Fuoss’ equation—

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. 

Assays. Nitrogen assays to determine 1-amidoethylene unit content were done by Kjeldahl method. Limiting viscosity numbers were determined from 4 or more viscosity measurements made on a Cannon-Fenske capillary viscometer at 30°C. Data was extrapolated to 0 g/dL polymer concentration using the Huggins equation(44) for nonionic polymers and the Fuoss equation(45) for polyelectrolytes. Equipment. Viscosities were measured using Cannon-Fenske capillary viscometers and a Brookfield LV Microvis, cone and plate viscometer with a CP-40, 0.8° cone. Capillary viscometers received 10 mL of a sample for testing while the cone and plate viscometer received 0.50 mL. 

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]/... 

Such calculations are based on the Bjerrum-Fuoss equation... 

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 order to achieve an adequately large KD, the adjustment of three variables is at our disposal, which are summarised in the Bjerrum-Fuoss equation (2) relating KD to D, the temperature T, and a quantity a which is determined by the distance of closest approach of the ions when paired ... 

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) ... 

By ussing the Fuoss equation distance (a) between ions constituting the ion-pair was calculated. 

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. 

K (26). The former value leads to an interlonlc distance i equal to 4.6 A according to the classical Fuoss equation. This value Is too small compared to the results obtained for cryptated living polypropylene sulfide (8) and for cryptated tetraphenyl-borides In THF (24). This might mean that either K is not located inside the cavity of the ligand or the oxanion can penetrate into the cavity of the cryptand. This last explanation is consistent with comparative conductivity data made on model compounds (17) as shown in Table III. 

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<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) ... 

The failure of the Fuoss equation to reproduce experimental data appears particularly evident by conductometric measurements of hydrochloric acid (18) and by spectrophotometric measurements of picric acid (21) which we have carried out in TMS at 35° and 30°C, respectively. 

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... 

In the case of LiCl, NaCl, and KC1, Figure 6 enables experimental values to be compared with theoretical ones calculated by the Fuoss equation, substituting for the a parameter the lowest value with a physical meaning (a = 2.413, the sum of Li+ and Cl" crystallographic radii), in order to get larger theoretical values for KA. As seen from Figures 5 and 6, the observed trends in almost the whole range of solvent composition are linear but with different slopes from the theoretical one in 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 ... 

Application of the Models. Plotted in Figures 11 and 12 are measured and model-predicted AG° and AS° values for 1 1 sulfate complexes against z z /(r + r ). Clearly, the Fuoss equation... 

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 Fuoss equation is a good predictor of AG and AS values for complexes in which the bonding is chiefly electrostatic (most 1 1, 1 2, 2 1, and 2 2 complexes formed by hard acids and hard bases, including in this study F , S04 , and HP04 complexes). 

Thermodynamic data, and especially AS values, are generally unreliable or lacking for important phosphate complexes. Until such AS data is measured, it can be estimated with fair accuracy using the Fuoss equation for monovalent and divalent-bonded complexes and the electrostatic model when trivalent and quadrivalent addends are associated. Unfortunately, published AG and AS data on HS , s2 , and Se and Te aquo-complexes are suspect or largely lacking (Barnes, H. L., Pennsylvania State University, personal communication, 1978). Both the stoichiometry and stability of such complexes remains in doubt. Once a few such data have been accurately measured, plots with EN (10) or Q (44) as a variable, or using hard and soft acid and base concepts (3, 40) should permit the useful estimation of many as yet un-... 

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). 

Especially important is the first point, which is frequently used to check whether a polymer is ionic. This is a characteristic behavior of polyelectrolyte solutions since the early observation by Staudinger on sodium polyacrylate in water [33], To explain this characteristic behavior, the following Fuoss equation was proposed [14] ... 

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) ... 

For the work term between charged ions in a polar solvent, the Fuoss equation is often used, giving... 

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