Affinity Spectra

Shuman, M.S., Collins, B.J., Fitzgerald, RJ. and Olson, D.L. (1983) Distribution of stability constants and dissociation rate constants among binding sites on estuarine copper-organic complexes rotated disk electrode studies and an affinity spectrum analysis of ion-selective electrode and photometric data. In Aquatic and Terrestrial Humic Materials (eds Christman, R.F. and Gjessing, E.T.). Ann Arbor Science, Ann Arbor, MI, pp. 349-370. 

The Henderson-Hasselbalch, Perdue-Lytle, and Scatchard models are special cases of an affinity spectrum model, defined by23,24... 

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). 

A common method of extracting f K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth fimction describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range... 

The model assumes the existence of a continuum of binding sites in which the stoichiometric concentration of binding sites with a particular pKa value is functionally related to the pKa value. In other words, the probability of occurrence of a binding site depends on the Gibbs free energy of dissociation of that site. The nature of the hypothesized distribution function is, of course, unknown however, the affinity spectrum technique is specifically designed to numerically estimate that function from observable titration data. 

As a first approximation, a value can also be derived from the absorption limit of the H02 ion in the near ultraviolet, if this is interpreted as an electron affinity spectrum as expressed by Eq. (16). Taking as absorption limit X 3200 A., which corresponds to an energy of 3.8 electron volts, one obtains from Eq. (16) ... 

It is possible to estimate (E0, + So,-) also from the long wave absorption limit of the 02 ion in aqueous systems if this is interpreted as an electron affinity spectrum, i.e.,... 

Binding affinity spectrum Narrow for monoclonal, hroad for polyclonal Narrow for covalent imprinting, hroad for non-covalent imprinting... 

A variety of alternative models more realistic than the Langmuir isotherm may be used to fit the isotherm data the most accurate but most demanding (in terms of amount and accuracy of data required) is the affinity spectrum approach, which yields a continuous spectrum of number of binding sites vs. [27]. The simplest approach, however, is to model the isotherm as resulting from just two classes of binding site ... 

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. 

In Equations 11.30,11.31, and 11.37, the intrinsic binding constants A) are true thermodynamic quantities, related to the free energy by Equation 11.3. However, when aqueous ion activities are replaced by concentrations and the ionic strength (and/or pH and/or other concentrations) is kept constant, or when (more frequently) surface electrostatic interactions are neglected, results for the binding constants can be found, but which are valid only in the same experimental conditions as the original data these are called conditional affinity spectra or conditional constants, a concept widely used in analytical chemistry (Garces, Mas, and Puy 2006 Orsetti, Andrade, and Molina 2009). Numerical methods for affinity spectrum extraction almost always yield conditional spectra. 

Equation 11.51 can be substituted into Equation 11.30 and the affinity spectrum found as... 

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 Lagrange multipliers A, are found from the constrains, which merely are the experimental data that is, the affinity spectrum should fit the experimental data through Equation 11.30 ... 

A recent model, based on the Donnan equilibrium concept, but with a different approach, is the elastic polyelectrolyte network (EPN) model. In the EPN model (Orsetti, Andrade, and Molina 2010), the HS particles are considered as divided in to two fractions an inner fraction gf, which behaves as a gel in Donnan equilibrium with the bulk solution, and an external fraction 1-g/, which is in equilibrium with the bulk (Figure 13.8) thus, it is assumed that a fraction gf of the total number of sites resides inside the gel phase, whereas a fraction f-g/ resides outside the gel given an affinity spectrum, it applies to the external sites in equilibrium with the bulk solutions, with activity a for ion i, whereas for the internal medium, it is applied the internal activity aP, in Donnan equilibrium with the bulk ... 

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