Fuoss expression

When only pair formation is observed in these solvents it is usually not necessary to evaluate the data with eqn. 5.3.4. The simpler Shed-lovsky or Fuoss expressions (eqn. 5.4.10) may be employed since the concentration of free ions is very small. However, Treiner and Justice have recently used the complete eqn. 5.3.4 with = / to evaluate the conductance of BU4NCIO4 in THF = 7.39) finding the data can be fitted with extreme precision up to c = 1.12 X 10 molar the value of corresponds to a = 4.4 A according to eqn. 5.3.8. 

The successive equilibria are characterized by K12 and K23, respectively, and when Kl2 (often denoted K0) cannot be directly determined, it may be estimated from the Fuoss equation (3), where R is the distance of closest approach of M2+ and 1/ (considered as spherical species) in M OH2 Um x) +, e is the solvent dielectric constant, and zM and zL are the charges of Mm+ and Lx, respectively (20). Frequently, it is only possible to characterize kinetically the second equilibrium of Eq. (2), and the overall equilibrium is then expressed as in Eq. (4) (which is a general expression irrespective of mechanism). Here, the pseudo first-order rate constant for the approach to equilibrium, koba, is given by Eq. (5), in which the first and second terms equate to k( and kh, respectively, when [Lx ] is in great excess over [Mm+]. When K0[LX ] <11, koba - k,K0[Lx ] + k.it and when K0[LX ] > 1, fc0bs + k l. Analogous expressions apply when [Mm+] is in excess. 

Further, in the case of virtually non-existent ion-solvent interactions (low degree of solvation), so that solute-solute interactions become more important, Kraus and co-workers47 confirmed that in dilute solutions ion pairs and some simple ions occurred, in more concentrated solutions triple ions of type M+ X M+ orX M+X andinhighly concentrated solutions even quadrupoles the expression triple ions was reserved by Fuoss and Kraus48 for non-hydrogen-bonded ion aggregates formed by electrostatic attraction. 

A comparison of experimental results with those calculated from the Fuoss (2) theory is presented in Table I. The theory 1s only valid approximately so that the order of magnitude agreement is fairly good, except in the cases of MgC03° and CaC03 . Stoichiometric association constants K are then obtained from the activity coefficients, expressions for K, and from equations for the conservation of mass. The latter express the total concentration of a given ion as the sum of the concentrations of the free ion and of the ion-pairs. Values of K and of the activity coefficients of free ions in ionic media depend only upon the effective ionic strength as is shown later. 

Various treatments of these effects have been developed over a period of years. The conductance equations of Fuoss and Onsager l, based on a model of a sphere moving through a continuum, are widely used to interpret conductance data. Similar treatments n 3, as well as more rigorous statistical mechanical approaches 38>, will not be discussed here. For a comparison of these treatments see Ref. 11-38) and 39>. The Fuoss-Onsager equations are derived in Ref.36), and subsequently modified slightly by Fuoss, Onsager and Skinner in Ref. °). The forms in which these equations are commonly expressed are... 

Data of sufficient precision to be treated by the Fuoss-Onsager conductance expression yield, in addition to values for A0 and KA, an ion size param-... 

Fuoss (42) has also derived an expression which allows computation of the critical approach distance a, in the ion pair (for singly charged ions). 

In order to obtain better agreement with experimental results, the concept of a distribution of correlation times was introduced in nuclear magnetic relaxation. Different distribution functions, G(i c), such as Gaussian, and functions proposed by Yager, Kirkwood and Fuoss, Cole and Cole, and Davidson and Cole (asymmetric distribution) are introduced into the Eq. (13), giving a general expression for... 

Organic ion radicals exist together with counterions and often form ion pairs. Since the pioneering works of Grunwald (1954), Winstein with co-authors (1954) and Fuoss and Sadek (1954), the terms contact, tight, or intimate ion pair and solvent-separated or loose ion pair have become well known in the chemical world. More recently, Marcus (1985) and Boche (1992) introduced other colloquial expressions, the solvent-shared ion pair and the penetrated ion pair. 

P relaxation is observed as a shoulder of the a relaxation. This behavior preclude the possibility to perform an exhaustive analysis of the j3 relaxation [33], The 5 and y relaxations are commonly deconvoluted for the Fuoss-Kirkwood [69] empirical expression ... 

As predicted by Fuoss and Shedlovsky, in more complicated cases, when the activity coefficient must be taken into account, the following expression may be valid for the relationship between the molar conductivity, the concentration, and the activity coefficient [33] ... 

Fuoss adopted the concept of the electrostatic contact ion-pair [60] and considered the anion as a point charge that may also penetrate the cation-conducting sphere of radius a. The final expression for the ion-pairing equilibrium constant,... 

The solvation and correlation are taken into account, thereby departing from the RPM. The dependence of solvation on distance was recently investigated [67], The expression of the ion-pairing equilibrium constant is analogous to that developed by Gilkerson or by Fuoss. Again, the solvation parameter avoids the linearity of In /Teh with Me. 

Association Phenomena According to the theoretical model of spheres in a dielectric continuum the ions are represented as rigid, charged spheres that do not interact with solvent, which is considered to be a medium without any kind of structure. The only interaction is that which occurs between the ions, and the formation of ion pairs is controlled only by electrostatic forces. On these bases, the association constant may be expressed by the Fuoss equation (29) ... 

Braun derived an expression for k E) from the 1934 Onsager theory by use of the expression given by Fuoss and Accascina (1959) and Eigen et al. (1964) for the zero-field equilibrium constant for the dissociation of an ion pair. Assuming a field-independent lifetime, Braun determined the field dependence of the charge-transfer state dissociation probability as... 

