Parallel plate condenser

In the case of a charged particle, the total charge is not known, but if the diffuse double layer up to the plane of shear may be regarded as the equivalent of a parallel-plate condenser, one may write... 

A number of more or less equivalent derivations of the electrocapillary Eq. V-49 have been given, and these have been reviewed by Grahame [113]. Lippmann based his derivation on the supposition that the interface was analogous to a parallel-plate condenser, so that the reversible work dG, associated with changes in area and in charge, was given by... 

Two parallel plates of conducting material separated by an insulation material, called the dielectric, constitutes an electrical condenser. The two plates may be electrically charged by connecting them to a source of direct current potential. The amount of electrical energy that can be stored in this manner is called the capacitance of the condenser, and is a function of the voltage, area of the plates, thickness of the dielectric, and the characteristic property of the dielectric material called dielectric constant. 

Let us consider now the amount of work required in the familiar problem of the charging of a parallel-plate condenser. Between the plates of the condenser there may be either a vacuum or a dielectric we shall first consider the case of a vacuum. With insulated parallel plates A and B we may imagine that electrons removed from A are conveyed across the gap to the parallel plate B, so that A acquires a positive and B a negative charge. When, at any stage of this process, a field of intensity X has been set up between the plates, the work required to convey the next installment of charge dq across the gap will be XI dq, if l is the distance between the plates. Thus the work to set up charges +g and -ij on the plates is... 

Two Types of Process Contrasted. Although the discussion of a parallel-plate condenser has been useful, it will be obvious, if Fig. 6 is compared with Fig. 1, that in the processes A, B, and C of Fig. 6 we have... 

The simplest structure of the double layer is the surface charge in one plane and the counter charge in a similar parallel plane. Then, to a first approximation, the double layer may be visualized as a parallel plate condenser of distance d between the two plates and with its capacitance, C... 

The electrostatic potential between a solid and an aqueous solution can be esplained in terms of a parallel plate condenser with a jiositiye excess charge on one phase and a negative excess charge on the other. The interfacial charge on the solid (electronic conductor) is usually carried by mobile excess electrons and holes, while it is carried by mobile excess hydrated ions on the side of aqueous solution (ionic conductor). 

A simple parallel plate condenser model (Fig. 5-12) gives the electric capacity Ch of the compact double layer as shown in Eqn. 5-8 ... 

In a range of potential where the interfacial charge is relatively small, the electric capacity, Cm, due to the electron tailing from the metal side is, to a first approximation, represented hy the capacity of a parallel plate condenser of thickness Xim as shown in Eqn. 5-30 ... 

Thus, according to this model, the interphase consists of two equal and opposite layers of charges, one on the metal ( m) the other in solution (q ). This pair of charged layers, called the double layer, is equivalent to a parallel-plate capacitor (Fig. 4.5). The variation of potential in the double layer with distance from the electrode is linear (Fig. 4.4). A parallel-plate condenser has capacitance per unit area given by the equation... 

Computers in electrochemistry, 1159, 1162 robotization to control experiments. 1162 pattern recognition analysis, 1162 Condenser, 1117 capacitance of. 861 model of parallel-plate, 873, 875, 961... 

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

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.)...

It appears that an electrified interface does not behave like a simple double layer. The parallel-plate condenser model is too naive an approach. Evidently some crucial secrets about electrified interfaces are contained in those asymmetric electrocapillaiy curves and the differential capacities that vary with potential. One has to think again. 

Fig. 6.80. A dipole layer is electrically equivalent to a parallel-plate condenser.

The electrical interaction with the field is a matter of the work of taking a charged ion through a distance Xj — jq, i.e., from the OHP to the IHP. The electrical field, X, in a parallel-plate condenser is q /EEq. From Eq. (6.20), the difference in energy in bringing a test charge z e0 from jq to x, in this electrical field is... 

The surfaces being considered are not planar, and therefore instead of Helmholtz-Perrin parallel-plate condensers, one has concentric-sphere capacitors Gouy-Chapman regions show radial instead of planar symmetry. All such points complicate the mathematics, but lead to few new truths. Hence, such details will be ignored in this very simple account of the dominating role of double layers in colloid chemistry. 

The charge in the diffuse layer can be considered equivalent to the Gouy charge density qd placed at a distance K-1 from the OHP. This gives rise to a parallel-plate condenser model. The potential at one plate—deep in the solution side—is taken at zero, while the potential at the other plate—which coincides with the OHP—is [f0. This latter potential is often referred to in the study of electrokinetic phenomena as the zeta ( ) potential. Thus,... 

The potential across a parallel-plate condenser is 100 V. If the distance between plates is 1 cm, calculate the charge of the plates of the condenser if (a) there is vacuum between the plates and (b) the dielectric constant between the plates is 80. (Gamboa-Aldeco)... 

Another electrical method involves an analysis of the current voltage characteristics of a parallel plate condenser through which the aerosol is passed continuously (2N). 

Surface film potential measurements can yield useful, if not absolute, information about the orientation of the film molecules. Treating the film as a parallel plate condenser leads to the approximate expression... 

From equation (7.9) it can be seen that, at low potentials, a diffuse double layer has the same capacity as a parallel plate condenser with a distance 1/k between the plates. It is customary to refer to 1/k (the distance over which the potential decreases by an exponential factor at low potentials) as the thickness of the diffuse double layer. 

In studying systems in an electrostatic field, we must consider two systems because of the dependence of the field on matter within the field. One system is a parallel-plate condenser in empty space. The area of the plates is designated by A, and the distance between the plates by /. The other is an identical condenser immersed in an isotropic, homogenous, dielectric medium. The conductivity of the medium is zero, so no free charges are present in the medium. Edge effects are neglected and rational units are used throughout. 

---