Editing Exact solution of the Percus Yevick integral equation for hard spheres
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The exact solution for the [[Percus Yevick]] | The exact solution for the [[Percus Yevick]] integral equation for the [[hard sphere model]] | ||
was derived by M. S. Wertheim in 1963 | 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 (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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and <math>R</math> is the hard sphere diameter. | and <math>R</math> is the hard sphere diameter. | ||
The | The equation of state is (Ref. 1 Eq. 7) | ||
:<math> | :<math>\beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math> | ||
Everett Thiele (1963 Ref. 4}) also studied this system, | |||
resulting in (Eq. 23) | resulting in (Eq. 23) | ||
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where (Eq. 24) | where (Eq. 24) | ||
:<math>a = \frac{( | :<math>a = \frac{(2x+1)^2}{(x-1)^4}</math> | ||
and | and | ||
:<math>b= - \frac{ | :<math>b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}</math> | ||
and | and | ||
:<math>c= \frac{ | :<math>c= \frac{x(2x+1)^2}{2(x-1)^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= | :<math>P=nkT\frac{(1+2x+3x^2)}{(1-x)^2}</math> | ||
and the compressibility route is | |||
:<math> | :<math>P=nkT\frac{(1+x+x^2)}{(1-x)^3}</math> | ||
==References== | ==References== | ||
#[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)] | |||
#[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)] | |||
#[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)] | |||
#[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)] | |||
[[Category: Integral equations]] | [[Category: Integral equations]] |