Ion-Pair Association Constants

One can write acid-base equilibrium constants for the species in the inner compact layer and ion pair association constants for the outer compact layer. In these constants, the concentration or activity of an ion is related to that in the bulk by a term e p(-erp/kT), where yp is the potential appropriate to the layer [25]. The charge density in both layers is given by the algebraic sum of the ions present per unit area, which is related to the number of ions removed from solution by, for example, a pH titration. If the capacity of the layers can be estimated, one has a relationship between the charge density and potential and thence to the experimentally measurable zeta potential [26]. 

Ion-pair association constants K A determined with the set of conductivity equations (7)—(15) agree with those obtained from Eq. (18) and (19) [100]. Salomon and Uchiyama have shown that it is also possible to extend the directly Fuoss-Hsia equation to include triple-ion formation [104],... 

The conductivity functions of such electrolytes can be evaluated at the level of limiting laws with the help of Eq. (18), permitting the determination of the tripleion-constant KT and the ion-pair association constant KA. 

Table 5 contains a selection of ion-pair association constants, triple ion formation constants, and limiting conductivities for various electrolytes which have been studied in connection with the optimization of battery electrolytes. It shows... 

Ion pair association constants measured by conductance have not often been verified by independent techniques, but where comparisons have been made agreement appears satisfactory. For example, a value of 70.2 0.5 was obtained for the association constant of silver nitrate in acetonitrile by conductance l0), and a value of 74 5 by potentiometric measurements, 44) of the cell... 

These new methods for determining the partition coefficients of ionized species are still e3q>erimenta1, but they are presented in a spirit that they may stimulate thinking and further refinement. Single-phase titrations with HCl in octanol have only recently been run. A possible concentration dependency of pKa in the single-phase titrations has been suggested by a referee and will be looked for. Further refinement of the two-phase titrations may incorporate ion-pair association constants. 

In the case of sulfate and dihydrogen phosphate where the ion pair association constants have been clearly identified, the anation rates are known for the 1 1 outer sphere complex. These rate values vary somewhat, and this perhaps indicates participation of the entering group. But there may be another way to interpret what is going on. These two rates of anion entry are smaller than the rate of water exchange of the aquopentammine complex. In fact, the univalent anion enters the complex from the outer sphere at approximately one-eighth of the rate of water exchange, and the divalent anion enters the complex from the outer sphere about twice as fast. 

There is one rather nasty twist in the ion pair evaluation where a negative value for free carbonate ion concentration results, because the initial value which must be used for the carbonate ion activity coefficient is much larger than it is under the new conditions. This demands some maneuvers with both the calcite solubility constant and the calcium carbonate ion pair association constant. These will not be gone into here. 

Mayer [22], the above correlations indeed work well and are quite useful for predicting values such as the free energy of salt solutions and complex formation in various solvents. Another typical example of the importance of the use of DN and AN as solvent parameters, instead of properties such as the dielectric constant, would be ion pair association constants in isodielectric solvents. For instance, as shown by Mayer [15], association constants of various perchlorates isocyanates, and halides (alkali metal, ammonium, and tetraalkyl ammonium cations) are very different in isodielectric solvents such as nitromethane (DN = 2.7), acetonitrile (DN = 14.1), and DMF (DN = 26.6), whose dielectric constant is around 26 at room temperature. 

The electrical conductivities of electrolyte solutions and the ion-pair association constant are both very sensitive to ion solvation and permit the calculation of solvation constants. 

The addition of polar molecules to electrolyte solutions effects a much larger change in conductance if low dielectric solvents are employed In a series of papers Gilkerson et a/. studied the ion-molecule interaction of tertiary and quarternary ammonium cations with Lewis bases in low dielectric solvents, like o-dichlorobenzene, chlorobenzene or 1,2-dichloroethane. The change of the ion-pair association constant with concentration of an additive L was attributed to the formation of 1 1 cation-molecule complexes ... 

Fig. 1. The variation of the ion-pair association constant with the dielectric constant of the solvent for the electrolyte tetrabutylammonium picrate in nitro-benzene-CCU mixtures (Hirsch and Fuoss, 1960).

TABLE 2. Ion pair association constants (logJiTas in DMSO at 25 °C) for enolates derived from 1,3-dicarbonyl compounds with alkali metal cations. Reprinted with permission from Reference 233. Copyright 1980 American Chemical Society... 

Examination of electrostatic principles allows some conclusions to be drawn regarding the effect of ion pairing on the selectivity of salt partitioning or, equivalently, on the driving force for cation exchange. As outlined in a standard text [234], treatments of Fuoss [235] or of Bjerrum [236] may be applied to estimate the ion-pair association constant /Ca.,soc- The Fuoss treatment assumes contact ion pairs and is conceptually simpler to use and apply. As the simplification will not affect the conclusions to be drawn here, it will be employed with the additional proviso that the effect of water in the solvent will be neglected for the moment. According to Fuoss, the ion-pair association constant at 298 K may be expressed in terms of the solvent dielectric constant 6 and the internuclear distance i m-x (in nm) between the cation and anion ... 

Figure 8 The ion-pair association constant (as log /Cassoc) as a function of cation radius for tetraphenylborate (left panel, r = 0.461 [127]) and perchlorate (right panel, r = 0.250 [127]) at four different solvent dielectric constants. The calculations employed Eq. (48) and apply to 298 K.

In sect. 5.1 it was mentioned that some of the most important information obtained from conductance measurements are the values of the ion-pair association constants, K. For symmetrical electrolytes refers to the equilibrium... 

The contributions of K i may be important. These can be factored into intermolecular and intramolecular contributions. The former amounts to an ion-pair association constant and is expected to be small in high dielectric media. The intramolecular coulombic term is the largest term for the (D )- A ) species in Equation (25). Some additional considerations are required if the reaction free energy is referenced to the limit in which /7da b. 

At low electrolyte concentrations the activity coefficient of ion pairs may be set equal to unity and Eq. (77) is linked to the ion-pair association constant and the mean activity coefficient of the free ions by the relationships I — O 1... 

At concentrations far below the conductivity minimum (c 10 mol dm in Fig. 6), triple-ion formation can be neglected. Data analysis is possible with the help of Eqs. (65), in agreement with pairwise additive potential functions, and yields the values of A in Table III. The ion-pair association constants of these plots agree well with the determined independently at higher concentrations with the help of Eq. (72), which takes into account both ion-pair and triple-ion formation. No method is known for the determination of the values of data analysis yields only the product which the quantity is commonly estimated to be 2 A /3. Both ion-pair... 

Figure 15. Simulation results for the isochoric temperature dependence (top) and isothermal density dependence (bottom) of the ion pair association constant in comparison with the corresponding experimental data from conductance experiments of Ho et al [241] and Zimmerman et al [242].

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