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 $B_{2}$ by Johannes Diderik van der Waals^[1], $B_{3}$ by Ludwig Eduard Boltzmann ^[2], and $B_{4}$ by Johannis Jacobus van Laar ^[3]. The calculation of $B_{5}$ had to wait for the Rosenbluths ^[4] 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^[5]

{\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.