Boltzmann distribution, Gouy-Chapman theory

We use the Gouy-Chapman theory for the diffuse layer which is based on the Poisson-Boltzmann (P.B.) equation for the potential distribution. Although the different corrections to the P.B. equation in double-layer theory have been investigated (20, 21, 22, 23), it is difficult to state precisely the range of validity of this equation. In the present problem the P.B. equation seems a reasonable approximation at 0.1M of a 1-1 electrolyte to 50mV for the mean electrostatic potential pd at the ohp (24) this upper limit for pd increases with a decrease in electrolyte concentration. All the values for pd calculated in Tables I-IV are less than 50 mV— most of them are well below. If n is the volume density of each ion type of the 1-1 electrolyte in the substrate, c the dielectric constant of the electrolyte medium, and... 

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

A commonly used model for describing counterion distribution at a charged surface is based on the Gouy-Chapman diffuse double-layer (DDL) theory. This model assumes that the surface can be visualized as a structurally featureless plane with evenly distributed charge, while the counterions are considered point charges in a uniform liquid continuum. In this simplified picture, the equilibrium distribution of counterions is described by the Boltzmann equation ... 

Let us now examine how the potential cp changes within the diffuse part of the EDL, assuming that cp=0 in the bulk of the dispersion medium. The theory describing this part of the EDL was developed by Gouy and Chapman, who compared the energy of the electrostatic interaction of the ions with the energy of their thermal motion, assuming that the concentration of ions in the EDL was consistent with the Boltzmann distribution ... 

There have been considerable efforts to move beyond the simplified Gouy-Chapman description of double layers at the electrode-electrolyte interface, which are based on the solution of the Poisson-Boltzmann equation for point charges. So-called modified Poisson-Boltzmann (MPB) models have been developed to incorporate finite ion size effects into double layer theory [61]. An early attempt to apply such restricted primitive models of the double layer to the ITIES was made by Cui et al. [62], who treated the problem via the MPB4 approach and compared their results with experimental data for the more problematic water-DCE interface. This work allowed for the presence of the compact layer, although the potential drop across this layer was imposed, rather than emerging as a self-consistent result of the theory. The expression used to describe the potential distribution across this layer was... 

This simple idea of Helmholtz, that a layer of ions from the solution becomes attached to the surface, was modified in 1910 by the French physicist Georges Gouy (1854-1926) and in 1913 by the British chemist David Leonard Chapman (1869-1958). These workers pointed out that the Helmholtz theory is unsatisfactory in neglecting the Boltzmann distribution of the ions. They suggested that on the... 

FIGURE 3.15 Dimensionless mean electrostatic potential (a) and surface-ion distribution function (b) as predicted by the Gouy-Chapman-Stern (GCS) and modified Poisson-Boltzmann (MPB) theories for a 1 1 electrolyte with a = 0.425 nm and c = 0.197 M. (Outhwaite, Bhuiyan, and Levine, 1980, Theory of the electric double layer using a modified Poisson-Boltzmann equation. Journal of the Chemical Society, Faraday Transactions 2 Molecular and Chemical Physics, 76, 1388-1408. Reproduced by permission of The Royal Society of Chemistry.)... 

The derivation of the Debye— Hiickel equation for the activity coefficient is based on the linearized Boltzmann equation for electrostatic charge distribution around an ion. This limits the applicability of Eq. (57) to solutes with low surface potentials, which occurs for solution concentrations of monovalent ions of < 0.01 M. However, it is important to note that die method used for deriving activity coefficient equation (25) is based on rigorous thermodynamics and is not limited by the Debye—Huckel theory. If, for example, the Gouy—Chapman equation [22] was... 

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