# Difference between revisions of "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]] | + | The exact solution for the [[Percus Yevick]] [[Integral equations |integral equation]] for the [[hard sphere model]] |

− | was derived by M. S. Wertheim in 1963 | + | was derived by M. S. Wertheim in 1963 <ref name="wertheim1" >[http://dx.doi.org/10.1103/PhysRevLett.10.321 M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters '''10''' 321 - 323 (1963)]</ref> (see also <ref>[http://dx.doi.org/10.1063/1.1704158 M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, '''5''' pp. 643-651 (1964)]</ref>), and for [[mixtures]] by Joel Lebowitz in 1964 <ref>[http://dx.doi.org/10.1103/PhysRev.133.A895 J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review '''133''' pp. A895 - A899 (1964)]</ref>. |

− | + | ||

− | The [[direct correlation function]] is given by ( | + | The [[direct correlation function]] is given by (Eq. 6 of <ref name="wertheim1" > </ref> ) |

:<math>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}</math> | :<math>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}</math> | ||

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and <math>R</math> is the hard sphere diameter. | and <math>R</math> is the hard sphere diameter. | ||

− | The equation of state is ( | + | The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" > </ref>) |

:<math>\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math> | :<math>\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math> | ||

− | Everett Thiele | + | where <math>\beta</math> is the [[inverse temperature]]. Everett Thiele also studied this system <ref>[http://dx.doi.org/10.1063/1.1734272 Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, '''39''' pp. 474-479 (1963)]</ref>, |

resulting in (Eq. 23) | resulting in (Eq. 23) | ||

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

− | :<math>P= | + | :<math>P=nk_BT\frac{(1+2x+3x^2)}{(1-x)^2}</math> |

− | and the compressibility route is | + | and the [[Compressibility equation |compressibility]] route is |

− | :<math>P= | + | :<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math> |

==References== | ==References== | ||

− | + | <references/> | |

− | + | ||

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[[Category: Integral equations]] | [[Category: Integral equations]] |

## Revision as of 11:56, 5 April 2011

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>`

tag; name "wertheim1" defined multiple times with different content Cite error: Invalid`<ref>`

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)