Mass transport spherical diffusion

It is possible that the pores of wetted catalyst particles eire filled with liquid. Hence, by virtue of the low values of liquid diffusivities (ca. 10 cm s" ), the effectiveness factor will almost certainly be less than unity. A criterion for assessing the importance of mass transfer in the trickling liquid film has been suggested by Satterfield [40] who argued that if liquid film mass transport were important, the rate of reaction could be equated to the rate of mass transfer across the liquid film. For a spherical catalyst particle with diameter dp, the volume of the enveloping liquid fim is 7rdp /6 and the corresponding interfacial area for mass transfer is TTdn. Hence... 

It should be noted here that while in catalytic systems the rate is based on the moles disappearing from the fluid phase - dddt, and the rate has the form ( —ru) = f(k, C), in adsorption and ion exchange the rate is normally based on the moles accumulated in the solid phase and the rate is expressed per unit mass of the sohd phase dqldt where q is in moles per unit mass of the solid phase (solid loading). Then, the rate is expressed in the form of a partial differential diffusion equation. For spherical particles, mass transport can be described by a diffusion equation, written in spherical coordinates r ... 

In the foregoing, the expressions needed to account for mass transport of O and R, e.g. eqns. (23), (27), (46), and (61c), were introduced as special solutions of the integral equations (22), giving the general relationship between the surface concentrations cG (0, t), cR (0, t) and the faradaic current in the case where mass transport occurs via semi-infinite linear diffusion. It is worth emphasizing that eqns. (22) hold irrespective of the relaxation method applied. Of course, other types of mass transport (e.g. bounded diffusion, semi-infinite spherical diffusion, and convection) may be involved, leading to expressions different from eqns. (22). 

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 ... 

A voltammetric experiment in a microelectrode array is highly dependent on the thickness of the individual diffusion layers, <5, compared with the size of the microelectrodes themselves, and with the interelectrode distance and the time experiment or the scan rate. In order to visualize the different behavior of the mass transport to a microelectrode array, simulated concentration profiles to spherical microelectrodes or particles calculated for different values of the parameter = fD Ja/r s can be seen in Fig. 5.17 [57] when the separation between centers of... 

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. 

Compared to conventional (macroscopic) electrodes discussed hitherto, microelectrodes are known to possess several unique properties, including reduced IR drop, high mass transport rates and the ability to achieve steady-state conditions. Diamond microelectrodes were first described recently diamond was deposited on a tip of electrochemically etched tungsten wire. The wire is further sealed into glass capillary. The microelectrode has a radius of few pm [150]. Because of a nearly spherical diffusion mode, voltammograms for the microelectrodes in Ru(NHy)63 and Fe(CN)64- solutions are S-shaped, with a limiting current plateau (Fig. 33a), unlike those for macroscopic plane-plate electrodes that exhibit linear diffusion (see e.g. Fig. 18). The electrode function is linear over the micro- and submicromolar concentration ranges (Fig. 33b) [151]. 

The particular advantages of microelectrodes were discussed in Section 5.5. The current density at a microelectrode is larger than that at a spherical or planar electrode of larger dimensions owing to radial and perpendicular diffusion. Mass transport is greater, and we observe differences in the experimental results obtained by the various electrochemical techniques relative to macroelectrodes. 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

Mass transport of developer to, and oxidized developer and halide ions from, the silver speck is calculated by the method of spherical diffusion. The surface concentrations of the active species are then used in the Nemst equation to calculate the surface potential of the developing nucleus. The rate equation (79) obtained is ... 

Electrocatalysts One of the positive features of the supported electrocatalyst is that stable particle sizes in PAFCs and PEMFCs of the order of 2-3 nm can be achieved. These particles are in contact with the electrolyte, and since mass transport of the reactants occurs by spherical diffusion of low concentrations of the fuel-cell reactants (hydrogen and oxygen) through the electrolyte to the ultrafine electrocatalyst particles, the problems connected with diffusional limiting currents are minimized. There has to be good contact between the electrocatalyst particles and the carbon support to minimize ohmic losses and between the supported electrocatalysts and the electrolyte for the proton transport to the electrocatalyst particles and for the subsequent oxygen reduction reaction. This electrolyte network, in contact with the supported electrocatalyst in the active layer of the electrodes, has to be continuous up to the interface of the active layer with the electrolyte layer to minimize ohmic losses. 

The transient mass transport in spheres plays an important role in many process engineering appheations. At adsorption the adsorptive moves through porosities and is aeeumulated on the internal surface of the spherical adsoibent. At regeneration of such adsorbents, as well as at drying of capillary active solids the opposite process occurs. Mass transport caused by transient diffusion is also existent in fluid particles, as long as no convection occurs in the particle. This applies for small viscous droplets. 

Finally, the cases of spherical, hemispherical and cylindrical electrodes will be tackled, which cover the use of wire electrodes, mercury drops and microhemispheres, and liquid-liquid interfaces. These geometries enable us to introduce the effects due to convergent diffusion on the mass transport and voltammetric response. Moreover, as in the case of planar electrodes, because of the symmetry of the mass transfer field the problems can each be reduced to only one dimension the distance to the electrode surface in the normal direction. 

For spherical microelectrodes in the limit of low scan rates T 1), the diffusive mass transport is able to keep the surface gradient constant with time and the steady state is attained. Under these conditions, a sigmoidal response is obtained in cyclic voltammetry with the current reaching a... 

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