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 $\rho$ is the number density. The virial coefficients are given by the fits

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 B_3^* = 10 + 13.094756 \alpha' - 2.073909\tau' + 4.096689 \alpha'^2 + 2.325342\tau'^2 - 5.791266\alpha' \tau',}

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 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

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 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

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 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 the page "Second virial coefficient".

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)