Fuoss’ equation— constant

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. 

Rotzinger then evaluated and H t as a function of the distance between the two reactant metal centers. He used the Fuoss equation to calculate the ion-pairing equilibrium constant to form the precursor complex at these internuclear distances. Assembly of these data then allowed the calculation of the self-exchange rate constants as a function of the internuclear distance in the transition state, the maximum rate being taken as the actual rate. 

If the ions are large, it is to be expected that the ratio of free ions to ion-pairs will be relatively great. For instance, it follows from the Fuoss equation [72] that if the interionic distance is 10 A, then in ethyl chloride at -78° (eT = 3.29 x 103) [73], the dissociation constant of ion-pairs is 2.5 x 10"3 mole/1. At a total concentration of electrolyte of 5 x 10 2 mole/1, the degree of dissociation is 0.2, and the ratio [cations]/... 

The limits to the validity of the Bjerrum-Fuoss equation (1) are set not so much by a breakdown of the model from which it is derived, as by the progressively increasing abundance of ternary and higher aggregates, as the dielectric constant of the medium is reduced. 

In the present context P+n A might be poly(iso-butyl)+ A1C14". For the simple case of spherical ions, the dissociation constant, KD, of ion-pairs is governed by the Bjerrum-Fuoss Equation (13) ... 

The Kq has been measured directly for Na+ + [222] as counterion in THF, and calculated from the interionlc distance a according to the Fuoss equation for other cryptatee, using the Stokes radius values R8+ obtained from conductimetric studies of the corresponding tetraphenylborides (24). The value of Kq for Na+ + [222] in THP was deduced from that found in THF assuming that the interionlc distance remains constant in both solvents. 

A second alternative which accords with first order kinetics consists in the formation of a low steady-state concentration of dissociated ions, followed by rate-determining attack of halide on the quasiphosphonium ion (k2<Fuoss equation which permits calculation of K from the mean ionic radius of the ions and the dielectric constant of the medium (7). For the present purpose we... 

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) ... 

Water-TMS Mixtures. Conductometric studies on Li, Na, and K chlorides and hydrochloric acid in water-TMS have shown association constants higher than the Fuoss equation predicts in these mixtures too. In the case of HCl, KA values equal to 26 5, 51 d= 9, and 76 4 corresponded to dielectric constant values of 61.42, 54.69, and 47.34, respectively. On the contrary, KA values for the same systems calculated on the basis of the Fuoss equation using the reasonable value of 4 A for the... 

Figure 6. Dependence of association constants on the dielectric constant for LiCl, NaCl, and KCl in water-TMS mixtures at 35°C (—-), association constants calculated from the Fuoss equation (a = 2.413 A) (O), LiCl (U), NaCl (A), KCl...

Ki may be calculated by the Fuoss equation and K2 becomes larger as the bond between the anion and the solvent molecule weakens. The association constant observed is therefore ... 

Further, the dielectric constant of water associated with a complex is known to decrease as cation and ligand more closely approach each other (19, 56). Thus, Choppin and Unrein (57) suggest "effective" e values of 57.0 for MF+2 and 40.8 for MF+3 complexes. The drop in both d and e should increase the stability of multivalent cation complexes over monovalent ones. That the Fuoss equation roughly predicts AG° for 3+ and 4+ cation complexes although ignoring real changes in d and e, must therefore be considered fortuitous. 

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 (37) has recently stressed that the determination of the distance between ions for such a multistep process may require an approach differing from the application of a simple dependence of Kp on the dielectric constants or reciprocal of the absolute temperature (i.e. the Fuoss equation). 

As is the case of polyelectrolyte aqueous solutions, the viscosity data from ionomer solutions apparently follow the Fuoss equation. A later study [50] has shown that the Fuoss equation is basically an empirical one and that the physical meaning of the constants A and B is not as clear as origi-... 

To calculate the effect of pressure on the formation or ionization constants of aqueous complexes, the partial molal volume change of the ionization reaction must be known. Standard partial molal volumes (25 C, 1 bar) of some aqueous complexes are known, and can be used together with the molar volumes of uncomplexed ions to calculate AV. The Fuoss equation can also be used to estimate the standard molal volume change in ionization reactions (20,21) ... 

The important parameters in these equations are the dielectric constant D and the viscosity n, characterizing the macroscopic properties of the medium, and the interionic distance which is a structural property of the ion-pair. The distance parameter , calculated from the dissociation constant with the Fuoss equation, was shown to be pressure-independent. The values of for the different solvent mixtures are given in Table I. 

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],... 

Under the aforementioned circumstances, the two-step reaction 4.53 and the associated eqns. 4.54-4.62 are equally valid on the understanding that HS represents Hcres, etc. further, it must be realized that during titration various amounts of HX and B are simultaneously present. Therefore, from previous measurement of the conductivities (k) of dilution series of the separate acids, bases and salts in m-cresol, the overall constants KHX, KB and KBH+X were calculated by the Fuoss and Kraus method66,67 (with the use of e = 12.5 and viscosity = 0.208 P for m-cresol). For C6H6S03H and HC1 it was necessary to calculate the equivalent conductivity at zero concentration from the equation... 

In all cases the key to assigning mechanism is the ability to detect and measure the equilibrium constant K. The equilibrium constant Kos can be estimated through the Fuoss-Eigen equation,10 as shown in equation 1.22. Usually, Kos is ignored in the case of L = solvent. 

Once again, in this paper, the electrochemical aspects of the ions and their equilibria are prominent. The Fuoss-Kraus equation is applied to the pairing of the carbenium ions with the anions. It is shown that since an increase in the solvent polarity reduces the propagation rate-constant, the increase in rate in changing from a less polar to a more polar solvent must be due to the increase in polarity augmenting the ratio of the concentrations of unpaired to paired cations, (here called Up and in later papers yip) on the assumption that as in anionic polymerisations, the unpaired ions propagate faster than the paired ions. 

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. 

Table 2 lists limiting equivalent conductance and association constant values for a number of 1 1 electrolytes in the solvents of Table 1, and Table 3 gives single ion mobility values. The data include results that appear to have sufficient precision to give meaningful values when treated by the Fuoss-On-sager conductance equation. In a few cases data of somewhat lower precision have been included to indicate the magnitude of the association constants, which can often be determined with fair accuracy from such data. 

An alternative approach to ion association was proposed by Fuoss. He defined the ion-pair as two oppositely charged ions that are in contact, i.e. at a distance of r=a, and derived the following equation for the ion association constant ... 

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