\displaystyle B_{5}/B_{2}^{4}}$ 0.33355604(1) 0.110252(1) 0.0357041(17) 0.0129551(13) 0.0075231(11) 0.0070724(10) 0.00743092(93) ${\displaystyle B_{6}/B_{2}^{5}}$ 0.1988425(42) 0.03888198(91) 0.0077359(16) 0.0009815(14) -0.0017385(13) -0.0035121(11) -0.0045164(11) ${\displaystyle B_{7}/B_{2}^{6}}$ 0.1148728(43) 0.01302354(91) 0.0014303(19) 0.0004162(19) 0.0013066(18) 0.0025386(16) 0.0034149(15) ${\displaystyle B_{8}/B_{2}^{7}}$ 0.0649930(34) 0.0041832(11) 0.0002888(18) -0.0001120(20) -0.0008950(30) -0.0019937(28) -0.0028624(26) ${\displaystyle B_{9}/B_{2}^{8}}$ 0.0362193(35) 0.0013094(13) 0.0000441(22) 0.0000747(26) 0.0006673(45) 0.0016869(41) 0.0025969(38) ${\displaystyle B_{10}/B_{2}^{9}}$ 0.0199537(80) 0.0004035(15) 0.0000113(31) -0.0000492(48) -0.000525(16) -0.001514(14) -0.002511(13) ${\displaystyle B_{11}/B_{2}^{10}}$ 0.000122 (4) ${\displaystyle B_{12}/B_{2}^{11}}$ 0.000027 (7)

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