Helmholtz condenser model

Further, we assume that the surface forming molecules, represented hy their net dipole moments are oriented in two separated layers like a condenser, which is identical to the Helmholtz condenser model (Fig. 2.3). From elementary physics it is well known that the potential difference AV of this condenser is given by the charge density separated plates and the dielectric constant e of the medium inside the condenser. 

The fundamental drawback of the classical Helmholtz picture is its inability to provide any explanation for the dependence of the electrode charge (or of the capacitance) on the electrolyte concentration, which is observed experimentally in sufficiently dilute solutions. Besides, there is a large group of electrokinetic phenomena that demonstrate a much greater extension of the EDL than it is assumed in the Helmholtz condenser model (see Sect. 5.4). 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

The Parallel-Plate Condenser Model The Helmholtz-Perrin Theory... 

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. 

Finally, as an illustrative exercise, let us calculate the variation of the Helmholtz free energy of the familiar parallel plate condenser model for two interacting colloidal particles. We choose coordinate x o sl to the surface of the condens6r plates, with x and x therefore lying in the plane of the plates at right angles to each other and normal to Xj. 

Fig. 6.62. The Helmholtz-Perrin parallel-plate model, (a) A layer of ions on the OHP constitutes the entire excess charge in the solution. (b) The electrical equivalent of such a double layer is a parallel-plate condenser, (c) The corresponding variation of potential is a linear one. (Note The solvation sheaths of the ions and electrode are not shown in this diagram nor in subsequent ones.)...

An important excess property is the excess Gibbs energy GE. Many models have been developed to describe and predict GE from the properties of the molecules in the mixture and their mutual interactions. GE models often refer to the condensed state, the solid and liquid phases. In case significant changes in the volume take place upon mixing, or separation, the Helmholtz energy A, defined as... 

As a typical example of CEDFT calculations, we present in Fig. 1 the capillary condensation isotherm of N2 in a cylindrical pore mimicking the pore channel in MCM-41 mesoporous molecular sieves. The isotherm is presented in co-ordinates adsorption N versus chemical potential p Calculations were performed at 77 K for the internal diameter of 3.3 nm up to the saturation conditions, point H. We used Tarazona s representation of the Helmholtz free energy [6] with the parameters for fluid-fluid and solid-fluid interaction potentials, which were employed in our previous papers [7]. We distinguish three regions on the isotherm. The adsorption branch OC corresponds to consecutive formation of adsorption layers. Note that the sharp transitions between the consecutive layers are not observed in experiments. They are caused by a well-known shortcoming of the model employed, which ignores intrinsic to real... 

The earliest theoretical studies of the behavior of an electrified interface were made by Helmholtz (1879). He discussed the adsorption of ions at a fixed double layer and he believed that this double layer formed the equivalent of a parallel-plate condenser. But this double layer model is an inadequate description of particles in electrolyte-containing systems. 

Three interface layers occur within the electrical or the diffuse double layer (DDL) of a clay particle the inner Helmholtz plane (IHP) the outer Helmholtz plane (OHP) with constant thicknesses of Xi and X2, respectively and third is the plane of shear where the electro kinetic potential is measured (Rg. 2.10). This plane of shear is sometimes assumed to coincide with the OHP plane. The IHP is the outer limit of the specifically adsorbed water, molecules with dipoles, and other species (anions or cations) on the clay solid surface. The OHP is the plane that defines the outer limit of the Stem layer, the layer of positively charged ions that are condensed on the clay particle surface. In this model, known as the Gouy-Chapman-Stera-Grahame (GCSG) model, the diffuse part of the double layer starts at the location of the shear plane or the OHP plane (Hunter, 1981). The electric potential drop is linear across the Stem layer that encompasses the three planes (IHP, OHP, and shear planes) and it is exponential from the shear plane to the bulk solution, designated as the reference zero potential. 

Historically, Frumldn-type models, which represent the Helmholtz region by a network of two or three condensers [449, 520-523] and classical thermodynamics based on a mean-field treatment [524-526], were apphed first to describe 2D phase transitions in organic adlayers at metal-electrolyte interfaces as a function of concentration, potential, and temperature (Sect. 33.2.2). In the simplest... 

In 1879, von Helmholtz proposed that all of the counterions are lined up parallel to the charged surface at a distance of about one molecular diameter (Figure 10.5). The electrical potential decreases rapidly to zero within a very short distance from the charged surface in this model. Such a model treated the electrical doublelayer as a parallel-plate condenser, and the calculations of potential decay were based on simple capacitor equations. However, thermal motion leads to the ions being diffused in the vicinity of the surface, and this was not taken into account in the Helmholtz model. 

In the GG model (Eigure 15.5a), the ions are not surface adsorbed in a condensed layer as considered by Helmholtz, but remain in solution because of their thermal motion. At equilibrium, the ion concentration profiles can be described in first approximation by the Boltzmann distribution. 

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