Discrete Affinity Spectra

To reduce the number of adjustable parameters, the A- and B-type proton dissociation constants are given, respectively, by 

For metal ion binding, there are different approaches monodentate metal com-plexation is treated as exchange with proton, such as 

Equations 13.6 and 13.7 are replaced with the introduction of new parameters ALKj, and AL/fg by  

Based on empirical correlations, in Model VI, the parameter ALATj was set in terms of the first formation constant for the complexation of the metal M + with NH3  

a correlation between and AT g (Equations 13.8 and 13.9) was found in the form 

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Discrete affinity spectra are composed of a few, well-characterized binding sites with known binding constants this is the case of common polyprotic acids, such as phosphoric acid... 

The total surface coverage for discrete affinity spectra is simply the weighted sum of the coverages of the diverse sites, as in Equation 4.53 ... 

Gustafsson 2001) that in this way a few higher affinity sites (N groups, thiols) are accounted for. Discrete affinity spectra models allow for multidentate binding. Here, the approaches are more different Model V considers that, when a cation binds simul-... 

FIGURE 11.6 Sorption isotherms of dodecylpiridinium on a soil material (EPA-12). (a) Points experimental data line fitted isotherms (almost indistinguishable) with discrete and continuous affinity spectra (b) discrete affinity spectrum, found using regularization for a small number of sites (c) continuous affinity spectrum, after regularizing for smoothness note that the smaller peaks are magnified by a factor of 30. (Reprinted with permission from Cernik, M. et al.. Environ. Sci. TechnoL, 29,2,413-425. Copyright 1995 American Chemical Society.)... 

The adsorption-desorption reaction in Eq. 4.3 has been applied to soils in an average sense in a spirit very similar to that of the complexation reactions for humic substances, discussed in Section 2.3.11 Although no assumption of uniformity is made, the use of Eq. 4.3 to describe adsorption or desorption processes in chemically heterogeneous porous media such as soils does entail the hypothesis that effective or average equilibrium (or rate) constants provide a useful representation of a system that in reality exhibits a broad spectrum of surface reactivity. This hypothesis will be an adequate approximation so long as this spectrum is unimodal and not too broad. If the spectrum of reactivity is instead multimodal, discrete sets of average equilibrium or rate constants—each connected with its own version of Eq. 4.3—must be invoked and if the spectrum is very broad, the sets of these parameters will blend into a continuum (cf. the affinity spectrum in Eq. 2.38). 

The affinity spectra or constant distribution is a matter that has attracted several researchers, especially in the case of HS (Koopal and Vos 1993 Borkovec and Koper 1994a Manunza et al. 1995 Borkovec et al. 1996 Lin and Rayson 1998 Avena, Koopal, and van Riemsdijk 1999 Lin, Drake, and Rayson 2002 Garces, Mas, and Puy 2004 Orsetti, Andrade, and Molina 2009 David et al. 2010). The proton affinity spectrum is treated separately from the spectra of other ions. The reason is that, as H" " binding is almost always present, for all other ions (either metal cations or anions), there is always competition with protons thus, competitive adsorption must be considered. Here, discrete and continuous affinity spectra are discussed, both for noncompetitive and for competitive cases, and in Section 11.3, methods to extract them from experimental data are presented. 

For a heterogeneous population of sites with a discrete affinity distribution, the overall degree of protonation, 0, is the weighted sum of the degrees of protonation of the different categories of sites for a continuous affinity spectrum, the summation is replaced by an integral... 

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