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. It is given by (Ref Eqn. 10).

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 Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }. }

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 Z } is the compressibility factor
• 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 p } is the pressure
• 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 V } is the volume
• 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 N } is the number of particles
• 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 k_B } is the Boltzmann constant
• $T$ 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.

The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 .

Virial expansion

It is interesting to compare the virial coefficients of the Carnahan-Starling equation of state (Eq. 7 of ) with the hard sphere virial coefficients in three dimensions (exact up to 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 B_4} , and those of Clisby and McCoy ):

$n$	Clisby and McCoy	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 B_n=n^2+n-2}
2	4	4
3	10	10
4	18.3647684	18
5	28.22451(26)	28
6	39.81515(93)	40
7	53.3444(37)	54
8	68.538(18)	70
9	85.813(85)	88
10	105.78(39)	108

Thermodynamic expressions

From the Carnahan-Starling equation for the fluid phase the following thermodynamic expressions can be derived (Ref Eqs. 2.6, 2.7 and 2.8)

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{p^{CS}V}{N k_B T } = \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 \frac{ \overline{\mu }^{CS}}{k_B T} = \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}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} -1 = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}}

where $\eta$ is the packing fraction.

Configurational Helmholtz energy function:

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{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}}

The 'Percus-Yevick' derivation

It is interesting to note (Ref 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.

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 Z = \frac{ p V}{N k_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. ).

Kolafa correction

Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy :

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 Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }. }

Liu correction

Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude :

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 Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }. }

See also

• Equations of state for hard spheres
• Kolafa-Labík-Malijevský equation of state