Segregation of molecules in binary solvent mixtures without H bonds. A quantitative treatment based on the theory of mobile order and disorder

Abstract

This treatment applies to binary liquid mixtures of volume fractions φ₁ and φ₂, but only in the absence of H bonding or ionisation. An environmental layer is defined around the individual volume v₁ of a given molecule 1. At a given instant, a fraction α₁₁ of this layer contains atoms belonging to molecules of the same kind as 1 whereas the complementary fraction α₁₂ contains atoms of molecules of kind 2. Due to the spontaneous displacements of v₁ in the liquid, α₁₁ fluctuates between the extreme values 1 and 0. However, after a long time t, the time fraction defined by the integral γ₁₁≡(1/t)∫₀^tα₁₁ dt no longer depends on the time and has the same value for all the molecules of the same kind. The four time fractions γ₁₁ and γ₁₂, γ₂₂, γ₂₁ (defined in a similar way) are characteristic of the equilibrium. They are directly related to the partition of the cohesive energy in 11, 22 and 12 interactions, ξ₁₁=γ₁₁φ₁; ξ₂₂=γ₂₂φ₂; ξ₁₂=2γ₁₂φ₁=2γ₂₁φ₂. Random mixing occurs when γ₁₁^rand=γ₂₁^rand=φ₁ and γ₁₂^rand=γ₂₂^rand=φ₂. In this case (ξ₁₂^rand)²/(ξ₁₁^randξ₂₂^rand)=4. However, in most cases, homogeneous environments are preferred and (ξ₁₂)²/(ξ₁₁ξ₂₂)=4K. The "‘environmental’' constant K is then smaller than unity. K is related to the molar volumes V₁ and V₂ and to the differences in standard Gibbs energies between the pure state and the state of infinite dilution in the other liquid. K can be estimated by the geometric mean rule. For liquids of similar polarities, K does not differ markedly from unity, but for mixtures of polar liquids and liquid alkanes, K is significantly smaller and an important segregation is predicted. Using the experimental solubilities of solid n-alkanes in pure solvents, the equations predict the solubilities in mixtures of the two solvents without any adapted parameter. The predicted values agree in this case, considerably better with the experimental ones than do those derived from the hypothesis of random mixing. The validity of time fractions thermodynamics has thus been experimentally verified.