Diffusion Nernst film

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). 

Even where ion exchange is not affected by the above factors, the Nernst-Planck equations are not very useful for diffusion phenomena in the film. After all, the Nernst film is somewhat enigmatic and there is a combination of diffusive and convective mass transfer that changes from the bulk solution to the particle surface. Nernst (1904) originally defined the outer limit of film only as the point where the concentration profile, if linearly extrapolated from the particle surface, reaches the concentration level of the bulk solution. 

Commonly, mass transfer in the liquid is modeled as diffusion in a fic-litious liquid film adhering to the particle Nernst film (Nernst, 1889 Planck, 1890). Intraparticle diffusion and film diffusion are the two possible rate-controlling steps (Helfferich, 1962a). 

On the assumption that there would be a thin stagnant film of liquid adjacent to the growing crystal face, through which molecules of the solute would have to diffuse, Nernst (1904) modified equation 6.14 to the form... 

At high velocities where turbulence dominates, the main body of flowing fluid is well mixed in the direction normal to the flow, minor differences in temperature and concentration can be neglected, and the film concept can be applied. This describes the flow as if all gradients for temperature and concentration are in a narrow film along the interface with the solid (Nernst 1904), and inside the film conduction and diffusion are the transfer mechanisms. This film concept greatly simplifies the engineering calculation of heat and mass transfer. 

When paint films are immersed in water or solutions of electrolytes they acquire a charge. The existence of this charge is based on the following evidence. In a junction between two solutions of potassium chloride, 0 -1 N and 0 01 N, there will be no diffusion potential, because the transport numbers of both the and the Cl" ions are almost 0-5. If the solutions are separated by a membrane equally permeable to both ions, there will still be no diffusion potential, but if the membrane is more permeable to one ion than to the other a diffusion potential will arise it can be calculated from the Nernst equation that when the membrane is permeable to only one ion, the potential will have the value of 56 mV. 

Steady diffusion across a thin film is mathematically straightforward but physically subtle. Dissolution film theory, suggested initially by Nernst and Brunner, is essentially based on steady diffusion across a thin film. 

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. 

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... 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

Also, in the late 1950s and 1960s some particularly seminal papers on ion exchange kinetics appeared by Helfferich (1962b, 1963, 1965) that are classics in the field. In this research it was definitively shown that the rate-limiting steps in ion exchange phenomena were film diffusion (FD) and/ or particle diffusion (PD). Additionally, the Nernst-Planck theories were explored and applied to an array of adsorbents (Chapter 5). 

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. 

Additional deviations from the Nernst law [Eq. (4)] can come from kinetic effects in other words, if the potential scan is too fast to allow the system to reach thermal equilibrium. Two cases should be mentioned (1) ion transport limitation, and (2) electron transfer limitation. In case 1 the redox reaction is limited because the ions do not diffuse across the film fast enough to compensate for the charge at the rate of the electron transfers. This case is characterized by a square-root dependence of the current peak intensity versus scan rate Ik um instead of lk u. Since the time needed to cross the film, tCT, decreases as the square of the film thickness tCT d2, the transport limitation is avoided in thin films (typically, d < 1 xm for u < 100 mV/s). The limitation by the electron transfer kinetics (case 2) is more intrinsic to the polymer properties. It originates from the fact that the redox reaction is not instantaneous in particular, due to the fact that the electron transfer implies a jump over a potential barrier. If the scan... 

The mechanism of dissolution was proposed by Nernst (1904) using a film-model theory. Under the influence of non-reactive chemical forces, a solid particle immersed in a liquid experiences two consecutive processes. The first of these is solvation of the solid at the solid-liquid interface, which causes the formation of a thin stagnant layer of saturated solution around the particle. The second step in the dissolution process consists of diffusion of dissolved molecules from this boundary layer into the bulk fluid. In principle, one may control the dissolution through manipulation of the saturated solution at the surface. For example, one might generate a thin layer of saturated solution at the solid surface by a surface reaction with a high energy barrier (Mooney et al., 1981), but this application is not commonly employed in pharmaceutical applications. 

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. 

The contribution of electric field to lithium transport has been considered by a few authors. Pyun et argued on the basis of the Armand s model for the intercalation electrode that lithium deintercalation from the LiCo02 composite electrode was retarded by the electric field due to the formation of an electron-depleted space charge layer beneath the electrode/electrolyte interface. Nichina et al. estimated the chemical diffusivity of lithium in the LiCo02 film electrode from the current-time relation derived from the Nernst-Planck equation for combined lithium migration and diffusion within the electrode. 

Why did adding CO slow up the reaction Bodenstein and Fink drew on earlier work by Walther Nernst to suggest an answer. They assumed that CO and O2 would only react rapidly at the platinum surface. The CO, they suggested, formed a film several molecules thick on that surface - the more CO, the thicker the film. And the thicker the film, the longer it took the O2 to diffuse through and get to the platinum, so the slower the reaction. ... 

Its substance is the diffusion ion migration through the Nernst layer (film) between the slip plane and active centres sXO- on the mineral surface. 

The rate of the mass transfer through the Nernst layer is defined by the rate of diffuse flow of reagents and products participating in chemical reactions on the interface. This rate was already reviewed as the adsorption rate in the process of film diffusion. It is based on the same Shchukarev equation (2.204). However, contrary to adsorption, here in the flow participate all ions formed in the process of dissolution, not just one ion. That is why in the equation (2.204) should be reviewed the flow of the mineral itself in solution, not of its individual ions, and its subscripts i should be replaced with ... 

The Nernst equation says that this increase in [Py ] will cause a corresponding increase in Eq After termination of the current pulse, these Py" " sites will diffuse (i.e. self-exchange) into the bulk of the film. Eqq will, therefore, decrease with time this transient will persist until the concentration of Py is uniform throughout the film. Because the rate of equilibration is controlled by diffusional transport of the Py sites, Dapp obtained from the Eq vs. t transient. 

Define (a) voltammograms, (b) hydrodynamic voltammetry, (c) Nernst diffusion layer, (d) mercury film electrode, (e) half-wave potential, and (f) voltammetric sensor. 

The same discussion with the same equations holds for anodic reactions with diffusion of the species Red as the rate-determining step. In the case of intense metal corrosion, the diffusion of cations from the electrode surface to the bulk may become the rate-determining step, with their aecumula-tion at the electrode surface and the final precipitation of a salt film. These processes are important for intense active metal dissolution and localized corrosion, as will be discussed in Sec. 1.5.4. At small current densities, a superposition of charge transfer and diffusion control is obtained. Therefore the current density increases exponentially in the vicinity of the Nernst potential ac-... 

A similar theoretical approach has been developed [75] except that reaction layer thicknesses are considered instead of characteristic currents. The location of these reaction layers reflects the relative rates of charge and substrate transport within the film, and the thickness of the layers conveys an idea of the fraction of the film usefully employed in the redox catalysis. Note that the two extremes here are the scenarios that the charge transport through the polymer is very fast (i.e., not rate-limiting) such that the reaction zone is essentially confined to the Nernst diffusion layer (Fig. 20.19a) and the opposite extreme that the reaction zone is located at the support electrode/polymer film boundary (Fig. 20.19b). 

In film diffusion controlled situations, a co-ion is present in the external solution. Consequently, the Nernst-Plank equation for co-ions needs to be considered along with the role of co lons in the electroneutrality and no electric current conditions (Helfferich, 1962) in deriving the exact flux... 

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