Nernst layer approximation

If instead of semi-infinite diffusion, some distance (5m acts as an effective diffusion layer thickness (Nernst layer approximation), then a modified expression of equation (63) applies where ro is substituted by 1 / (1 /Vo + 1 /<5m ) (see equation (38) above). For some hydrodynamic regimes, which for simplicity, are not dealt with here, the diffusion coefficient might need to be powered to some exponent [57,58],... 

Using linear approximation known as the Nernst layer approximation, that is,... 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

Here C is the specific differential double layer capacitance. The two terms on the left side of Eq. (4) describe the capacitive and faradaic current densities at a position r at the electrode electrolyte interface. The sum of these two terms is equal to the current density due to all fluxes of charged species that flow into the double layer from the electrolyte side, z ei,z (r, z = WE), where z is the direction perpendicular to the electrode, and z = WE is at the working electrode, more precisely, at the transition from the charged double layer region to the electroneutral electrolyte. 4i,z is composed of diffusion and migration fluxes, which, in the Nernst-Planck approximation, are given by... 

FIGURE 2.3. Schematic representation of the substrate concentration profile (normalized with respect to bulk substrate concentration) as a function of distance from the surface of a rotating disk. The profile calculated via analytical solution of the convective diffusion equation, which is calculated from the Nernst diffusion layer approximation. Both calculations are shown. 

The steady state limiting /lim current The steady state limiting current can be used to estimate the inter-electrode gap based on the Nernst diffusion layer approximation 7/j = nFDAcjd where is the limiting current, n is the number of electrons, F is Faraday s constant, D is the diffusion coefficient, A is the electrode area, c is the concentration and 6 is the inter-electrode gap. 

The modeling of the current distribution in a general-geometry cell nearly always requires a numerical solution. The following discussion focuses on the thin boundary layer approximation, with the overpotential components lumped within a thin boundary layer which may be of a varying thickness. The Laplace equation for the potential with nonlinear boundary conditions must be solved. Similar considerations typically apply to the more comprehensive solution of the Nernst-Planck equation (10) however, the need to account for the convective fluid flow in the latter case makes the application of the boundary methods more complex. We focus our brief discussion on the most common methods the finite-difference method, the finite-element method, and the boundary-element method, schematically depicted in Fig. 4. Since the finite-difference method is the simplest to implement and the best known technique, it is discussed in somewhat more detail. 

Barkey, Tobias and Muller formulated the stability analysis for deposition from well-supported solution in the Tafel regime at constant current [48], They used dilute-solution theory to solve the transport equations in a Nernst diffusion layer of thickness S. The concentration and electrostatic potential are given in this approximation... 

In this approximation, therefore, one can consider that the diffusion occurs across a region parallel to the interface, i.e., across a Nernst diffusion layer of effective thickness 8. 

Let us suppose that there is a layer of solution close to the electrode within which all concentration changes due to electrode reaction occur and that transport within this layer is entirely by diffusion. For hydro-dynamic electrodes, this approximation is reasonable since the diffusion layer is very thin owing to the effects of forced convection. Following Nernst, we assume that the concentration varies linearly within the diffusion layer such that the flux, j, at the electrode is... 

The mathematical solution was first studied by Girina et al. [274] based on the same approximation as that for potential step studies, by dropping the highest-order convective term [237], and by Fried and Elving by using the Nernst diffusion-layer concept [275]. 

To evaluate + for each metal ion, values of p8 are required at each concentration. While this can often be evaluated from electrophoretic mobility data, the high ionic strengths—Le., pH < 2—preclude meaningful measurement of mobilities. However, it can be seen that when ij/s and cf)+ are equal and opposite then adsorption is reduced to zero. The adsorption of Na+ is reduced to zero at the z.p.c. since, in this case, + is negligibly small. With Ni2+ and Cu2+ the pH must be reduced—i.e., made more positive—by 1.3 pH units to effect zero adsorption. Since near the z.p.c. ips and i//0, the total double layer potential, are approximately equal and given by the Nernst Equation, then... 

