Exact solution of the Percus Yevick integral equation for hard spheres

The exact solution for the Percus Yevick integral equation for the hard sphere model was derived by M. S. Wertheim in 1963 (see also ), and for mixtures by Joel Lebowitz in 1964 .

The direct correlation function is given by (Eq. 6 of )

$C(r/R)=-{\frac {(1+2\eta )^{2}-6\eta (1+{\frac {1}{2}}\eta )^{2}(r/R)+\eta (1+2\eta )^{2}{\frac {(r/R)^{3}}{2}}}{(1-\eta )^{4}}}$

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 = \frac{1}{6} \pi R^3 \rho}

and 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 R} is the hard sphere diameter. The equation of state is given by (Eq. 7 of )

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}{\rho} = \frac{(1+\eta+\eta^2)}{(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 \beta} is the inverse temperature. Everett Thiele also studied this system , resulting in (Eq. 23)

$\left.h_{0}(r)\right.=ar+br^{2}+cr^{4}$

where (Eq. 24)

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 a = \frac{(2x+1)^2}{(x-1)^4}}

and

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= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}}

and

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 c= \frac{x(2x+1)^2}{2(x-1)^4}}

and 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 x=\rho/4} .

The pressure via the pressure route (Eq.s 32 and 33) is

$P=nk_{B}T{\frac {(1+2x+3x^{2})}{(1-x)^{2}}}$

and the compressibility route is

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=nk_BT\frac{(1+x+x^2)}{(1-x)^3}}