Statistical thermodynamics of regular solutions and solubility parameters

Abstract

Solubility parameters, developed originally for regular solutions, have been applied to solutions beyond the presumed weak non-ideality, implying that the true foundation of the solubility parameters may be more general than the regular solution theory. To assess the root 2 of regularity on rigorous statistical thermodynamics, here we reexamine the classical iodine dissolution experiments by Shinoda and Hildebrand, who concluded that the entropy of mixing is ideal regardless of solute-solvent size ratio. We show that iodine solubility is concerned with the limit of infinite dilution, while the regular solution theory is a scheme to describe the dependence on the solute concentration. This means that the solubility of iodine cannot be a foundation of the regular solution; it is further shown that the differences in the solvation free energy among organic solvents are dominated by enthalpy with negligible role of the entropic component. In addition, the validity of the regular solution concept, i.e., the enthalpic nature of the solution non-ideality, can now be examined quantitatively by expressing the Margules model in terms of the Kirkwood-Buff integrals, which incorporate the excluded volume effects and the potential of mean force nature of interactions that were beyond the reach of the classical thermodynamic models. Such insights into the physical basis of solubility parameters may be useful for improving solubility prediction.

Sorption from Solution: A Statistical Thermodynamic Fluctuation Theory

“…On this foundation, we have introduced the Kirkwood-Buff χ parameter as the measure of net self-interaction at the interface and in the solution. Unlike the Flory χ based on the lattice model, the Kirkwood-Buff χ is assumption-free and appears widely in the solution theory, such as in the activity coefficient and cooperative solubilization by hydrotropes and surfactants. ,, Hence the use of Kirkwood-Buff χ establishes a common language between sorption and solution.…”

“…The key to achieving this goal is the relationship between the concentration fluctuations (eqs and ) and the Kirkwood-Buff integrals ( G ij , between the species i and j , see Supporting Information section C for derivation) via where we have introduced the Kirkwood-Buff χ parameter via eq , which will be used for * and II . Note the involvement of sorbate–sorbate ( G 22 ), sorbate–solvent ( G 12 ), and solvent–solvent ( G 11 ) Kirkwood-Buff integrals in eq , as compared to gas (vapor) sorption for which only G 22 is present. , What is crucial for a molecular-based interpretation is the relationship between G ij and the distribution function between the species i and j , g ij ( r ) with their relative configuration r , via The term, i.e., “the Kirkwood-Buff χ parameter”, has been inspired by its relationship to the activity coefficient, γ 1 , in dilute binary solutions, where x 2 is the mole-fraction of species 2 and χ ∞ is the limiting value at x 2 → 0; eq is analogous to the role of the Flory–Huggins χ parameter, χ FH , present in the following equation: where ϕ 2 is the volume fraction of species 2, z is the number of contacts, and w ij is the contact energy between species i and j , yet, in practice, the mole fraction x 2 is widely used in place of ϕ 2 . (Note that we have not incorporated the factor 1/2, that are present in both the Kirkwood-Buff and Flory–Huggins theories, into the definition of χ in eq simply to keep our subsequent equations simpler.)…”

Cooperativity in Sorption Isotherms

“…N 22 + 1, when viewed in the constant μ 1 ensemble, represents the net self-interaction (i.e., the difference between self-interactions, G 11 and G 22 , and mutual interaction, G 12 ). The common measure for net self-interaction is the Flory χ parameter, which is restricted to the lattice model of solutions, – yet can be generalized beyond the lattice model as χ = ⟨ n 1 ⟩ v ( G 11 + G 22 − 2 G 12 ) based on a correspondence in activity coefficients between the lattice model and the Kirkwood–Buff theory of solutions . Using eq , N 22 can be expressed as N 22 = ⟨ n 2 ⟩ ⟨ n 1 ⟩ ( χ + 1 ) This marks a departure from the sorbent/gas interface, in which only the sorbate–sorbate interaction contributes to N 22 + 1.…”

Dichloromethane replacement: towards greener chromatography via Kirkwood–Buff integrals

“…Herein, through a statistical thermodynamic framework, we will quantify the driving molecular interactions that determine empirical TLC measurements, aiding in the further understanding of solvation behaviours. [25][26][27][28][29] Within this framework, we are able to deduce R f value dependence on polar eluent concentration between the stationary and mobile phases in the presence of a low-polarity mediating solvent. The competitive strength of the analyte-eluent and analyte-solvent interactions will be quantied through the Kirkwood-Buff integrals (KBIs) [25][26][27][28][29] of the radial distribution functions, describing the dispersion of molecules around a given analyte.…”

“…[25][26][27][28][29] Within this framework, we are able to deduce R f value dependence on polar eluent concentration between the stationary and mobile phases in the presence of a low-polarity mediating solvent. The competitive strength of the analyte-eluent and analyte-solvent interactions will be quantied through the Kirkwood-Buff integrals (KBIs) [25][26][27][28][29] of the radial distribution functions, describing the dispersion of molecules around a given analyte. The resulting KBIs at the dilute eluent limit will allow us to interpret the chromatographic data of green replacement solvents in the practical substitution of DCM.…”