Although the transition between stagnant and flowing solution is considered to be abrupt in this example, the transition is in reality gradual. Consequently, the profiles will be rounded as shown by the dotted line in Figure 3.37C. However, the hypothetical situation of an abrupt transition is a useful approximation in mathematical treatments as shown below. (See Ref. 6, Chap. 4, for a critique of the Nernst diffusion layer.)... 

Here the state with [ZA] = [Z] is taken as a standard state of the adsorbed layer thus, in the case when only one gas is adsorbed, the layer is in the standard state at the coverage 1 /2. It can be easily seen that 1 /a is the equilibrium pressure at [ZA] = [Z], i.e., at the standard state of the adsorbed substance. This value may be called desorption pressure we shall denote it as b. It is analogous to vapor pressure or dissociation pressure in monovariant systems (24). Indeed, in the case of equilibrium of liquid with its vapor, the surface from which evaporation occurs is equal to the surface for condensation the same equality is realized at the adsorption equilibrium if the fraction of the occupied surface is equal to that of the free surface. This analogy explains the applicability of the Nernst approximate formula to desorption pressure (24) ... 

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... 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

Consider the process of plating copper on a plane electrode. Near the electrode, copper ions are being discharged on the surface and their concentration decreases near the surface. At some point away from the electrode, the copper ion concentration reaches its bulk level, and we obtain a picture of the copper ion concentration distribution, shown in Fig. 6. The actual concentration profile resembles the curved line, but to simplify computations, we assume that the concentration profile is linear, as indicated by the dashed line. The distance from the electrode where the extrapolated initial slope meets the bulk concentration line is called the Nernst diffusion-layer thickness S. For order of magnitude estimates, S is approximately 0.05 cm in unstirred aqueous solution and 0.01 cm in lightly stirred solution. 

Well-defined hydrodynamic conditions, with high rate of mass transport, are essential for a successful use of electrochemical detectors. According to the Nernst approximate approach, the thickness of the diffusion layer (8) is empirically related to the solution flow rate (U) via... 

The kinematic viscosity of dilute aqueous solutions is ca. 0.01 cm s. So Nernst diffusion layers 5/v between 200 and 30 pm result at flow velocities between 5 and 100 cm s. For small molecules like benzene, tetrachloroethene, etc. the diffusion coefficients in water and in the PDMS membrane are of the same order. Then the time constant is approximately ... 

However, for most applications, the Nernst-approximated profile is used, in which the concentration is assumed to vary linearly in a diffusion layer with thickness, <5rde, given by Eq. 100. 

The relation between the interfacial and bulk concentrations depends on mass transport, most often by diffusion (i.e., thermal motion) and/or convection (mechanical stirring). Often a stationary state is reached, in which the concentrations near the electrode can be described approximately by a diffusion layer of thickness 8. For a constant diffusion layer thickness the Nernst equation takes the form... 

Consider what happens when a potential step of magnitude E is applied to an electrode immersed in a solution containing a species O. If the reaction is nernstian, the concentrations of O and R at x = 0 instantaneously adjust to the values governed by the Nernst equation, (1.4.12). The thickness of the approximately linear diffusion layer, grows with time (Figure 1.4.5). At any time, the volume of the diffusion layer is... 

In gaseous reactions the diffusion steps (1) and (5) are very fast and are rarely, if ever, rate determining. For very fast reactions in solution the rate may be limited by diffusion to or from the surface of the catalyst. If diffusion is the slow step, then the concentration c of the diffusing species at the surface will differ from the concentration c in the bulk. In Fig. 34.1, the concentration is plotted as a function of the distance from the surface. This curve is conveniently approximated by the two dashed lines. The distance 3 is the thickness of the diffusion layer. This approximation was introduced by Nernst, and the layer in which the concentration differs appreciably from that in the bulk is called the Nernst diffusion layer. The concentration gradient across the diffusion layer is given by (c — c )/3, so that the rate of transport per square metre of the surface is... 

The Nernst film model is used to quantify diffusional transport through the static boundary layer. This model approximates the low velocity boundary layer as a thin, static film between the surface and the free fiowing solution. Pick s first law gives the diffusional flux (/, mol/m sec) through this film (Figure 7.6). 

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