Nernst diffusion-layer model

Nernst Diffusion-Layer Model This model assumes that the concentration of Ox has a bulk concentration up to a distance 8 from the electrode surface and then falls off linearly to Ox x = 0) at the electrode (neglecting the double-layer effect). The Nernst diffusion-layer model is illustrated in Figure 6.11. 

On the other hand, as the Nernst diffusion layer model is applied to an unstirred solution, it is expected that the passage of current will cause formation of the depletion layer (Fig. 7.1), whose thickness 5o will increase with time. In time, this layer will extend from the electrode surface to the bulk of the solution over tens of pm. In order to estimate the time-dependence of So, we can use the approximate Einstein... 

FIGURE 6 Nernst diffusion-layer model. The solid line represents the actual concentration profile, and the dashed line for c0 the Nernst model concentration profile. 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

The simplest description may be obtained by applying the Nernst diffusion layer model (Bard and Faulkner, 1980) to this hydrodynamic situation. 

According to the Levich equation (4.105) the limiting current density for a rotating disk electrode is proportional to Cg. On the other hand, the Nernst model (equation 4.81) indicates that the limiting current is proportional to the product Cb Db- By combining these two equations we find that 5has a Db dependence. This result reveals the artificial character of the Nernst diffusion layer model. Every dissolved species that reacts in an electrochemical system has its own speeifie Nernst diffusion layer. 

We assume that the Nernst diffusion layer model is applicable and therefore within the diffusion layer only diffusion and migration contribute to mass transport. Equation (4.118) then gives for the cation flux N+ and the anion flux N at the electrode surface ... 

Fig. 4.8 - Concentration profile for the electroactive species using the Nernst diffusion layer model for transport to a rotating disc electrode.

It has already been noted that the flux of material to the rotating disc electrode is uniform over the whole surface, and it is therefore possible to discuss the mass transport processes in a single direction, that perpendicular to the surface (i.e. the z direction). Furthermore, it has been noted that the velocity of movement of the solution towards the surface, is zero at the surface and, close to the surface, proportional to Hence, even in the real situation it is apparent that the importance of convection drops rapidly as the surface is approached. In the Nernst diffusion layer model this trend is exaggerated, and one assumes a boundary layer, thickness 6, wherein the solution is totally stagnant and transport is only by diffusion. On the other hand, outside this layer convection is strong enough for the concentration of all species to be held at their bulk value. This effective concentration profile must, however, lead to the same diffusional flux to the surface (and hence current density) as it found in the real system. 

It is also clear from the previous section that the rate of convective diffusion to the disc is strongly dependent on the rotation rate of the disc, but this is readily taken into account in the Nernst diffusion layer model by noting that the stagnant layer thickness will decrease as the rotation rate is increased. In fact, a quantitative relationship has been deduced [3], i.e. 

Using the Nernst diffusion layer model, derive an equation which describes how the flux (/a) varies with overpotential for a one-electron reduction. 

In quiescent solution, a familiar peak-shaped voltammogram is seen as expected for a macro electrode under planar diffusion-only transport. In contrast, inso nation establishes a strong convective flow as a result of acoustic streaming such that mass transport to the electrode is enhanced significantly. Under the latter conditions, applying the simple Nernst diffusion layer model gives an expression for the limiting current ... 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

Along with a group of models that have shown themselves useful, their particular normalisations will be presented. The first model, the Cottrell system, will also serve to introduce the concept of the Nernst diffusion layer. 

This idealized model does not capture all of the essential details of corrosion deposits. As indicated in Fig. 17, the influence of local chemistry within the deposit (especially pH effects) is likely to separate the corrosion site (at the material/deposit interface) from the site at which the deposit forms (deposit/ environment interface). Consequently, diffusion processes within the porous deposit must be involved if corrosion is to be sustained. Under simple steady-state conditions, diffusion can be treated simply using the Nernst diffusion layer approach i.e., the flux, J, of a species dissolving in a pore will be given by... 

