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)∫0tα11 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 γ11rand=γ21rand=φ1 and γ12rand=γ22rand=φ2. In this case (ξ12rand)2/(ξ11randξ22rand)=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.