Carnahan-Starling equation of state

The equation of Carnahan-Starling is an approximate 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.

References

1. N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" J. Chem. Phys. 51 , 635-636 (1969)