Nernst thickness

The diffusion flux J, in moles per square centimeter, is proportional to the concentration gradient and inversely proportional to the diffusion layer s effective thickness SN (also called Nernst thickness). The proportionality constant D is the diffusion constant hence... 

Fig. 1.3.1 Variation of the concentration (c) of a species as a function of the disttince from the electrode surface (x) in the case of a steady-state electrode reaction. 1 True concentration profiie 2 fictitious profile of Nernst. Thickness of diffusion layer (5) depends on the rate of stirring, while c(x = 0) depends on the overpotential...

Volt mmetiy. Diffusional effects, as embodied in equation 1, can be avoided by simply stirring the solution or rotating the electrode, eg, using the rotating disk electrode (RDE) at high rpm (3,7). The resultant concentration profiles then appear as shown in Figure 5. A time-independent Nernst diffusion layer having a thickness dictated by the laws of hydrodynamics is estabUshed. For the RDE,... 

Ion-selective bulk membranes are the electro-active component of ion-selective electrodes, which sense the activity of certain ions by developing an ion-selective potential difference according to the Nernst equation at their phase boundary with the solution to be measured. The main differences to biological membranes are their thickness and their symmetrical structure. Nevertheless they are used as models for biomembranes. 

Although it is possible to control the dissolution rate of a drug by controlling its particle size and solubility, the pharmaceutical manufacturer has very little, if any, control over the D/h term in the Nernst-Brunner equation, Eq. (1). In deriving the equation it was assumed that h, the thickness of the stationary diffusion layer, was independent of particle size. In fact, this is not necessarily true. The diffusion layer probably increases as particle size increases. Furthermore, h decreases as the stirring rate increases. In vivo, as GI motility increases or decreases, h would be expected to decrease or increase. In deriving the Nernst-Brunner equation, it was also assumed that all the particles were... 

Fig. 2. Concentration profile of the reacting ion at an electrode. The so-called Nernst diffusion layer thickness is indicated by <5n. ...

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. 

In a stagnant solution, free convection usually sets in as a density gradient develops at the electrode upon passing current. The resulting convective velocity, which is zero at the wall, enhances the transfer of ions toward the electrode. At a fixed applied current, the concentration difference between bulk and interface is reduced. For a given concentration difference, the concentration gradient of the reacting species at the electrode becomes steeper (equivalent to a decrease of the Nernst layer thickness), and the current is thereby increased. 

The dependence of the limiting current density on the rate of stirring was first established in 1904 by Nernst (N2) and Brunner (Blla). They interpreted this dependence using the stagnant layer concept first proposed by Noyes and Whitney. The thickness of this layer ( Nernst diffusion layer thickness ) was correlated simply with the speed of the stirring impeller or rotated electrode tip. 

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... 

We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are /),yi and 2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance <5,- (diffusion layer thickness) from the surface where c, = 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst-Planck equation (10), it can be seen that the flux at any surface is ... 

In passing, it is good to emphasise that the above analysis illustrates the limitations of the widely used Nernst diffusion layer concept. This concept assumes that there is a certain thin layer of static liquid adjacent to the solid plane under consideration at x = 0. Inside this layer, diffusion is supposed to be the sole mechanism of transport, and, outside the layer, the concentration of the diffusing component is constant, as a result of the convection in the liquid. We have seen that, in contradiction with this oversimplified picture, molecular diffusion and liquid motion are not spatially separated, and that the thickness... 

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],... 

Figure 9. Formation of Stern plane and diffuse layer on particle surface ( I 0 = surface or Nernst potential, = potential of inner Flelmholtz plane, I 5 = Stern potential, l = thickness of Stern plane, ZP = zeta potential at surface of shear, d = distance from particle surface).

Conversely, after the voltage is switched, Tl" ions will be oxidized to according to the reaction, Tl Tl + + 2e . Very soon, the electrode will be surrounded with a layer of solution that contains no Tl. It is said then to be depleted of T1+. The volume around the electrode, which now contains no Tl" ", is called the depletion region, the depletion layer or sometimes, the Nernst layer. Its thickness is often given the symbol 5. As the length of time increases following the potential being stepped, i.e. as the extent of electrolysis increases, so the thickness of the Nernst layer (5) increases. 

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,... 

These two thin liquid films, which are also called diffusion films, diffusion layers, or Nernst films, have thicknesses that range between 10 and 10 cm (in this chapter centimeter-gram-second (CGS) units are used, since most published data on diffusion and extraction kinetics are reported in these units comparison with literature values is, therefore, straightforward). 

The stagnant film theory was developed by Nernst (1904). In this theory, a stagnant film exists on both sides of the interface, as illustrated in Figure 8.8. The thickness of the film is controlled by turbulence and is constant. 

Based on the Nernst-Planck flux equation and Eyring s rate theory, a simple theoretical model was evolved for the description of the transport of ions through thick carrier membranes5 (see also Ref. 15). The primary... 

As with other matters concerned with transport to electrodes, detailed treatments were set up very early. Various boundary layers at interfaces under flow were suggested by Prandtl as early as 1904. Three are shown in Fig. (7.93). The 8y is die well-known diffusion layer due to Nernst (Section 7.9.9). The 8 is the thermal boundary layer and the 8V signifies the thickness of the layer (Prandtl s layer) in a flowing liquid in which the velocity slows an approach peipendicular to the surface. 

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. 

Fig. 7.95. The Nernst diffusion-layer thickness is obtained by extrapolating the linear portion of the concentration change to the bulk concentration value.

The film model referred to in Chapters 2 and 5 provides, in fact, an oversimplified picture of what happens in the vicinity of interface. On the basis of the film model proposed by Nernst in 1904, Whitman [2] proposed in 1923 the two-film theory of gas absorption. Although this is a very useful concept, it is impossible to predict the individual (film) coefficient of mass transfer, unless the thickness of the laminar sublayer is known. According to this theory, the mass transfer rate should be proportional to the diffusivity, and inversely proportional to the thickness of the laminar film. However, as we usually do not know the thickness of the laminar film, a convenient concept of the effective film thickness has been assumed (as... 

Turbulent flow comprises the solution bulk. (2) As the electrode surface is approached, a transition to laminar flow occurs. This is a nonturbulent flow in which adjacent layers slide by each other parallel to the electrode surface. (3) The rate of this laminar flow decreases near the electrode due to frictional forces until a thin layer of stagnant solution is present immediately adjacent to the electrode surface. It is convenient, although not entirely correct, to consider this thin layer of stagnant solution as having a discrete thickness 5, called the Nernst diffusion layer. 

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 equivalent Nernst diffusion layer thickness is T (4/3) times d. Eq. (3-1) becomes ... 

Fig. 6-11. Scheme of the modified electrode (polymer film thickness 0) in contact with a solution containing a redox substrate. <5 is the Nernst layer thickness defined for a rotating disc electrode. From [85]. 

As the skin is relatively thick compared to the space-charge layers at its boundaries, the bulk of the membrane may be expected to be electroneutral [56,57], The Nernst-Planck equation can be solved, therefore, by imposing the electroneutrality condition C,/C= C. /C, where the subscripts j and k refer to positive and negative ions, respectively, and C is the average total ion concentration in the membrane. In the case of a homogenous and uncharged membrane bathed by a 1 1 electrolyte, the total ion concentration profile across the membrane is linear and the resulting steady-state flux is described by... 

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