Vega equation of state for hard ellipsoids

The Vega equation of state for an isotropic fluid of hard (biaxial) ellipsoids is given by (Ref. 1 Eq. 20):

$Z=1+B_{2}^{*}y+B_{3}^{*}y^{2}+B_{4}^{*}y^{3}+B_{5}^{*}y^{4}+{\frac {B_{2}}{4}}\left({\frac {1+y+y^{2}-y^{3}}{(1-y)^{3}}}-1-4y-10y^{2}-18.3648y^{3}-28.2245y^{4}\right)$

where $Z$ is the compressibility factor and $y$ is the volume fraction, given by $y=\rho V$ where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho } is the number density. The virial coefficients are given by the fits

$B_{3}^{*}=10+13.094756\alpha '-2.073909\tau '+4.096689\alpha '^{2}+2.325342\tau '^{2}-5.791266\alpha '\tau ',$

$B_{4}^{*}=18.3648+27.714434\alpha '-10.2046\tau '+11.142963\alpha '^{2}+8.634491\tau '^{2}-28.279451\alpha '\tau '-17.190946\alpha '^{2}\tau '+24.188979\alpha '\tau '^{2}+0.74674\alpha '^{3}-9.455150\tau '^{3},$

and

$B_{5}^{*}=28.2245+21.288105\alpha '+4.525788\tau '+36.032793\alpha '^{2}+59.0098\tau '^{2}-118.407497\alpha '\tau '+24.164622\alpha '^{2}\tau '+139.766174\alpha '\tau '^{2}-50.490244\alpha '^{3}-120.995139\tau '^{3}+12.624655\alpha '^{3}\tau ',$

where

$B_{n}^{*}=B_{n}/V^{n-1}$,

$\tau '={\frac {4\pi R^{2}}{S}}-1,$

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 \alpha' = \frac{RS}{3V}-1.}

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} is the volume, 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 S} , the surface area, 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 R} the mean radius of curvature.

For $B_{2}$ see B_2 for any hard convex body.

References

1. Carlos Vega "Virial coefficients and equation of state of hard ellipsoids", Molecular Physics 92 pp. 651-665 (1997)
2. Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria (2007)