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).

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

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:

$\chi _{T}-1={\frac {1}{kT}}\left.{\frac {\partial P^{CS}}{\partial \rho }}\right\vert _{T}={\frac {8\eta -2\eta ^{2}}{(1-\eta )^{4}}}$

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 pp. 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)
3. Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry 93 pp. 6916-6919 (1989)