Also the choice of the electrostatic model for the interpretation of primary surface charging plays a key role in the modeling of specific adsorption. It is generally believed that the specific adsorption occurs at the distance from the surface shorter than the closest approach of the ions of inert electrolyte. In this respect only the electric potential in the inner part of the interfacial region is used in the modeling of specific adsorption. The surface potential can be estimated from Nernst equation, but this approach was seldom used In studies of specific adsorption. Diffuse layer model offers one well defined electrostatic position for specific adsorption, namely the surface potential calculated in this model can be used as the potential experienced by specifically adsorbed ions. The Stern model and TLM offer two different electrostatic positions each, namely, the specific adsorption of ions can be assumed to occur at the surface or in the -plane. 

In reality, as one moves away from the interface towards the bulk solution, the contribution of convection to transport increases while that of diffusion decreases. Rather than treating simultaneously transport by diffusion and convection, the Nernst model makes a clear separation between the two transport mechanisms a total absence of convection inside the Nernst diffusion layer (y < S), and an absence of diffusion outside the Nernst diffusion layer (y > S). The intensity of convection affects the flux at the electrode by fixing the thickness of the Nernst diffusion layer. For the remainder of this book, the Nernst diffusion layer will simply be called the diffusion layer. 

Fig.l Diagram of the Nernst Boundary Layer (cone, of reacting species vs. distance from the electrode) 1. Nernst model concentration profile, 2. true concentration profile, 6 = Nernst diffusion layer, 6 = true diffusion layer... 

Looking again at Eq. (2.8), is identical with the Nernst diffusion layer thickness. Unfortunately, this quantity is often discussed and used as if it had physical reality when as we have seen from Fig. 2.5 it is a mathematical artifact arising from a particular mass transfer model. For this reason diffusion layer thickness is not normally found in design equations but Eq. (2.8) is written in the form ... 

Plot, on an impedance plane format, the impedance obtained for a Nernst stagnant diffusion layer and the impedance obtained for a rotating disk electrode under assumption of an infinite Schmidt number. Show that, while the behaviors of the two models at high and low frequencies are in agreement, the two models do not agree at intermediate frequencies. Explain. 

Concentration Polarisation is the accumulation of solute due to solvent convection through the membrane and was first documented by Sherwood (1965). It appears in every pressure dri en membrane process, but depending on the rejected species, to a very different extent. It reduces permeate flux, either via an increased osmotic pressure on the feed side, or the formation of a cake or gel layer on the membrane surface. Concentration polarisation creates a high solute concentration at the membrane surface compared to the bulk solution. This creates a back diffusion of solute from the membrane which is assumed to be in equilibrium with the convective transport. At the membrane, a laminar boundary layer exists (Nernst type layer), with mass conservation through this layer described by the Film Theory Model in equation (3.7) (Staude (1992)). cf is the feed concentration, Ds the solute diffusivity, cbj, the solute concentration in the boundary layer and x die distance from the membrane. 

Interestingly, Shchukarev equation, which included the separation surface area and equated the concentration on the inner side of the diffusion layer to the solubility value, was published in April 1896. But it was not noticed abroad where the priority was ascribed to A. Noyes and W. Witney who published their equation without considering the area only by the end of 1897. In 1900 their error was corrected, and in 1904 Nernst expanded the upgraded equation on all heterogeneous processes. For this reason this model mass transfer subsequently was treated as Nernst-Noyes equation. 

Figure 5.2 Model of stationary diffusion in a stirred electrolyte. The Nernst-Brunner diffusion layer forms on an electrode surface of thickness because the viscosity of the electrolyte creates a gradient of convection between the bulk electrolyte and the electrode surface.

In hydrodynamic systems Planar diffusion to a uniformly accessible electrode, e.g. for rotating disk electrodes (hypothetical Nernst model with S = diffusion layer thickness)... 

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