# 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 [1] (see also [2]), and for mixtures by Joel Lebowitz in 1964 [3].

The direct correlation function is given by (Eq. 6 of [1] )

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

where

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

and $R$ is the hard sphere diameter. The equation of state is given by (Eq. 7 of [1])

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

where $\beta$ is the inverse temperature. Everett Thiele also studied this system [4], resulting in (Eq. 23)

$\left.h_0(r)\right. = ar+ br^2 + cr^4$

where (Eq. 24)

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

and

$b= - \frac{12\eta + 12\eta^2 + 3\eta^3}{2(\eta-1)^4}$

and

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

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

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

and the compressibility route is

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

## A derivation of the Carnahan-Starling equation of state

It is interesting to note (Ref [5] 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.

$Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }$

The reason for this seems to be a slight mystery (see discussion in Ref. [6] ).