# 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[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)

This table is taken directly from Table 1 in Ref.[10]

##  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. 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)