The Gouy-Chapman theory

The limitations imposed on this theory, have been discussed (see for example Sposito, 1984). 

Equality of electrochemical potential, jl (= p + zFY) of every ion, regardless of position 

A simple but surprisingly good model for the metal-solution interface was developed by Gouy [1] and Chapman [2] as early as 1910. The 

To be specific we consider a planar electrode in contact with a solution of a z — z electrolyte (i.e., cations of charge number z and anions of charge number -z). We choose our coordinate system such that the electrode surface is situated in the plane at x = 0. The inner potential f (x) obeys Poisson s equation  

The ionic densities must in turn depend on the potential f x). We choose (j) po) = 0 as our reference, and apply Boltzmann statistics  

Strictly speaking the exponents should not contain the inner potential 4 but the so-called potential of mean force, but this subtlety is only important at high electrolyte concentrations and high potentials, where other weaknesses of this theory also become important. Substituting Eqs. (3.3) and (3.2) into Eq. (3.1) gives  

Ld = 1/A is the Debye length Table 3.1 shows values for several concentrations of a 1-1 electrolyte in an aqueous solution at room temperature. The solution compatible with the boundary condition f oo) = 0 has the form 4 (x) = Aexp(—kx), where the constant A is fixed by the charge balance condition  

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Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

To our knowledge this is quite a new formula for the differential capacitance. It is vahd whenever charging is equivalent to a shift in space of the position of the wall. We can verify that it is fulfilled for the Gouy-Chapman theory. One physical content of this formula is to show that for a positive charge on the wall we must have g (o-) > (o-) in order to have a positive... 

In Fig. 8 density profiles are presented for several values of charge density a on the wall and for the wall potential h = — and h= Fig. 9 contains the corresponding ionic charge density profiles. For the adsorptive wall potential h < 0) the profiles q z) in Fig. 9(a) and j (z) in Fig. 8(a) are monotonic, as in the Gouy-Chapman theory. For a wall which is neutral relative to the adsorption A = 0 the density profiles are monotonic with a maximum at the wall position. This maximum also appears on the charge... 

It is natural to consider the case when the surface affinity h to adsorb or desorb ions remains unchanged when charging the wall but other cases could be considered as well. In Fig. 13 the differential capacitance C is plotted as a function of a for several values of h. The curves display a maximum for non-positive values of h and a flat minimum for positive values of h. At the pzc the value of the Gouy-Chapman theory and that for h = 0 coincide and the same symmetry argument as in the previous section for the totally symmetric local interaction can be used to rationalize this result. 

A full mathematical treatment of the Gouy-Chapman theory and the derivation of these equations is given in Appendix B.)... 

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

According to the Gouy-Chapman theory, the surface charge densities on the aqueous and membrane sides, and o , respectively, can be expressed as... 

We recently synthesized several reasonably surface-active crown-ether-based ionophores. This type of ionophore in fact gave Nernstian slopes for corresponding primary ions with its ionophore of one order or less concentrations than the lowest allowable concentrations for Nernstian slopes with conventional counterpart ionophores. Furthermore, the detection limit was relatively improved with increased offset potentials due to the efficient and increased primary ion uptake into the vicinity of the membrane interface by surfactant ionophores selectively located there. These results were again well explained by the derived model essentially based on the Gouy-Chapman theory. Just like other interfacial phenomena, the surface and bulk phase of the ionophore incorporated liquid membrane may naturally be speculated to be more or less different. The SHG results presented here is one of strong evidence indicating that this is in fact true rather than speculation. 

When the ITIES is polarized with a potential difference 0, there is a separation of electrical charge across it. According to the Gouy-Chapman theory, the charges in the aqueous and organic diffuse layers are related to the potential drops and A0 in the respective layers by the equations... 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

The Gouy-Chapman theory relates electrolyte concentration, cation valence, and dielectric constant to the thickness of this double layer (see Equation 26.2). This theory was originally developed for dilute suspensions of solids in a liquid. However, experience confirms that the principles can be applied qualitatively to soil, even compacted soil that is not in suspension.5... 

Figure 2.10 Schematic representation of the variation of the excess, or nett, surface charge density as a function of the distance away from the electrode surface, according to the Gouy-Chapman theory. The distance of nearest approach of the ions, with their associated solvation...

Figure 2.11 According to the Gouy-Chapman theory, the capacity of the electrode/electrolyte interface should be a cosh function of the potential difference across it (see text). Concentration of electrolyte in (b) > than that in (a).

At low electrolyte concentrations, up to about a 10 3 M solution, the Gouy-Chapman theory agrees quite well with experimental values of... 

The Thomas-Fermi model of a metal is similar to the Gouy-Chapman theory for electrolytes. In this model the surface-charge density o is... 

The surface concentrations c x and cjed differ from those in the bulk even if the surface region and the bulk are in equilibrium. Using the same arguments as in the Gouy-Chapman theory, the surface concentration cs of a species with charge number z is ... 

Figure 12.3 Capacity of the interface between a solution of NaBr in water and TBAs/TPB in nitrobenzene. The upper points are for 0.1 M solutions, the lower for 10 2 M in both phases. The two curves have been calculated from the Gouy-Chapman theory. The sign convention for the potential is A = (py, — (po + const., where the index w stands for the aqueous and o for the organic phase. Data taken from Ref. 1.

On the whole, the Gouy-Chapman theory seems to work well for ITIES, indicating that any contribution from the dipole potential is small. In particular the interfacial capacity exhibits a minimum at the potential of zero charge for low electrolyte concentrations (see Fig. 12.3). 

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