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

• 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 T } is the absolute temperature

• 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 \eta } is the packing fraction:

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

• 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 \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):

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

Configurational chemical potential:

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

Isothermal compressibility:

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 \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 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 \eta} is the packing fraction.

References

1. N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics51 , 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)