# 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)

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

and

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

and

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

and where \(x=\rho/4\).

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

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

and the compressibility route is

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

## A derivation of the Carnahan-Starling equation of state[edit]

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]} ).

## References[edit]

- ↑
^{1.0}^{1.1}^{1.2}M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters**10**321 - 323 (1963) - ↑ M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics,
**5**pp. 643-651 (1964) - ↑ J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review
**133**pp. A895 - A899 (1964) - ↑ Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics,
**39**pp. 474-479 (1963) - ↑ G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics
**54**pp. 1523-1525 (1971) - ↑ Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry
**93**pp. 6916-6919 (1989)