Diffusion Nernst 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. 

Figure 8.13. Scheme of the Nernst diffusion iayer model at an electrode surface in the presence of US. (Reproduced with permission of Elsevier, Ref. [133].)... 

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. 

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

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

This can be illustrated in the example of a mechanical stirring device (forced convection) when used for homogenising the liquid electrolyte, mainly away from the zones that are next to the interfaces. The same case applies to analytical chemistry when a rotating disc electrode is involved or when a system is installed in industrial electrolysers in order to force the circulation of the electrolyte. By following a simplified model called the Nernst model one can define the thickness of the diffusion layer (often denoted by 5)... 

According to the Nernst model, the concentration profile of the electroactive species within the diffusion layer is described by the following system of equations ... 

A steady state with a non-zero current is reached very quickly in these forced convection conditions. The interfacial slope is identical to that in the Nernst model as shown in figure 4.20. The Nernst layer thickness is lower than that of the diffusion layer since, at that distance from the electrode, the actual profile still has a significant slope with a relative concentration difference of 11% with respect to the initial concentration. The diffusion layer thickness is equal to 1.5 for a target accuracy of 1% and to 1.8 NERNsyfor a target accuracy of 0.1%. 

Figure A.27 - Steady-state concentration profiles for Fe (grey) and Fe (black) for a diffusion-convection mode according to the Nernst model...

Chronopotentiometry with diffusion-convection according to the Nernst model 311 Chronoamperometry with steady-state unidirectional diffusion. 312... 

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. 

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.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... 

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. 

Both the semi-infinite diffusion and Nernst s steady diffusion layer concept produce the same results when the diffusion front shift is not yet large as compared to In these conditions, the semi-infinite diffusion model, which is simpler from the mathematical point of view, is to be applied. However, the Nernst model is more general because the regularities of the first model follow from it at 5j.j oo. 

Though in the general case, mathematical expressions of the Nernst model are more complicated than of those semi-infinite diffusion, stationary mass transport is described by a rather simple Eq. (3.12). In this connection, there occurs an interesting possibility to use superposition of both models, which is convenient to apply when i is the periodic time function. Perturbation signals of this type are considered in the theory of electrochemical impedance spectroscopy. In this case, i(t)... 

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 ... 

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