Constant capacitance model protonation

In Fig. C microscopic acidity constants of the reaction AlOHg =AIOH + H+ for y-AI203 are plotted as a function of AIOH. The data are for 0.1 M NaCICV This figure illustrates (within experimental precision) the conformity of the proton titration data to the constant capacitance model. Calculate the capacitance. 

Nature of the Surface Complexes. The constant capacitance model assumes an inner-sphere molecular structure for surface complexes formed in reactions like equation 5a or 7. But this structure does not manifest itself explicitly in the composition dependence of Kc everything molecular is buried in which is an adjustable parameter. This encapsulating characteristic of the model was revealed dramatically by Westall and Hohl (13), who showed that five different surface speciation models, ranging from the Gouy-Chapman theory to the surface complex approach, could fit proton adsorption data on AL O., equally well, despite their mutually contradictory underlying molecular hypotheses [see also Hayes et al. (19)]. They concluded that "... no model will yield an unambiguous description of adsorption. .. . To this conclusion one may add that no model should provide such a description,... 

The intrinsic equilibrium constants for the diffuse layer model are similar to those for the constant capacitance model where P is replaced by Equations (6.10) and (6.11) describe surface protonation and dissociation, respectively. Metal surface complexation is described by two constants similar to tliat defined in Eq. (6.12) for strong and weak sites ... 

In the triple layer model the surface reactions for protonation and dissociation of the surface functional group are Eqs. (6.6) and (6.7) as written for the constant capacitance model, where h is replaced by I, . The reactions for adsorption of the background electrolyte in the P-plane are... 

Generalized composite approaches have also been used in application of the constant capacitance model to describe molybdenum (Goldberg et al., 1998) and arsenate adsorption by soil (Goldberg and Glaubig, 1988) and sediments (Gao et al., 2006) and the triple layer model to describe calcium and magnesium adsorption by soil (Charlet and Sposito, 1989). In these applications the electrostatic terms and protonation-dissociation reactions were retained. 

For proton-promoted dissolution, the adsorption of protons can be predicted using the constant capacitance model for the formation of protonated sites on oxide surfaces (Sposito, 1983 Hayes and Leckie, 1987). The rate can then be related to the quantity of protonated sites, [= MOH2 ], by the equation... 

The surface protonation equilibria are interpreted according to the constant-capacitance model of Schindler and Stumm (1987). The two model parameters, the intrinsic constant pK x (intr.) at zero surface charge and the integral capacitance of the flat electric double layer C( = 2 Fm"2) are determined from titration curves. The methods of the acidimetric titration of kaolinite suspensions are discribed by Wieland (1988). 

THE NET PROTON CHARGE. In the Constant capacitance model, the surface complexation reactions that involve or OH alone are special cases of Eq. 5.37a ... 

The conformity of the proton titration data for Y-AI2O3 to the constant capacitance model (within experimental precision) is evident in Fig. 5.2. 

Eq. 5.34. Thus the protonation (and proton dissociation) of the inorganic hydroxyl surface is described in the objective model just as it is in the constant capacitance model. However, in the objective model, cth is given the explicit mathematical representation... 

FIG. 4 Experimental points of net proton consumption from forward titration with 0.1 M KOH for purified 5-AI2O3 dispersed in indifferent electrolyte (KNO3) solutions at room temperature. The continuous lines are numerically fitted [32] using the constant capacitance model (C = 1.2 F/m ). 

Figure 4 Comparison of sorption models. Several commonly used sorption models are compared with respect to the independent constants they require. These constants are valid only under spedlic conditions, which must he specified in order to properly use them. In other words, the constants are conditional with respect to the experimental variables described in the third column of the figure. is the radionuclide distrihution constant K and n are the Freundlich isotherm parameters and f3 are surface complexation constants for protonation and deprotonation of surface sites K, K-, 13, are surface complexation constants for sorption of cations and anions in the constant capacitance model and TLM, respectively C, Ci, and are capacitances for the electrical double layers rr, ov, and oj, are surface charges at different surface planes (Me) and (S) are concentrations of the sorbing ions and the surface sites, (M), (L) are concentrations of other cations and ligands in solution, respectively I is the ionic strength of the background electrolyte and 5a are the site density and specific surface of the substrate, respectively. The requirements of the DLM are similar to those of the constant capacitance model.

The electrostatic models discussed in Sections 2.9.2.1 through 2.9.2.5 apply to a simple chemical model involving one reaction (Reaction 2.14) of the transfer of one species (proton) from the solution to the surface. More complex chemical models involving the transfer of two or more species from the solution to the surface and/or allowing various distances of the adsorbed solution species from the surface require more complex electrostatic models. Usually, three or more capacitors in series are considered. All the capacitors but one have constant capacitances, and the relationship between and Vd in one capacitor (diffuse layer) is expressed by Equation 2.18. 

A specific example of the relationship between the microscopic subreactions required to model experimental observations of metal removal and the macroscopic proton coefficient is shown for the case of Cd(II) adsorption onto a-A f (Figure 3). One variation of the surface coordination concept is used to describe the system subreactions the Triple Layer Model of Davis et al., (1,20). The specific subreactions which are considered, the formation constants and compact layer capacitances, are shown in Table IV. Protons are assigned to the o-plane (the oxide surface) and Cd(II) surface species and electrolyte ions to the 8-plane located a distance, 8, from the o-plane. 

First, assume that the surface charge on the membrane particles does not interact with the mobile protons (no proton release or uptake). An ion step will result in an increase in the double-layer capacitances of the particles and consequently in a decrease of the surface potentials fr, because the charge densities remain constant. The ISFET will measure a transient change in the mean pore potential. As a result of the potential changes, an ion redistribution will take place and the equilibrium situation is re-established. The theoretical maximum ion step response is the change in the mean pore potential. This is comparable with the Donnan model where the theoretical maximum is determined by the change in the Donnan potential at the membrane solution interface. 

The model parameters are the capacitances Q and Cj (as in the TLM), the intrinsic protonation and complexation constants, and f. The model has shown good predictive capabilities, as shown in Figure 12.15, where data from different authors (Hiemstra and Van Riemsdijk 1996 Bowden et al. 1980) are seen to fit well by the CD-MUSIC model with the same parameter values. 

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