The Fuoss Model. Fuoss ( ) observed that because the BJerrum d value would generally exceed the sum of the radii of the addends , Bjerrum s so-called "ion pairs" might not be in contact. Accordingly, he developed an equation based on the assumption that an "ion pair" existed only when oppositely charged ions were in contact. Based on this and other arguments (12, ), Fuoss derived the expression... 

For polyelectrolytes (charged polymers), a plot of rjgp/c versus c may be a curve. An alternate expression of Fuoss and Strauss (1948) can be used (Chamberlain and Rao, 2000) ... 

For polyelectrolytes (charged polymers), Tam and Hu (1993) utilized the expression for specific viscosity in the equation of Fuoss-Strauss ... 

The sequential equilibria in Equation (4) are characterized by 12 = 12/ 21 (often denoted as K0) and K23 = k23/k32, respectively. When Kn cannot be directly determined it is often estimated using the electrostatic Fuoss equation.215 Usually, it is only possible to characterize the kinetics of the second equilibrium of Equation (4) so that the overall equilibrium is expressed as in Equation (5) irrespective of the intimate mechanism of ligand substitution. The pseudo-first-order rate constant for the approach to equilibrium, kabs, is given by Equation (6)... 

Fuoss developed a new theory of ion association in 1958 [27] which overcame some of the difficulties associated with the Bjerrum approach. The cations in the solution were assumed to be conducting spheres of radius a and the anions to be point charges. The ions are assumed to be immersed in a dielectric continuum of permittivity Sj. Only oppositely charged ions separated by the distance a are assumed to form ion pairs. The resulting expression for the association constant is... 

For theories later than the Fuoss-Onsager 1932 treatment it is useful to express the effects of electrophoresis, relaxation and other contributions in a form showing how they modify the external field under which the ions are migrating. 

Also take note the base for the logarithmic term in Equation (12.52) is not specified. This is because the Fuoss-Onsager equation (Equation 12.52) can be quoted in terms of logio or loge. This will affect the expression for Ei, E2, J, J2, and their values (see Appendix 2, Table 12.3). 

Fuoss has stressed that his equations can only be legitimately applied for concentration ranges where kci < 0.2. This corresponds to approximately 4 x 10 moldm for 1-1 electrolytes, 1 X 10 moldm for 2-2 electrolytes and 4 x 10 moldm for 3-3 electrolytes. The theory also only applies to symmetrical electrolytes, though later workers were able to obtain expressions for unsymmetrical electrolytes. These are much more complex than that of Fuoss and Onsager. 

Table 12.3 Expression for the constants in the Fuoss-Onsager 1957 equation along with formulae for calculating their values...

The expression for J is based on the 1957 un-amended equation, i.e. the equation where the logarithmic term is given as EiJsP — E2. Fuoss has not given any corrections for / in the corrected version which uses — 2E2-... 

The equilibria considered up to now have all involved inner sphere complexes. There is the possibility that an inner sphere complex may react with free ligands in solution this includes the solvent itself, to give an outer sphere complex where the ligand enters the secondary solvation shell of the inner sphere complex. If the two species involved in this type of interaction are of opposite sign, which is the situation where this type of complex formation is expected to be most effective, the outer sphere complex is called an ion pair. Fuoss has derived an expression (equation 38) for the ion pair formation constant, Xjp, from electrostatic arguments ... 

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 ... 

This expression was obtained by Leist and gives the first order correction to the limiting law 5.2.1 due to the finite size of the ions. Equation 5.2.21 appears as the first order correction in both the Pitts and Fuoss-Onsager treatments. 

The expression for AX/X is obtained by a series of successive approximations yielding the terms of different order which contribute to the relaxation field. The first order term arises from eqn. 5.2.13 putting all the T terms equal to zero. Fuoss and Onsager obtain, in this way, an equation equivalent to 5.2.15. The first order expression for is then replaced in the T terms and a further approximation to the perturbed distributions and ionic potentials is calculated. 

The differences between Pitts (P) and Fuoss-Onsager (F-O) are first, the above mentioned omission by F-O of the effect of asymmetric potential on the local velocities of the solvent near the ions second, the use of the more usual boundary conditions 5.2.28b by F-O compared to the P assumption that perturbations cease to be important at r = a. Pitts, Tabor and Daly, who have analysed in detail both treatments, concluded that the discrepancy due to the different boundary conditions is small but has the effect of reducing ionic interactions in the P treatment with respect to the F-O. This is confirmed by the analysis of data with both theories. Usually P requires a smaller value of the a parameter than F-O. The third discrepancy between the theoretical treatments is in the expression of Vj, in eqn. 5.2.5, for which F-O add a term which involves the effect of the asymmetry of the ionic atmosphere upon the central ion surrounded by such atmosphere. The last difference lies in the hydrodynamic approaches and the corresponding boundary conditions. P imposes the condition that the velocity of the smoothed... 

Another contribution has been added by Onsager and Fuoss to their original expression for the relaxation field. Since the ionic atmosphere is deformed by the displacement of the central ion, there are more anions (cations) behind a central cation (anion) than in front of it. Hence collisions from behind a central cation (anion) are more frequent than in front of it. This will result in an increased velocity of the central ion. The calculated effect has a minor effect on the conductance, contributing only terms linear in the concentration. Valleau has cast doubts on the reality of this effect. 

The expressions for the A terms are given in Table 5.2.1 according to the Pitts (P) and Fuoss and Hsia (F-H) treatments. Another theoretical treatment of conductances has been given by Kremp and by Kremp, Kraeft and Ebeling. Their result has been approximated by Kraeft to an equation of the form 5.2.31 with = 0 the expression for /i has been included in Table 5.2.1. 

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