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 )

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \eta ={\frac {1}{6}}\pi R^{3}\rho }

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

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

where $\beta$ 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a={\frac {(2\eta +1)^{2}}{(\eta -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{12\eta + 12\eta^2 + 3\eta^3}{2(\eta-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{\eta(2\eta+1)^2}{2(\eta-1)^4}}

The pressure via the pressure route (Eq.s 32 and 33) 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+2\eta+3\eta^2)}{(1-\eta)^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+\eta+\eta^2)}{(1-\eta)^3}}

A derivation of the Carnahan-Starling equation of state

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 via the compressibility route, to one third via the pressure route, i.e.

$Z={\frac {pV}{Nk_{B}T}}={\frac {2}{3}}\left[{\frac {(1+\eta +\eta ^{2})}{(1-\eta )^{3}}}\right]+{\frac {1}{3}}\left[{\frac {(1+2\eta +3\eta ^{2})}{(1-\eta )^{2}}}\right]={\frac {1+\eta +\eta ^{2}-\eta ^{3}}{(1-\eta )^{3}}}$

The reason for this seems to be a slight mystery (see discussion in Ref. ).