Kirkwood correlation factors in liquid mixtures from an extended Onsager-Kirkwood-Frohlich equation

Two approaches for applying the Onsager-Kirkwood-Frohlich equation to liquid mixtures are revisited at the light of recent developments leading to the estimation of relative permittivities and refractive indices of thermodynamically ideal liquid mixtures. From the one-liquid approach, the squared permanent dipole moment of the mixture molecular-equivalent species M is demonstrated to be a mole-fraction average of squared permanent dipole moments of the components. An expression is obtained for calculating the ideal Kirkwood correlation factor of M at any composition by using only pure-constituent properties. From the two-liquid approach (Bottcher's equation), equations are obtained to describe the dependence on composition of the Kirkwood correlation factor of both components in the ideal mixture, even in mixtures of Onsager liquids. This dependency is tentatively ascribed to London dispersion forces acting between unlike molecules. It is demonstrated that Bottcher's equation can only be applied to mixtures where the relative permittivity of each component is larger than the squared refractive index of the other component. From the interplay of one-and two-liquid approaches, the ideal Kirkwood correlation factor of M and of both constituents are inter-related. Thermodynamic expressions are given for the calculation of excess Kirkwood correlation factors. In the case where permanent dipole moments are unknown, the ratio excess/ideal, termed the relative excess Kirkwood correlation factor for components and species M can still be evaluated. These ratios are related to more conventional excess properties. Density, relative permittivity and refractive index data are reported for binary mixtures of 2,2,2-trifluoroethanol with mono-, di-, tri- or tetra-glyme over the whole composition range at 288 K and 298 K. For these systems, ideal, excess and relative excess and Kirkwood correlation factors are calculated and discussed. In particular, by regarding Kirkwood correlation factors as a measure of order/molecular organisation in liquid mixtures, it is found that the formation of ideal mixtures entails a decrease of order which, for the present binary systems, is almost cancelled out upon passage to the corresponding real mixtures. It is concluded that the present formulation permits to estimate Kirkwood correlation factors of each constituent of liquid mixtures and thereby to draw information on their molecular organisation.