Equilibrium constants counterions

The equilibrium constants KNa+ and Kci- introduced here characterize the extent of counterion complexation that occurs. Two other constants characterize the potential generation that results from this complexation, namely the capacitances CNa+ and ( Cl-- These are the capacitances between the planes of counterion complexation and the surface plane where ao is located. The potentials rpNa+ and rpcl are the electrostatic potential at the location in the double layer where the ions adsorb and form a surface complex. 

Co/pH and V o/pH results are sensitive to different aspects of the surface chemistry of oxides. Surface charge data allow the determination of the parameters which describe counterion complexation. Surface potential data allow the determination of the ratio /3 —< slaDL- Given assumptions about the magnitude of the site density Ns and the Stern capacitance C t, this quantity can be combined with the pHp2C to yield values of Ka and Ka2. Surface charge/pH data contain direct information about the counterion adsorption capacitances in their slope. To find the equilibrium constants for adsorption, a plot such as those in Figures 7 and 8 can be used, provided that Ka and Kai are independently known from V o/pH curves. 

The standard deviation has been determined as ct = j where v is the number of degrees of freedom in the fit. The parameters for the molecular interaction /3, the maximum adsorption Too, the equilibrium constant for adsorption of surfactant ions Ki, and the equilibrium constant for adsorption of counterions K2, are thus obtained. The non-linear equations for the Frumkin adsorption isotherm have been numerically solved by the bisection method. 

A common technique for measuring the values has been to employ species that produce anions with useful ultraviolet (UV) or visible (vis) absorbances and then determine the concentrations of these species spectropho tome trie ally. Alternatively, NMR measurements could be employed, but generally they require higher concentrations than the spectrophotometric methods. A hidden assumption in Eq. 5 is that the carbanion is fully dissociated in solution to give a free anion. Of course, most simple salts do fully dissociate in aqueous solution, but this is not necessarily true in the less polar solvents that are typical employed with carbanion salts. For example, dissociation is commonly observed for potassium salts of carbanions in DMSO because the solvent has an exceptionally large dielectric constant (s = 46.7) and solvates cations very well, whereas dissociation occurs to a small extent in common solvents such as DME and THE (dielectric constants of 7.2 and 7.6, respectively). In these situations, the counterion, M+, plays a role in the measurements because it is the relative stability of the ion pairs that determines the position of the equilibrium constant (Eq. 6). 

As an example, let us take an anionic stationary phase in which an E species is in equilibrium between the mobile phase and the stationary phase. This species, called a counterion, is present in high abundance in the mobile phase. Although the OH-species would be a simple and logical choice for this counterion, hydrogenated carbonate forms are preferred (C03 and HCOJ at 0.003 M). Carbonated species are much more efficient at displacing the ions to be separated. As an anionic species A is transported by the mobile phase, and a series of reversible equilibria are produced. These equilibria are dependent on the ionic equilibrium constant K (Fig. 4.5). 

The solubility product is the equilibrium constant for the dissolution of a solid salt into its constituent ions in aqueous solution. The common ion effect is the observation that, if one of the ions of that salt is already present in the solution, the solubility of a salt is decreased. Sometimes, we can selectively precipitate one ion from a solution containing other ions by adding a suitable counterion. At high concentration of ligand, a precipitated metal ion may redissolve by forming soluble complex ions. In a metal-ion complex, the metal is a Lewis acid (electron pair acceptor) and the ligand is a Lewis base (electron pair donor). 

Measurement of the rate of polymerization of 4-methyl-1-pentene have shown that in polar solvent the rate is first order in monomer. This is evidently an example of Case 2 (K<41). It is postulated that K, the equilibrium constant, would be small in polar solvent since the concentration of the ion-counterion pair,... 

The Mass Action Model The mass action model represents a very different approach to the interpretation of the thermodynamic properties of a surfactant solution than does the pseudo-phase model presented in the previous section. A chemical equilibrium is assumed to exist between the monomer and the micelle. For this reaction an equilibrium constant can be written to relate the activity (concentrations) of monomer and micelle present. The most comprehensive treatment of this process is due to Burchfield and Woolley.22 We will now describe the procedure followed, although we will not attempt to fill in all the steps of the derivation. The aggregation of an anionic surfactant MA is approximated by a simple equilibrium in which the monomeric anion and cation combine to form one aggregate species (micelle) having an aggregation number n, with a fraction of bound counterions, f3. The reaction isdd... 

