Diffuse-layer, model

Fig. 20.8 Gouy-Chapman diffuse layer model of the double layer...

Fig. 15 Two of the simplest theories for the dissolution of solids (A) the interfacial barrier model, and (B) the diffusion layer model, in the simple form of Nemst [105] and Brunner [106] (dashed trace) and in the more exact form of Levich [104] (solid trace). c is the concentration of the dissolving solid, cs is the solubility, cb is the concentration in the bulk solution, and x is the distance from the solid-liquid interface of thickness h or 8, depending on how it is defined. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., from Ref. 1, p. 478.)...

The diffusion layer model satisfactorily accounts for the dissolution rates of most pharmaceutical solids. Equation (43) has even been used to predict the dissolution rates of drugs in powder form by assuming approximate values of D (e.g., 10 5 cm2/sec), and h (e.g., 50 pm) and by deriving a mean value of A from the mean particle size of the powder [107,108]. However, as the particles dissolve, the wetted surface area, A, decreases in proportion to the 2/3 power of the volume of the powder. With this assumption, integration of Eq. (38) leads to the following relation, known as the Hixon-Crowell [109] cube root law ... 

To be useful in modeling electrolyte sorption, a theory needs to describe hydrolysis and the mineral surface, account for electrical charge there, and provide for mass balance on the sorbing sites. In addition, an internally consistent and sufficiently broad database of sorption reactions should accompany the theory. Of the approaches available, a class known as surface complexation models (e.g., Adamson, 1976 Stumm, 1992) reflect such an ideal most closely. This class includes the double layer model (also known as the diffuse layer model) and the triple layer model (e.g., Westall and Hohl, 1980 Sverjensky, 1993). 

The Gouy Chapman diffuse layer model has been shown to describe adequately the electrostatic potential produced by charges at the surface of the membrane [137]. For a symmetrical background electrolyte, a and i// are related by ... 

Combustion, 27 189, 190 reaction, sites for, 33 161-166 reaction scheme, 27 190, 196 Commercial isomerization, 6 197 CoMo catalysts, 40 181 See also Cobalt (nickel)-molybdenum-sulfide catalysts Compact-diffuse layer model, 30 224 Compensation behavior, 26 247-315 active surface, 26 253, 254 Arrhenius parameters, see Arrhenius parameters... 

Diffusion-Layer Model Let us consider again the general electrochemical reaction (6.6). Initially, at time before electrolysis, the concentration of the solution is homogeneous at all distances x from the electrode, equal to the bulk concentration of reactant Ox. In a more rigorous consideration, one would say that the concentration of the solution is homogeneous up to the outer Helmholtz plane (OHP), that is, up to x = xqhp-When a constant current is applied to the test electrodes and counterelectrodes such that the reaction... 

Nernst Diffusion-Layer Model This model assumes that the concentration of Ox has a bulk concentration up to a distance 8 from the electrode surface and then falls off linearly to Ox x = 0) at the electrode (neglecting the double-layer effect). The Nernst diffusion-layer model is illustrated in Figure 6.11. 

The earliest models used to describe the distribution of charges in the edl are, besides the Helmholz model, the Gouy-Chapman diffuse layer model and the Stem-Graham model. Details of these models are given in Westall and Hohl (1980), Schindler (1981, 1984) and Schindler and Stumm (1987). 

DDI DISSOL DLM DLVO DOC DOE DTA DU Distilled, deionized water Thermodynamic simulation model Diffuse layer model named after Derjaguin, Landau, Vervey, Overbeek Dissolved organic carbon Department of Energy Differential thermal analysis Depleted uranium... 

After the initial jubilation that the diffuse-layer model has overcome the weakness of constant capacity with change of potential of the parallel-plate model, one has to... 

In many cases, a less exact treatment suffices, e.g. to obtain the mean concentrations mentioned above. With the diffusion layer model, eqns. (144) are replaced by... 

Diffusion-Layer Model. Let us consider again the general electrochemical reaction (6.6). Initially, at time /0, before electrolysis, the concentration of the solution is... 

Fortunately, in most cases, the salt form under serious consideration exhibits a faster dissolution rate than the corresponding parent drug at an equivalent pH. This dissolution phenomenon can be explained in light of the parameters that govern the dissolution rate, as found in the diffusion layer model of Brunner (1904) ... 

FIGURE 17.1 (a) Diffusion-layer model of dissolution, (b) Interfacial barrier model of dissolution. 

Two of the simplest theories to explain the dissolution rate of solutes are the interfacial barrier model and the diffusion-layer model (Figures 17.1 and 17.2). Both of these theories make the following two assumptions ... 

On the other hand, as the Nernst diffusion layer model is applied to an unstirred solution, it is expected that the passage of current will cause formation of the depletion layer (Fig. 7.1), whose thickness 5o will increase with time. In time, this layer will extend from the electrode surface to the bulk of the solution over tens of pm. In order to estimate the time-dependence of So, we can use the approximate Einstein... 

FIGURE 6 Nernst diffusion-layer model. The solid line represents the actual concentration profile, and the dashed line for c0 the Nernst model concentration profile. 

With FITEQL numeric procedure Hayes et al. fitted edl parameters to the three models of electric double layer DLM (diffuse layer model), CCM (constant capacity model) and TLM (three layer model) for the following oxides a-FeOOH, AI2O3 and TiC>2 in NaNC>3 solutions [51]. The fitting was performed for surface reaction constants, edl capacity and the densities of the hydroxyl groups on the surface of the oxides. The quality of the fitting was evaluated by the minimization of the function of the sum of the square deviations of the calculated value from the standard error of measured charge. The lower value of the function the better was the fit... 

Early studies in this field of research formulated two main models for the interpretation of the dissolution mechanism the diffusion layer model and the... 

Although the diffusion layer model is the most commonly used, various alterations have been proposed. The current views of the diffusion layer model are based on the so-called effective diffusion boundary layer, the structure of which is heavily dependent on the hydrodynamic conditions, fn this context, Levich [102] developed the convection-diffusion theory and showed that the transfer of the solid to the solution is controlled by a combination of liquid flow and diffusion. In other words, both diffusion and convection contribute to the transfer of drug from the solid surface into the bulk solution, ft should be emphasized that this observation applies even under moderate conditions of stirring. 

Figure 5.2 Schematic representation of the dissolution mechanisms according to (A) the diffusion layer model, and (B) the interfacial barrier model.

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