Equation Gouy-Chapman

Several features of the behavior of the Gouy-Chapman equations are illus-... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Nakagaki1U) has given a theoretical treatment of the electrostatic interactions by using the Gouy-Chapman equation for the relation between the surface charge density oe and surface potential /. The experimental data for (Lys)n agrees very well with the theoretical curve obtained. 

Soils containing polyvalent cations having high valence and high electrolyte concentration have a high conductivity, whereas the soils containing monovalent cations, such as sodium, have a low k. Distilled water at the extreme end of the spectrum is free of electrolytes. In the Gouy-Chapman equation, the electrolyte concentration na would be 0. The denominator, therefore, would go to 0 and the T value to infinity. 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

The surface charge density of the diffuse part of the double layer is given by the Gouy-Chapman equation ... 

The applicability of Equation 11 improves when the rate of aggregation is not too slow, e.g., when the surface potentials (calculated from the modified Gouy Chapman equation (see Refs. 28 and 29)) are not too high. These are the only cases of interest for quantitative comparison with experiment, since detection becomes impossible when the rate of aggregation is too slow. We point out below how we estimate rates for other cases. 

Although each SCM shares certain common features the formulation of the adsorption planes is different for each SCM. In the DDLM the relationship between surface charge, diffuse-layer potential, d, is calculated via the Gouy-Chapman equation (Table 5.1), while in the CCM a linear relationship between surface potential, s, is assumed by assigning a constant value for the inner-layer capacitance, kBoth models assume that the adsorbed species form inner-sphere complexes with surface hydroxyls. The TLM in its original... 

Finally, it may be noted that, since lim, -, p = 0, o-Ac= may be related to through Eq, [3] (which reduces to the Gouy—Chapman equation when p = 0). Substituting this relation into [9] ... 

Drzymala and coworkers proposed the method of the calculation of surface hydroxyl group complexion constants, where the concentrations of the group from Eqs. (25) and (26) are calculated in the similar to Schindler s and Gamsjager way, whereas the surface potential value is determined from the Gouy-Chapman equation [111]. The logarithm of such obtained values of the constants, differs from that of obtained by Schindler s method by a few tenths of the pK unit. 

Complementary is the method of complexation constant calculations based on the adsorption measurements of the 1 1 background electrolyte. The density of this adsorption consists of = SOH An+ or = SO Ct+ complex adsorption and a part, connected with the compensation of the surface charge in the diffuse layer of edl. To estimate the densities, the ions adsorbed in IHP layer, Sprycha assumed the background electrolyte ion density located in the diffuse layer of edl equals to the diffuse layer charge that may be calculated from Gouy-Chapman equation, when potential value is known. Then, the density of the ions that form surface complexes will be equal to ... 

If all ions are assumed to have the same charge q, (i.e., a symmetric salt) and the colloid is treated as a flat and infinitely large surface in contact with an infinite salt reservoir, the PB equations can be solved and the solution is denoted the Gouy-Chapman equation. Under these conditions Eq. (4) can be rewritten as... 

The linearization of the PB equation is often called the Debye-Hiickel approximation and it is valid when qfo/kT < 1. At room temperature this corresponds to surface potentials, 0o, below 25 mV. In the case of flat surfaces and if symmetry is considered as in the Gouy-Chapman equation... 

The surface potential of a bilayer is a result of having charged lipids in this bilayer (for reviews see References 63 and 66). In fluid phase bilayers, rapid translational diffusion of the lipids allows the surface charge associated with the lipids to be considered a smeared charge and the electrostatic potential at the surface of the bilayer, J o, is well described by the Gouy-Chapman equation ... 

Figure 34. Determination of micellar charge from equilibrium and kinetic measurements. The decrement of micellar charge as a function of sodium dodecyl sulfate added to Brij 58 micelles was calculated from the pK shift according to Gouy-Chapman equation (/ = lOm/W) (A) or from the second-order rate constant of protonation using Debye s equation (Eigen etal., 1964) for rates measured in the presence of ionic screening (O) at/ = 10mM,or from rates extrapolated to / = 0 ( ) (Gutman et al., 1981a).

One basic difficulty with the Gouy-Chapman Theory in systems involving an impenetrable flat surface or electrode is that, since the ions have finite size, the distance of closest approach of their centers to the surface is finite. Thus, the potential which appears in the Gouy-Chapman equation is not equal to the surface potential A(0), but is the potential in the plane of closest approach of the counterions to the surface. 

In one respect Derjag.uin s method is superior-to Levine and. Dube s. It is possible to. apply the principle of Der-j a g u i n s method to the complete Gouy-Chapman equation, whereas in Levine s and Dube s treatment it is unavoidable to introduce the linear approximation of Debye and Huekel. The theory for large spherical particles, surrounded by a thin double layer may therefore be made almost as exact as that of flat plates, but in the case of small particles with an extended double layer the situation is less favourable. 

If we wish to apply the complete Gouy-Chapman equations instead of the linear approximation, we have only to replace the value of (/J — ) as it is given by equation (55)... 

In the Gouy-Chapman equation, the electrolyte ions are considered to be point charges that interact with... 

The adsorption of counterions at the plane d from the surface can be described by the Gouy-Chapman equation as ... 

Equation (4.37), referred to as the Gouy-Chapman equation, is valid for any value of the surface potential cjjQ. However, for c )o less than —50 mV, the difference in the cj) values found from the Debye-HUckel approximation and the Gouy-Chapman equation is insignificant (Fig. 4.11). 

Figure 4.11 Comparison of the predictions of the Debye-Huckel and Gouy-Chapman equations for the potential 4> in the electrical double layer. The Debye-Hiickel equation can be used with insignificant error up to —50 mV. (From Ref. 3.)...

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