Carnahan-Starling equation of state

The Carnahan-Starling equation of state is an approximate (but quite good) equation of state for the fluid phase of the hard sphere model in three dimensions. (Eqn. 10 in Ref 1).

$Z=\frac{pV}{N{k}_{B}T}=\frac{1+\eta +{\eta }^2-{\eta }^3}{(1-\eta {)}^3}.$

where:

• $p$ is the pressure

• $V$ is the volume

• $N$ is the number of particles

• $k_B$ is the Boltzmann constant

• $T$ is the absolute temperature

• $\eta$ is the packing fraction:

$\eta =\frac{\pi }{6}\frac{N{\sigma }^3}{V}$

• $\sigma$ is the hard sphere diameter.

Thermodynamic expressions

From the Carnahan-Starling equation for the fluid phase the following thermodynamic expressions can be derived (Eq. 2.6, 2.7 and 2.8 in Ref. 2)

Pressure (compressibility):

$\frac{\beta p^{CS}}{\rho }=\frac{1+\eta +{\eta }^2-{\eta }^3}{(1-\eta {)}^3}$

Configurational chemical potential:

$\beta {\overline{\mu }}^{CS}=\frac{8\eta -9{\eta }^2+3{\eta }^3}{(1-\eta {)}^3}$

Isothermal compressibility:

${\chi }_T-1=\frac{1}{kT}{\frac{\partial P^{CS}}{\partial \rho }|}_T=\frac{8\eta -2{\eta }^2}{(1-\eta {)}^4}$

where $\eta$ is the packing fraction.

References

1. N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics 51 , 635-636 (1969)
2. Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics 103 pp. 9388-9396 (1995)