Simple Helmholtz layer

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. 

As a result of these experiments Smith concludes that (a) the simple Helmholtz theory of the double layer is insufficient to account for all the observed facts. The potential difference mercury-electrolyte is not purely electrostatic, but depends on the nature of the ions, as, according to Nemst s theory, it should do. This theory, it will be remembered, involves the " solution pressure of the ions, which varies with their chemical nature. (6) The potential difference mercury-electrolyte is not necessarily zero when the interfacial tension is a maximum, although in the particular case of dilute KC1 this condition is very nearly fulfilled. 

Consider a simple interfacial region at a mercury/solution interface. The electrolyte is 0.01 M NaF and the charge on the electrode is 10 iC negative to the pzc. The zeta potential is -10 mV on the same scale. What is the capacitance of the Helmholtz layer and that of the diffuse layer Galculate the capacitance of the interfaces. Take the thickness of the double layer as the distance between the center of the mercury atoms and that of hydrated K+in contact with the electrode through its water layer. (Bockris)... 

Experimental attempts to verify the dependence of the transfer coefficient on the electrode potential have been made with simple outer sphere redox electrode reactions (see refs. 5—19 in ref. 70a). Corrections to experimental values of the apparent transfer coefficient due to double layer effects are performed by the use of eqn. (109), but the value of a calculated from experimental data depends on the assumptions about the location of the centre of charge in the transition state in the Helmholtz layer [70b]. 

The electron transfer reactions at the semiconductor/electrolyte interface occur either via the conduction band or the valence band. The total current is therefore given by the sum of four partial currents, denoted as represent electron transfer via the conduction anc valence bands, respectively, and the superscripts, a and c, indicate anodic anc cathodic processes, respectively. Let us assume nereafter that the electron transfer occurs only via the conduction band. In a simple case where the concentration of the electrolyte is sufficiently high and only the overvoltages at the Helmholtz layer (tjh) and in the space charge layer (rjsc) are important, the ica and cc can be given as follows4)... 

These terms are based on a simple geometric model of the interface. One distinguishes between an inner and an outer Helmholtz layer. The inner Helmholtz layer comprises all species that are specifically adsorbed on the electrode surface. If only one type of molecule or ion is adsorbed, and they all sit in equivalent positions, then their centers define the inner Helmholtz plane. The outer Helmholtz layer comprises the ions that are closest to the electrode surface, but are not specifically adsorbed. They have kept their -> solvation spheres intact, and are bound only by electrostatic forces. If all these ions are equivalent, their centers define the outer Helmholtz plane. 

The most simple model for the distribution of the charges at the electrode-electrolyte interface consists of an electrode surface layer with excess electrons or ions and a layer of an equivalent number of ions in the electrolyte. This layer of ions has a thickness 6 corresponding to the radius of the ions. It is called the Helmholtz layer. 

Figure 29. Calculated current-potential characteristics for direct (dashed lines, 0/cm ) and surface state mediated electron transfer between an -type semiconductor electrode and a simple redox system. The plots show the transition from ideal diode behavior to metallic behavior with increasing density of surface states at around the Fermi-level of the solid (indicated in the figures). This is also clear from the plots below, which show the change of the interfacial potential drop over the Helmholtz-layer (here denoted as A(Pfj) with respect tot the total change of the interfacial potential drop (here denoted as A(p). Results from D. Vanmaekelbergh, Electrochim. Acta 42, 1121 (1997).

As a result, for sufficiently concentrated solutions, about 1 M or higher, the diffuse-layer contribution in Eq. (20) represents in most cases only a minor correction to the compact-layer term. Then, one can even use the simple Helmholtz model for qualitative interpretation of the data. In particular, far from the p.z.c., this approximation is acceptable for aU concentrations. On the contrary, the diffuse... 

If a Helmholtz layer does not have charges, potential drops in the two diffuse layers have a simple relation ... 

In most realistic situations, however, deviations from the ideal behavior, described in Eqs. 2-5, exist. Typically, semiconductors exhibit surface states of varying density and, for increased doping levels, the total CPD can be distributed more evenly between the semiconductor band bending and the potential drop in the Helmholtz layer. In that case, the simple relation between current density and voltage (Eq. 5) does not hold anymore. A schematic for Fermi level pinning at the electrolyte interface is shown in Fig. 4. 

The water molecules sometimes contain the specifically adsorbed anions. The water molecules form the inner Helmholtz layer. The line drawn through the center of these molecules is called the inner Helmholtz plane. The outer Helmholtz plane pz (OHP) represents the locus of the electrical centers of the positive charges. This plane resides at a fixed distance from the metal due to the water molecules that are between the surface of the metal and ions. The outer Helmholtz plane (OHP) is significantly affected by hydrated cations (hydrated). In the simple model of the Helmholtz double layer developed earlier, the adsorption of dipoles was not considered. When a metal surface has a slight excess charge, the dipoles are adsorbed. This process contributes significantly to the potential difference across the double layer. Two factors seriously affect the potential difference across the double layen... 

In simple salt solutions, metal ions are present in bulk solution as hydrated ions the metal ion when hydrated is represented as M(H20)/, where x is the number of water molecules in the primary hydration sheath. The reactions involved in the discharge process (Raub and Muller, 1967) of ions under the influence of an electric field are the transport of hydrated ions towards the cathode surface, the alignment of water molecules in the diffusion layer, the removal of water molecules in the Helmholtz layer, discharge followed by adsorption of the ions at the cathode surface as adatoms , surface diffusion and the incorporation of adatoms into the crystal lattice at the growth point. A schematic representation of these steps is given in Fig. 5.1. 

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [120], Gouy and Chapman [121,122], and Stern [123], and in most analytic studies, the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g. as a simple dipolar fluid. Computer simulations... 

In the simple case of electrostatic attraction alone, electrolyte ions can approach to a distance given by their primary solvation sheaths, where a monomolecular solvent layer remains between the electrode and the solvated ions. The plane through the centres of the ions at maximum approach under the influence of electrostatic forces is called the outer Helmholtz plane and the solution region between the outer Helmholtz plane and the electrode surface is called the Helmholtz or compact layer. Quantities related to the outer Helmholtz plane are mostly denoted by symbols with the subscript 2. 

PMS stars with M < 0.35 M0 have a simple structure - they are fully convective balls of gas all the way to the ZAMS. As the star contracts along its Hayashi track the core heats up, but the temperature gradient stays very close to adiabatic except in the surface layers. Li begins to burn in p, a reactions when the core temperature, Tc reaches c 3x 106 K and, because the reaction is so temperature sensitive (oc Tc16-19 at typical PMS densities) and convective mixing so very rapid, all the Li is burned in a small fraction of the Kelvin-Helmholtz timescale (see Fig. 1). 

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

---