Exact solution of the Percus Yevick integral equation for hard spheres: Difference between revisions
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was derived by M. S. Wertheim in 1963 Ref. 1 (See also Ref. 2) | was derived by M. S. Wertheim in 1963 Ref. 1 (See also Ref. 2) | ||
(and for mixtures by in Lebowitz 1964 Ref. 3). | (and for mixtures by in Lebowitz 1964 Ref. 3). | ||
The direct correlation function is given by (Ref. 1 Eq. 6) | The [[direct correlation function]] is given by (Ref. 1 Eq. 6) | ||
:<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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:<math>\eta = \frac{1}{6} \pi R^3 \rho</math> | :<math>\eta = \frac{1}{6} \pi R^3 \rho</math> | ||
and | and <math>R</math> is the hard sphere diameter. | ||
The equation of state is (Ref. 1 Eq. 7) | The equation of state is (Ref. 1 Eq. 7) | ||
Line 33: | Line 33: | ||
and where <math>x=\rho/4</math>. | and where <math>x=\rho/4</math>. | ||
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=nkT\frac{(1+2x+3x^2)}{(1-x)^2}</math> | :<math>P=nkT\frac{(1+2x+3x^2)}{(1-x)^2}</math> |
Revision as of 18:56, 5 March 2009
The exact solution for the Percus Yevick integral equation for the hard sphere model was derived by M. S. Wertheim in 1963 Ref. 1 (See also Ref. 2) (and for mixtures by in Lebowitz 1964 Ref. 3). The direct correlation function is given by (Ref. 1 Eq. 6)
where
and is the hard sphere diameter. The equation of state is (Ref. 1 Eq. 7)
Everett Thiele (1963 Ref. 4}) also studied this system, 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
- 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)