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Abstract: We study the molecular correlations in a lattice model of a solution of alow-solubility solute, with emphasis on how the thermodynamics is reflected inthe correlation functions. The model is treated in Bethe-Guggenheimapproximation, which is exact on a Bethe lattice Cayley tree. The solutionproperties are obtained in the limit of infinite dilution of the solute. With$h {11}r$, $h {12}r$, and $h {22}r$ the three pair correlation functionsas functions of the separation $r$ subscripts 1 and 2 referring to solvent andsolute, respectively, we find for $r \geq 2$ lattice steps that$h {22}r-h {12}r \equiv h {12}r-h {11}r$. This illustrates a generaltheorem that holds in the asymptotic limit of infinite $r$. The threecorrelation functions share a common exponential decay length correlationlength, but when the solubility of the solute is low the amplitude of thedecay of $h {22}r$ is much greater than that of $h {12}r$, which in turn ismuch greater than that of $h {11}r$. As a consequence the amplitude of thedecay of $h {22}r$ is enormously greater than that of $h {11}r$. Theeffective solute-solute attraction then remains discernible at distances atwhich the solvent molecules are essentially no longer correlated, as found insimilar circumstances in an earlier model. The second osmotic virialcoefficient is large and negative, as expected. We find that thesolvent-mediated part $Wr$ of the potential of mean force between solutes,evaluated at contact, $r=1$, is related in this model to the Gibbs free energyof solvation at fixed pressure, $\Delta G p^*$, by $Z-2 W1 + \Delta G p^*\equiv p v 0$, where $Z$ is the coordination number of the lattice, $p$ thepressure, and $v 0$ the volume of the cell associated with each lattice site. Alarge, positive $\Delta G p^*$ associated with the low solubility is thusreflected in a strong attraction large negative $W$ at contact, which is themajor contributor to the second osmotic virial coefficient. In this model, thelow solubility large positive $\Delta G p^*$ is due partly to an unfavorableenthalpy of solvation and partly to an unfavorable solvation entropy, unlike inthe hydrophobic effect, where the enthalpy of solvation itself favors highsolubility, but is overweighed by the unfavorable solvation entropy.

Autor: Marco A. A. Barbosa, B. Widom


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