On the Theory of Ion Pairs in Solutions

Abstract

Following the general lines of Fuoss' early analysis, two alternate definitions of ion partners (1) excluding and (2) including partners composed of ions having the same electrical sign are formulated. The resulting sets of coupled integral equations for the ion-partner radial distribution functions Gij(r), differ from Fuoss' by the inclusion of a factor for the probability that the orbital ion does not have a partner closer than the central ion and are valid for electrolytes of any symmetry type or any mixture of electrolytes at any concentration for which the ordinary pair correlation functions gij(r) for the various types of pairs of ions are known. By the presented iterative numerical method, the set of equations resulting from either definition (1) or (2) may be solved even for a general mixture. Explicit analytical solutions for Gij(r) in terms of an arbitrary (except for an easily relaxed restriction of a hard core) gij(r) are presented for the following electrolytes: definition (1) symmetrical and unsymmetrical binary; definition (2) symmetrical binary with equal ionic radii. For examples of these electrolytes, graphs of Gij(r) resulting from the use of the Debye-Hückel model and the infinite dilution form of gij(r) are presented (cations, 10—4M; D=20; T=25°C; interionic distance of closest approach, 4.6 A). Numerically obtained Gij(r) are also presented for a 1–1 electrolyte with unequal ionic radii [definition (2); conditions as above except ionic radii of 1.6 A and 3.0 A] and a mixture of two 1–1 electrolytes with a common anion [definition (1); interionic distance of closest approach of the second electrolyte, 7.0 A]. More extensive ion pair formation of the smaller cation shown in the latter case qualitatively explains anomalously low conductance in mixtures. The Gij(r), markedly smaller than the corresponding Fuoss curve at large separation, lead to the same association constant K—1 if the same critical distance is used to terminate the ion pair range; a constant critical distance, however, leads to a K—1 which is a continuous function of DT in qualitative agreement with some conductance data and is compatible with a ``contact ion pair'' concept. An equation for the activity coefficient ratio is presented and degrees of association are calculated in a variety of solutions. The formalism presented easily accepts more realistic gij(r) functions and can be extended to entirely different types of systems. The usefulness of the ion (or particle) pair approach is briefly discussed.