Constant-capacitance

Dielectric Constant. Capacitance, C, refers to the abiHty of two conductors to store charge, in the presence of a potential difference, V ... 

Fig. 20.7 Differential capacitance/mercury electrode potential relationships for potassium chloride at different concentrations showing (a) how minima are obtained only at low concentrations and (6) the constant capacitance at negative potentials (after Bockris and Drazic )...

It is possible on the basis of this model (arrangement O) to explain the constant capacitance region on the negative side of the C vs. E curve (Fig. 20.7), and why the capacitance in this region is independent of the nature of the cations in the solution. The model of the double layer is shown in Fig. 20.12 in which it can be seen that the surface of the electrode and the... 

The division of the interface into an inner layer and a diffuse layer has been a matter of discussion in view of the molecular dimensions of the inner layer.122-126,279-285 However, the contribution of a constant capacitance is an experimental fact. Furthermore, molecular theories for electrolytes near a charged hard wall282 as well as phenomenological nonlocal electrostatic theories283 predict such a component without artificial introduction of any inner layers. This turns out to be an effect of the short-range structure of the solvent.279-285... 

Goldberg, S. Reanalysis of boron adsorption on soils and soil minerals using the constant capacitance model. Soil Sci Soc Am J 1999 63 823-829. 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

A. Poghossian, M.H. Abouzar, F. Amberger, D. Mayer, Y. Han, S. Ingebrandt, A. Offenhauser, and M.J. Schoning, Field-effect sensors with charged macromolecules characterisation by capacitance—voltage, constant capacitance, impedance spectroscopy and atomic-force microscopy methods. Biosens. Bioelectron. 22, 2100-2107 (2007). 

Alternatively, in the literature, the constant capacitance model and the Stern model were used to describe the dependence of the surface charge density on the surface potential. In the constant capacitance model, the surface charge is defined as ... 

Vaughan PJ, Suarez DL. Constant capacitance model computation of boron speciation for varying soil water content. Vadose Zone J. 2003 2 253-258. 

A Simplified Double Layer Model (Constant Capacitance)... 

The diffuse double layer model is used to correct for Coulombic effects. The constant capacitance model depends on the input of a capacitance but the result obtained is not very different. 

In addition to the diffuse double layer and the constant capacitance model dis-... 

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. 

The term F2/CsRT is obtained from the constant capacitance model (Chapter 3.7). Fig. 4.6 gives a plot of the linear free energy relation between the rate constants for water exchange and the intrinsic adsorption rate constant, kads. 

Two models of surface hydrolysis reactions and four models of the electrical double layer have been discussed. In this section two examples will be discussed the diprotic surface group model with constant capacitance electric double layer model and the monoprotic surface group model with a Stern double layer model. More details on the derivation of equations used in this section are found elsewhere (3JL). ... 

Diprotic Surface Groups. Most of the recent research on surface hydrolysis reactions has been interpreted in terms of the diprotic surface hydrolysis model with either the triple layer model or the constant capacitance model of the electric double layer. The example presented here is cast in terms of the constant capacitance model, but the conclusions which are drawn apply for the triple layer model as well. 

An example of the use of this method with the constant capacitance model on the data for TiC>2 in 0.1 M KNO is illustrated in Figure 6. It appears from the figure that the problem is perfectly well determined, and that unique values of Ka and Ka2 can be determined. However, as is shown below, the values of Ka and Ka2 determined by this method are biased to fulfill the approximations made in processing the data (i) on the acidic branch, nx+, nx nx-, which yields a small value for Ka2, and (ii) on the basic branch, nx-, nx nx+, which yields a large value of Ka. Thus the approximation used to find values for Qa and Qa2 leads to values of Ka and Ka2 consistent with the approximation of a large domain of predominance of the XOH group. This constraint arose out of the need for mathematical simplicity, not out of any physical considerations. 

The parameters in the chemical and electrostatic models which appear in Equation 27 can be related to the experimentally observable slope of the avs. log a + curve by combining Equation 27 with Equation 25 and the constant capacitance constraint CTqH = C t/>q to yield ... 

Figure 7. Covariability between values of C and Kd yielding best fit of diprotic surface hydrolysis model with constant capacitance model to titration data for TiC>2 in 0.1 M KNOj (Figure 5). The line is consistent with Equation 29. The crosses represent values of C and log found from a nonlinear least squares (NLLS) fit of the model to the data, with the value of capacitance imposed in all cases the fit was quite acceptable. The values of and C found by Method I (Figure 6) also fall near the line consistent with Equation 29. The agreement between these results supports the use of the linearized model (Equation 29) for developing an intuitive feel for surface reactions.

Here it has been shown that the conclusion about is related to the mathematical approximation used in interpreting the data by way of Method I it is easy to show that the second and third conclusions are also dependent on the initial assumptions in Method I. Rearrangement of Equation 28 and substitution of the constant capacitance constraint yields a relationship between i/>q and In a + ... 

The discussion above pertains to the diprotic acid chemical model and the constant capacitance electrostatic model. It is interesting to note that in some applications of the triple layer model with site binding of electrolyte ions at the IHP, the... 

Equilibrium Calculations. The computer program SURFEQL (29) was used to calculate the equilibrium distrubution of chemical-species. The constant capacitance model (30, 1) was used for the surface equilibria calculations. The equilibrium constants used in these calculations are given in Davies (26). 

As shown in Figure 1, the adsorption of Mn(II) on y-FeOOH can be successfully described using a constant capacitance model. In these calculations the hydrolysed surface complex =FeO-Mn-OH was not considered. The reason for not considering both the bidentate (sS0)2Mn and hydrolysed surface species is that both have virtually the same pH dependence, so it is impossible using the available data to make anything other than an arbitrary choice about the relative proportions of these two species. Based on the model calculations, in the pH range 8-9, the predominant Mn(II) species on the y-FeOOH surface is the bidentate surface complex or the hydrolysed surface complex. 

Figure 1. Mn(II) adsorption as a function of pH. The solid lines are calculated using the constant capacitance model.

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