The second approach postulates a new chemical species and an equilibrium constant that relates the activity of the new species to the established species. The mechanism of nonspecific adsorption is believed to be due to long-range electrostatic forces on counterions near the charged Ti02 surface. The extent of nonspecific adsorption can be calculated once the electrical potential on the TiOz surface is known (Stone et al., 1993). 

In the previous discussion, activity coefficients have been totally neglected, for simplicity, but they should be included in a proper treatment. However, for ionic amphiphiles, where activity corrections are expected to be most important, additional complications arise. If the counterions are explicitly incorporated in the equilibria the number of possible chemical species is greatly enhanced making a detailed analysis even more complex. Furthermore a description of a process in terms of an equilibrium constant is only really suitable when the forces involved are of a short range... 

An alternative suggestion is that the coordinated monomer dissociates to give an oxyanion stabilized by the nucleophile s counterion, most commonly, tetrabutylammonium or cesium ions,69 by analogy to the mechanism proposed for activation of tributylstannyl ethers by nucleophiles.57 This proposal is summarized in Fig. 14, which also shows the product mixture obtained for methyl (2S,3R)-2,3-dihydroxybutanoate. For this compound, reaction on the oxygen atom of the inherently less acidic hydroxyl is favored. Both anions, E and F, are in equilibrium with the coordinated monomer, and the less populated (but more reactive) anion E reacts to a greater extent, or in other words, the difference in the rate constants for trapping is greater than the difference in equilibrium constants. 

Figure 9 shows evolution of molecular weights with conversion for a hypothetical system, in which ions and ion pairs have the same reactivities (kp+ = kp = 10s mol L sec-1), covalent species are inactive, the ionization equilibrium constant is Ki = 10 5 mol-, L, and the dissociation constant is Kd = 10 7 mol/L. Kf is defined by the ratio of the rate constant of ionization to that of recombination of counterions within the ion pair (Kj = kj/kr). Kd is defined by the ratio of the rate constants of the dissociation of ion pair and that of the association of free ions (Kd... 

A comparison of Eq. (22) with Eq. (23) reveals that the apparent rate constant, k, equals the product kp K. Thus, the rate of polymerization is affected by the concentrations of monomer, initiator, and Lewis acid, as well as by the rate and equilibrium constants. Although the ionic propagation rate constant is not very sensitive to the nature of the counterion, solvent and temperature, the equilibrium constant K usually depends strongly on the temperature, solvent, Lewis acid, and the leaving group X. 

It is possible to distinguish between free ions from associated and covalently bonded species by conductivity measurements, because only free ions are responsible for electrical conductivity in solution [136, 399], Spectrophotometric measurements distinguish between free ions and ion pairs on the one hand, and covalent molecules on the other, because in a first approximation the spectroscopic properties of ions are independent of the degree of association with the counterion [141], The experimental equilibrium constant. Kexp, obtained from conductance data, may then be related to the ionization and dissociation constants by Eq. (2-16). 

A m, the apparent constant, is the product of an intrinsic constant (a constant valid for a hypothetical uncharged surface) and a Boltzmann factor. / is the surface potential, F the Faraday constant, and AZ the change in the charge number of the surface species of the reaction for which the equilibrium constant is defined (in this case AZ = -bl). The intrinsic constant is experimentally accessible by extrapolating experimental data to the surface charge where op = 0 and where l/ = 0. The correction, as given above, assumes the classical diffuse double-layer model (a planar surface and a diffuse layer of counterions). 

Equation (6.20) represents the formation of a cationic micelle from N surfactant ions D+ and (N-p) firmly held counterions X. Whenever the thermodynamics of a process is under consideration, it is important to define the standard states of the species. In this example, the standard states are such that the mole fractions of the ionic species are unity and the solution properties are those of the infinitely dilute solutions. The equilibrium constant may be written in the usual way... 

Figure 6.24 Concentration of micelles, monomers and counterions against total concentration (arbitrary units) calculated from equation (6.21) for an aggregation number (N) of 100, micellar equilibrium constant (fCJ of 1, and wilfi 85% of ifie counterions bound to ifie micelle.

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