Diffusion stationary mass transport

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)... 

Table 1.4 Mass transport coefficients m,, for different experimental conditions. The values of m, correspond to the application of a constant potential. The expressions corresponding to the Rotating Disc Electrode (convective mass transport) under stationary conditions and to Dropping Mercury Electrode with the expanding plane model (diffusive-convective mass transport) have also been included...

Fig. 1. Voltaimrogram observable with stationary mass transport to the electrode. A and B, oxidi2able compounds C, reducible compound f, diffusion or limiting current ff, half-vrave potential.

When the two liquid phases are in relative motion, the mass transfer coefficients in either phase must be related to the dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive transfer to the Schmidt number. Another complication is that such a boundary cannot in many circumstances be regarded as a simple planar interface, but eddies of material are transported to the interface from the bulk of each liquid which change the concentration profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most industrial circumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass transfer model must therefore be replaced by an eddy mass transfer which takes account of this surface replenishment. 

The experiment is carried out under stationary conditions (i.e. the solution is kept unstirred) in order to ensure that the mass transport is purely diffusive. 

One of several possibilities to classify elec-troanalytical methods is based on the quantity that is controlled in the experiment, that is, current or potential. Alternatively, since diffusion is an important mode of mass transport in most experiments, we distinguish techniques with stationary or nonstationary diffusion. Finally, transient methods are different from those that work in an exhaustive way. 

Microcylindrical electrodes are easier to constract and maintain than microdisk electrodes [37]. Mass transport to a stationary cylinder in quiescent solution is governed by axisymmetrical cylindrical diffusion. For square-wave voltammetry the shape and position of the net current response are independent of the extent of cyhn-drical diffusion [38]. The experiments were performed with the ferri-ferrocyanide couple using a small platinum wire (25 pm in diameter and 0.5 -1.0 cm in length) as the working electrode [37]. 

It follows from Equation 6.12 that the current depends on the surface concentrations of O and R, i.e. on the potential of the working electrode, but the current is, for obvious reasons, also dependent on the transport of O and R to and from the electrode surface. It is intuitively understood that the transport of a substrate to the electrode surface, and of intermediates and products away from the electrode surface, has to be effective in order to achieve a high rate of conversion. In this sense, an electrochemical reaction is similar to any other chemical surface process. In a typical laboratory electrolysis cell, the necessary transport is accomplished by magnetic stirring. How exactly the fluid flow achieved by stirring and the diffusion in and out of the stationary layer close to the electrode surface may be described in mathematical terms is usually of no concern the mass transport just has to be effective. The situation is quite different when an electrochemical method is to be used for kinetics and mechanism studies. Kinetics and mechanism studies are, as a rule, based on the comparison of experimental results with theoretical predictions based on a given set of rate laws and, for this reason, it is of the utmost importance that the mass transport is well defined and calculable. Since the intention here is simply to introduce the different contributions to mass transport in electrochemistry, rather than to present a full mathematical account of the transport phenomena met in various electrochemical methods, we shall consider transport in only one dimension, the x-coordinate, normal to a planar electrode surface (see also Chapter 5). 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

The first applied potential is set at a value E at a stationary spherical electrode during the interval 0 < t < i. The diffusion mass transport of the electroactive species toward or from the electrode surface is described by the following differential equation system ... 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

Complex mathematical formulae will be minimized here for the purpose of simplicity since there are numerous texts that deal with detailed theory of mass transport in chromatography1[2,1,22 The flow of mobile phase through a packed column bed is shown schematically in Figure 2.1. There are two transport mechanisms in progress. Firstly, the convectional flow around the particles and secondly, the diffusion in and out of the pores of the stationary phase. 

Fortunately, the effects of most mobile-phase characteristics such as the nature and concentration of organic solvent or ionic additives the temperature, the pH, or the bioactivity and the relative retentiveness of a particular polypeptide or protein can be ascertained very readily from very small-scale batch test tube pilot experiments. Similarly, the influence of some sorbent variables, such as the effect of ligand composition, particle sizes, or pore diameter distribution can be ascertained from small-scale batch experiments. However, it is clear that the isothermal binding behavior of many polypeptides or proteins in static batch systems can vary significantly from what is observed in dynamic systems as usually practiced in a packed or expanded bed in column chromatographic systems. This behavior is not only related to issues of different accessibility of the polypeptides or proteins to the stationary phase surface area and hence different loading capacities, but also involves the complex relationships between diffusion kinetics and adsorption kinetics in the overall mass transport phenomenon. Thus, the more subtle effects associated with the influence of feedstock loading concentration on the... 

As discussed in Section 2 material may reach the electrode surface by diffusion or convection. In cyclic voltammetry at a stationary electrode, and assuming that migration can be neglected, diffusion is the sole form of mass transport. However, material may additionally be transported to the electrode by convection. This genre of voltammetry, where convection is a dominant form of mass transport, is described as hydrodynamic voltammetry. The focus in Section 4 will be on the use of rotating disc and channel electrodes in studies... 

The special case V - 0 corresponds to a stationary medium, which can now be defined more precisely as a medium whose mass-average velocity is zero. Therefore, mass transport in a stationary medium is by diffusion only, aud zero mass-average velocity indicates that there is no bulk fluid motion. 

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... 

Lichtner, P.C., 1991. The quasi-stationary state approximation to coupled mass transport and fluid-rock reaction local equilibrium revisited, in J. Ganguly, ed. Diffusion, Atomic Ordering and Mass Transport, Advances in Physical Geochemistry vol. 8, pp. 454-562. 

Therefore, the model of mass transport in a MREF waveform can be illustrated using a simple model of duplex diffusion layer , which was developed by Ibl 111121 for pulse plating. As shown in Figure 2, the diffusion layer may be divided into two parts, a pulsating diffusion layer of thickness 8p and a stationary diffusion layer. At the end of a pulse, the pulsating diffusion layer thickness 8p (under low duty cycle) is given by ... 

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