The Gouy-Chapman Equation

In UnderstandingVoltammetry, the problem of the diffuse double layer as conceived in the Gouy-Chapman theory is posed as  

the flux /, of any species i is constant through space. Since for an ideally polarisable electrode, the flux of any species into or out of the electrode must be zero (since the species does not absorb, adsorb or react at this boundary), Ji = 0 everywhere. Therefore from the Nernst-Planck equation 

is non-zero in bulk, we can assume that it is not zero anywhere, and therefore it is valid to divide through both sides of the equation by c,  

The elevation of the concentration of negative ions at an electrode of positive potential occurs because the positive charge on the electrode attracts anions and repels cations, thus forming an equilibrated double layer where the net negative charge on this layer exactly cancels the positive charge on the electrode. 

The exponential nature of the accumulation of oppositely charged ions means that Gouy-Chapman theory predicts local concentrations which are much larger than that in bulk if the potential difference applied between the electrode and the solution greatly exceeds RT/F. In practice, the Stern layer of adsorbed solvent molecules and the specific adsorption of ions mediates the potential perceived by the diffuse component of the double layer to a great extent. 

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

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. 

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 ... 

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 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). 

And substitution of (8.89) into the Gouy-Chapman equation gives the function of the surface potential cp upon the surface charge density o in fatty acid monolayer and the packing density of molecules at different ionic concentration in subphase ... 

We remark that in Eq.(1.63) (and also in Eq.(1.64), if the fluctuation terms, (2) and 0i(2) are neglected, then we get back the Gouy Chapman equation Eq.(1.50), which has a known analytical solution. In the BGY based theories the pair cor-relationfunction hy(l,2) must be given by some approximation. The interesting... 

Two potential improvements compared to the common practice are introduced. Both refer to the description of the diffuse layer. The commonly applied surface complexation models involve the Poisson-Boltzmann approximation for diffuse-layer potential of the electric double layer (resulting in the Gouy-Chapman equation for flat plates in most apphcations). 

Figures 21 and 22 show the influence of using the HNC approximation instead of the Gouy-Chapman equation for a purely difiuse-layer model. Figure 21 shows the charge potential relationship, which in the purely difiuse-double-layer model has a direct bearing on the ealeulation of surfaee-charge density (Fig. 22). Thus, it is apparent that even at low ionic strength, the influenee is quite strong, indicating that for this kind of model, the HNC approximation is preferable.

With multilayer models, the features in comparisons between the HNC approximation and the Gouy-Chapman equation were found to be similar to the results with the Stem model for the conditions tested. 

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