Mass transport rate constants

After perturbation, the surface concentrations of the reactants, intermediates, and products attain new values. In a genuine stationary technique, these new values are maintained by providing a constant rate of mass transport (hydrodynamic techniques). The surface concentration, c[ of a species i is related to its flux, J, by... 

The data demand the conclusion that reduction of Fe(Cp2+)surf, can become limited partially by mass transport and partially by ket at some point in the reaction (large fractional consumption of (FeCp2+)surf.) in the dark. This seems reasonable in view of the expected rate law, equation (8), the declining (FeCp2+) surf.] and a constant mass transport rate of fresh solution reductant. 

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 increased rate of mass transport associated with shrinking electrode size means that electrode processes which appear electrochemically reversible at large electrodes may show quasi- or irreversible electrode kinetics when examined using both steady-state and transient mode microelectrode methods. The latter represents a powerful approach for the determination of fast heterogeneous electrode kinetics. Rate constants in excess of lcms have been reported (Montenegro, 1994). 

The experimental technique and electrode geometry should be selected to match the kinetic time-scale (the time domain over which a chemical process occurs, e.g. 1/k, where k is a first-order rate constant) of the reaction being studied. This is achieved by varying the rate of mass transport via convection, electrode size/shape or potential scan rate. 

The general approach for modelling catalyst deactivation is schematically organised in Figure 2. The central part are the mass balances of reactants, intermediates, and metal deposits. In these mass balances, coefficients are present to describe reaction kinetics (reaction rate constant), mass transfer (diffusion coefficient), and catalyst porous texture (accessible porosity and effective transport properties). The mass balances together with the initial and boundary conditions define the catalyst deactivation model. The boundary conditions are determined by the axial position in the reactor. Simulations result in metal deposition profiles in catalyst pellets and catalyst life-time predictions. 

We started this book with a schematic presentation (Fig. lA) of the current-potential relationship in an electrolytic cell from the region where no current is flowing, in spite of the applied potential, to the region where the current rises exponentially with potential, following an equation such as Eq. 8D and through the limiting current region, where the current has a constant value, determined only by the rate of mass transport to the electrode surface or away from it. 

An understanding of the kinetics of ion exchange reactions has application in two broad areas. Firstly, it helps to elucidate the nature of the various fundamental ionic transport mechanisms which control or contribute to the overall exchange rate. Secondly derived numerical parameters such as rate constants , mass transfer coefficients, or diffusion coefficients found from a rate investigation are of value when making projections concerning the dynamic behaviour of columns and in process design. 

Levels of volatility that would lead to unacceptable rates of vapor transport-driven sintering, attrition of catalytically-active materials, or corrosion of catalytic materials or support oxides by transport from contaminants or substrate materials can be estimated given equilibrium vapor pressures and a few assumptions about evaporation rates and mass transport. In particular, the rate of condensation of a vapor species on its source solid phase at high temperatures is almost certainly non-activated and may show little configurational restriction. Therefore, using the principle of microscopic reversibility, we can take the rate constant for condensation to be approximately equal to the collision frequency. 

The current at any point in the voltammetry experiment described in Figure 23-5 is determined by a combination of (1) the rate of mass transport of A to the edge of the Nemst diffusion layer by convection and (2) the rate of transport of A from the outer edge of the diffusion layer to the electrode surface. Because the product of the electrolysis P diffuses away from the surface and i.s ultimately swept away by convection, a continuous current is required to maintain the surface concentrations demanded by the Nernst equation. Convection, however, maintains a constant supply of A at the outer edge of the diffusion layer. Thus, a steady-state cuirent results that is determined by the applied potential. 

In contrast to the two systems above, the effective rate of mass transport to the channel electrode is dependent upon both the solution velocity and the length of the electrode. We may therefore anticipate that a greater range of rate constants should be measurable at this hydrodynamic system compared with both the rotating-disc and (in principle) the microelectrode. 

The stripping time is normally orders of magnitude less than the accumulation time when adsorptive accumulation is employed. During the accumulation period, which is normally carried out at a constant potential, the oxidant is being reduced, thereby creating a diffusion layer dox- When the oxidation of the adsorbate is controlled by the rate of mass transport of the oxidant the stripping time is given by... 

A decrease in the characteristic dimension of the system (see schematic of parallel plate microreactor in Figure 10.4c) increases the rate of mass transport from the bulk gas to the reactor walls and changes Da. When Da <0.1, surface reaction is limiting and when Da > 10, mass transfer is limiting. The pseudo-first-order reaction rate constant is estimated from k, = a S/C, where o is the rate of fuel consumption (coming from a detailed model), a = 2/dis the catalyst area per unit volume and C is the concentration of the fuel. 

In the case of micro-ITIES, the rate of mass transport is enhanced when compared to linear diffusion, and higher rate constant values can be reached. To measure the rate of ion-transfer reactions, one can use nano-ITIES as proposed by Cai et al. in 2004 [109]. From a steady-state analysis using 25-nm-radius pipettes. Equation 1.19 yields a rate constant value of about 2.2 cm-s" (Figure 1.10) ... 

This shows that the electrochemical rate constants for the one electron oxidation of Fe kox) and for the reduction of Fe kred) depend exponentially on the electrode potential k increases as the electrode is made more positive relative to the solution whilst kred increases as the electrode is made more negative relative to the solution. It is clear that changing the voltage affects the rate constants. However, the kinetics of the electron transfer is not the sole process which can control the electrochemical reaction in many circumstances it is the rate of mass transport to the electrode which controls the overall reaction, which we dUigently explore later. 

In Fig. 2.10 it is evident that as the standard electrochemical rate constant, kf, is either fast or slow, termed electrochemically reversible or electrochemicaUy irreversible respectively, changes in the observed voltammetry are striking. It is important to note that these are relative terms and that they are in relation to the rate of mass transport to the electrode surface. The mass transport coefficient, mj-, is given by ... 

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