Carnahan-Starling equation of state

The Carnahan-Starling equation of state 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.

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:

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

Carnahan-Starling equation of state

Thermodynamic expressions

References

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