# Exact solution of the Percus Yevick integral equation for hard spheres

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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]} )

where

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

where is the inverse temperature. Everett Thiele also studied this system ^{[4]},
resulting in (Eq. 23)

where (Eq. 24)

and

and

and where .

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

and the compressibility route is

## References

- ↑
^{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) Cite error: Invalid`<ref>`

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tag; name "wertheim1" defined multiple times with different content - ↑ 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)