Fuoss model

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... 

The Fuoss model yields more accurate AS° values for the sulfate complexes than the Bjerrum model. Most remarkable, however, is the even better overall accuracy of AS° calculated by the simple electrostatic model. 

For the trivalent and quadrivalent actinide and lanthanide fluoride complexes, Choppin and Unrein (57) suggest an equation for AGO which contains z+z /(r]yj + rp)e, with e the adjustable parameter. They give e = 79.8, 57.0, and 40.8 for MF" , MF " , and MF " " complexes. Introducing these values in the electrostatic or Fuoss model equations greatly improves the agreement between predicted and empirical data. However, it must be remembered that these e values also include the effect of decreasing d values in the complex with increased valence of the cation. 

Figure 14. Empirical and model-predicted AS° values for 1 1 metal fluoride complexes fl = 0) plotted against z z./(ym + f)- (F) Fuoss model.

Figure 15. Empirical —aG° values for metal cation HPOj, complexes (1 = 0) plotted against t.+7.J(ym + ri/po j, with thpo = 3.15 A based on Izatt et al. (60) and Wells (27). — aG° for AlHPOj, is an estimate. The smooth plotted curve has no statistical significance. (F) Fuoss model.

The stability constants of ion pairs (their log /Cassoc values) have been shown to be proportional to the electrostatic function ZMzJd, where z Z/. are the charge of metal cation and ligand, and d rM + ri, the sum of their crystal radii (cf. Fig. 3.5). Mathematical models for predicting ion pair stabilities generally assume this proportionality and include the simple electrostatic model, the Bjerrum model, and the Fuoss model (cf. Langmuir 1979). Such models can predict stabilities in fair agreement with empirical data for monovalent and divalent cation ion pairs. 

In summary, the models discussed in this chapter focus on the physical aspects of electrolyte solutions but they ignore the chemical aspects. This is especially apparent in the treatment of ion solvation where an empirical correction to the MSA model was applied to treat the differences in behavior seen for cations and anions in water. The same problem arises in using classical electrostatics to describe ion pairing. In spite of the fact that the Bjerrum and Fuoss models give a good qualitative description of an ion association, this phenomenon can only be understood in detail by using quantum-mechanical methods. Needless to say, such calculations in condensed media are much more difficult to carry out. 

The parameter Kp can be estimated from the Eigen Fuoss model, which was originally derived to estimate the ion pair formation constant (see section 3.10). One of the ions is assumed to have a radius a which is equal to the distance between the reactants at the reaction site. Other ions of the redox couple which are within or on a sphere of radius a can react. The estimate of Kp also takes into consideration any electrostatic work done to bring the reactants to the reaction site. The equation for Kp in units of M is... 

The 1 1 ion-pair dissociation constants are determined by use of conductivity measurements or are estimated theoretically, for example, by use of the Fuoss model of ion pairs (eq 12) [61]. 

An interesting case of ion pairing where the Bjerrum and the Fuoss models fail to predict the association found experimentally, is that in which solvent-separated and contact ion pairs may be formed. B5hm and Schulz observed that the association of NaBPh4 in tetrahydro-pyran decreases seven times when the temperature goes from 45°C to... 

Diffusion coefficients of ammonium salts in aqueous solutions are theoretically estimated from the Onsager-Fuoss model. The influence of the ion size parameter a (mean distance of closest approach of ions), as well as of both the thermodynamic and the mobility factors on the variation of diffusion coefficients with concentration, is discussed. The aim of this chapter is to contribute to a better knowledge of the structure of these systems. 

The estimation of the diffusion coefficients of the ammonium salts in aqueous solutions can be made on the basis of the Onsager-Fuoss model (Eq. (1)) [9], by taking into account that D is a product of both a kinetic (or molar mobility coefficient... 

In Eq. (15) 2 qB/r is the coulombic part of the mean force potential, and Wjj is the noncoulombic part. The earlier association constants of Fuoss, Prue, and Bjerrum are special cases of this general chemical model [15]. The importance of noncoulombic interactions is proved [ 16] by ... 

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 preceeding section mention was made of ion association (ion-pairing) which, for the purposes of this paper, will refer to coulombic entities with or without cosphere overlap. Experimental support for ion-pairing has come from sound attenuation (2). Raman spectroscopy (2) and potentiometry (2, 2). Credibility has resulted from the model of Fuoss (2) applied by Kester and Pytkowicz (2). 

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... 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

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. 

In this system the a relaxation can be analyzed by the symmetric equation of Fuoss-Kikwood and a new model which is similar to Havriliak- Negami equation used in the analysis of dielectric spectroscopy. According to the Tg values calculated for these systems, the free volume can be appropriately described by the free volume theory. The analysis of these families of poly(methacrylate)s allow to understand in a good way the effect of the structure and nature of the side chain on the viscoleastic behavior of polymers [33],... 

The analysis in the subglass region allow to fit the loss factor permittivity by empirical equations. As in systems previously analyzed a reliable model to represent secondary relaxation is that of Fuoss-Kirkwood [9], Assuming that the two overlapped contributions for 8 and y relaxation are additive Sanchis and coworkers [64] have proposed the following equation ... 

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) ... 

Table XV.5 shows the rather dramatic change in Xeq for the dissociation of tetrisoamyl ammonium nitrate, (i-Am4N)+N03 ", with dielectric constant in mixtures of H2O and dioxane. Although it is possible to get much better agreement with the conductance data by using slightly different values of the case shown is used to emphasize the essential correctness of the method. Note also that no account has been taken of the preferential solvation of ions by one of the two solvents. The Fuoss and Kraus treatment also gives a simple model for the calculation of ion triplet and quadruplet concentrations.

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... 

Figure 11. Empirical — aG° data for 1 1 metal sulfate complexes (1 = 0) plotted against z z./(ym + r o J, where z+ and z. are the valence of cation and sulfate ion, Ym is the crystallographic radius in Angstroms of the cation in sixfold coordination (3S), and ysoj, = 3.05 A (59). The locus of — aG° values computed for the complexes by the simple electrostatic model is shown as a dashed line, and computed by the Fuoss and Bjerrum equations as lines labeled (F) and (B), respectively.

Sn ", and Zn " ") are ignored. Without these cations, the equation of the regression line through the data is -AG =-9.96 + 12.50 [z+z /(r]yj + rp)] with r = 0.95. The hard 3+ and 4+ cation complexes are much more stable than predicted by the Fuoss or electrostatic models. These complexes evidently owe an important part of their stability to covalent bonding. 

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-... 

Fuoss and Accascina [49] have shown, however, that for ionic concentrations as low as 10 M, Ostwald s equation is not exact because of its neglect of long range interionic attraction upon the conductance and activities of ions. Maintaining the concept of the sphere in continuum model, in which ions are regarded as hard spheres immersed in a continuous medium, Fuoss corrected the equation from first principles and derived the relationship... 

After reading the pertinent discussion in Langmuir (1979), contrast the applicability of the electrostatic, Bjerrum, and Fuoss thermodynamic models for predicting the stabilities of ion pairs. 

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