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

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The exact solution for the [[Percus Yevick]] [[Integral equations |integral equation]] for the [[hard sphere model]]
The exact solution for the [[Percus Yevick]] integral equation for [[hard sphere]]s
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>.
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 of <ref name="wertheim1" /> )
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 <math>R</math> is the hard sphere diameter.
and ''R'' is the hard sphere diameter.
The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" />)
The equation of state is (Ref. 1 Eq. 7)


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


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>,
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{(2\eta+1)^2}{(\eta-1)^4}</math>
:<math>a = \frac{(2x+1)^2}{(x-1)^4}</math>


and
and


:<math>b= - \frac{12\eta + 12\eta^2 + 3\eta^3}{2(\eta-1)^4}</math>
:<math>b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}</math>


and
and


:<math>c= \frac{\eta(2\eta+1)^2}{2(\eta-1)^4}</math>
:<math>c= \frac{x(2x+1)^2}{2(x-1)^4}</math>


The [[pressure]] via the pressure route (Eq.s 32 and 33) is
and where <math>x=\rho/4</math>.
The pressure via the pressure route (Eq.s 32 and 33) is


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


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


:<math>P=nk_BT\frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>
:<math>P=nkT\frac{(1+x+x^2)}{(1-x)^3}</math>
 
==A derivation of the Carnahan-Starling equation of state==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048    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)] </ref>  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.
 
:<math>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 }</math>
 
The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 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)]</ref> ).


==References==
==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/
#[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/
#[JMP_1964_05_00643]
#[JCP_1963_39_00474]


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