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.

The 'Percus-Yevick' derivation

It is interesting to note (Ref 3 Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the exact solution of the Percus Yevick integral equation for hard spheres via the compressibility route, to one third via the pressure route, i.e.

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

The reason for this seems to be a slight mystery (see discussion in Ref. 4).

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. G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics 54 pp. 1523-1525 (1971)
4. 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)