The Fuoss

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. 

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... 

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. 

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. 

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. 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

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]/... 

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. 

A comparison of stoichiometric association constants calculated from the Fuoss (2) model with Debye radii and from the measurements of Johnson and Pytkowicz (2). 

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... 

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. 

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-... 

At much higher concentrations (c greater than say c ) increases again with c. In the intermediate concentration region where decreases with c, the experimental data at low salt concentrations are fitted empirically to the Fuoss law,... 

An alternative procedure uses the Fuoss conductance-concentration function to relate the measured conductance to the ionic concentrations at equilibrium (8). 

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. 

The Fuoss-Onsager-Skinner equation satisfactorily describes the electrolytic conductance of lithium bromide in acetone. Values of 198.1 0.9 Q l cm2 eq l and (3.3 0.1) X I03 are established for A0 and KA, respectively, at 25°C furthermore, a value of 2.53 A is obtained for the sum of the ionic radii ( ). When bromosuccinic acid is added to 10 5 N lithium bromide in acetone, there is a decrease in the specific conductance of lithium bromide rather than the increase that is observed at higher concentrations. As the concentration of bromosuccinic acid is increased, the values obtained for A0 and KA decrease, while those for a increase when the bromosuccinic acid and acetone are considered to constitute a mixed solvent. These results do not permit any simple explanation. When bromosuccinic acid and acetone are considered a mixed solvent, the Fuoss-Onsager-Skinner theory does not describe the system. 

This study was undertaken to determine whether or not the electrolytic conductance of the lithium bromide-bromosuccinic acid-acetone system can be described by the Fuoss-Onsager-Skinner equation (FOS equation)—Equation 2—by treating the system as lithium bromide in a mixed solvent, and to establish values for Ao and KA for lithium bromide in anhydrous acetone with the same equation. The equation requires knowledge of the concentration and corresponding equivalent conductance along with the dielectric constant and viscosity of the solvent and the temperature that is,... 

Reynolds and Kraus (17) obtained conductance for 14 salts in acetone at 25°C, and used the Fuoss method to calculate their equivalent conductances at infinite dilution. Among the salts were tetra-n-butylammonium fluorotri-phenylborate, tetra-n-butylammonium picrate, lithium picrate, and tetra-n-butylammonium bromide. They then derived ionic equivalent conductances at infinite dilution by the method of Fowler (18) using tetra-n-butylammonium fluorotriphenylborate as the reference electrolyte and obtained a value of 188.7 12 1 cm2 eq-1 for Aq for lithium bromide. 

In 1953 Olson and Konecny (19) studied the conductance of lithium bromide in acetone-water mixtures at 25°C and 35°C. They calculated KD and Ao in the acetone-rich solvents by the Fuoss method and Ao in the water-rich solvents by extrapolation of the phoreogram. They found that as the water content increases Kd increases, Ao decreases but then undergoes an increase, and a increases from slightly less than the sum of the crystal ionic radii to the sum of the radii of the fully hydrated ions. Extrapolation of their data for A0 to zero water content is not reliable because of the large concave upward negative slope however, it would appear to lead to a value of about 220 U l cm2 eq-1. Similar extrapolations of values for Kd and a yield 2.0 X 10 4 and 2.2 A, respectively. 

Two years later Nash and Monk (20) also measured conductances at 25°C using aqueous acetone (12.5 wt % water) as the solvent. For KD they obtained values of 1 X 10 3 and 6 X 10 3 for lithium bromide and hydrogen bromide, respectively, by the Davies (21) method and 101.1 Q 1 cm2 eq-1 and 117.1Q-1 cm2 eq-1 for A0 for lithium bromide and hydrogen bromide, respectively, by the Fuoss method. 

Measurement of Kd for any ion pair using the Fuoss formulation enables the degree of dissociation of that pair to be assessed, and this knowledge is essential in evaluating reactivity. The theory automatically provides a value for A0 for the salt and, if Kd is determined at a series of temperatures, then the corresponding enthalpy and entropy of dissociation can also be calculated. Use of the Denison and Ramsey equation then conveniently provides for an estimate of Kd in other... 

Pt[P(iso-Pr)3]3 tends to release one of the ligands even in the solid state due to the ligand bulk. The dissociation of its dilute solution (< 0.01M) is nearly complete in coordinating solvents like pyridine. The solution behavior of Pt[P(iso-Pr)3]8/H20 is more complex than that ofPt-(PEt3)3/H20. Following essentially the Fuoss treatment, the con-... 

Sometimes, the conductivity of the solution may decrease due to the formation of electroneutral ion pairs. Under these conditions, the Fuoss-Onsager equation can be used to calculate the molar conductivity (A) of associated electrolytes [57] ... 

The results provided by Eq. (1.153) can be improved by using the Fuoss-Hsia equation modified by Femandez-Prini (for 1 1 electrolytes [58-60]) ... 

Figure 2 JO Schematic representation of the Fuoss-Mead osmometer (a) vertical cross-section (b) inner surface qf each half-cell...

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... 

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 ... 

The relaxation process associated with the dynamic glass transition, the a relaxation, and the P relaxation, as a shoulder of the a relaxation can be observed in all these figures. At low temperatures another relaxation labeled as y relaxation can be observed. In the case of PCHpM, the maximum of the y relaxation is well away from the temperature range. Heijboer and Pineri [36,57] have reported that the maximum for this polymer is at about 100 K for 1 Hz. In the case of PCHpMM and PCOcM, the y relaxation can be observed which may be analyzed by using the Fuoss-Kirkwood (F-K) equation ... 

Density functions used earlier to interpret the relaxation data of polymers were the Cole-Cole function,70 the Fuoss-Kirkwood function,71 and the log 2) function.72 These functions, particularly the skewed log ( 2) distribution, were accounted for by 13C T, and n.O.e. data of some polymers, but the physical significance of the adjustable parameters has been questioned by some authors.68... 

The value of the dielectric constant is incorporated into precise conductance treatments. However, as noted previously, there is some disagreement concerning the value of the dielectric constant of pure NMA. Most authors have used values of the dielectric constant consistent with the work of Bass et a/.93 and that of Lin and Dannhauser26. Alternatively, Kortiim and co-workers have used Bonner s higher value of e for NMA at 35 °C13 in conjunction with the Fuoss-Onsager equation128 to treat data for several salts. This approach has only very slightly affected the deter-... 

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) ... 

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