Mass transport rate coefficient

The mass transport rate coefficient, kd, for a RDE at the maximum practical rotation speed of 10000 per min"1 is approximately 2 x 10-2 cms-1 [28], which sets a limit of about 10 3 cms 1 for the electrode reaction kinetics. For the study of very fast electrode processes, such as some outer sphere redox reactions on noble metal electrodes under stationary conditions, higher mass transport rates in the solution adjacent to the electrode must be employed. 

Mass transport coefficient (also called mass transport rate coefficient, mass transfer coefficient, heterogeneous diffusion rate constant) is defined as... 

According to the classical treatment by Randles [47] for a simple electroreduction of O to R in solution and assuming, for simplicity, that the mass transport rate coefficients, kdj for the oxidized and reduced species are the same, the net current density under steady state conditions is... 

The analogy in the mass transport effects in electrode reaction and homogeneous second-order fast reactions in solution becomes clear. In electrode kinetics, however, the charge-transfer rate coefficient can be externally varied over many orders of magnitude through the electrode potential and kd can be controlled by means of hydrodynamic electrodes. For instance the mass transport rate coefficient, kd, for a rotating disc electrode at the maximum practical rotation speed of 10 000 per min is approximately 2 x 10... 

Important hints on the reaction site can be gained by the Hatta numbers (Ha) of mass transport at the G/L- and L/L-phase boundaries. These numbers are also essential in order to estimate mass transport rates and concentration profiles within the boundary layer. Since the main resistance of mass transport is in the aqueous phase, mass transport coefficients and Ha numbers mentioned in the text are related to the aqueous phase. 

Reactions carried in aqueous multiphase catalysis are accompanied by mass transport steps at the L/L- as well as at the G/L-interface followed by chemical reaction, presumably within the bulk of the catalyst phase. Therefore an evaluation of mass transport rates in relation to the reaction rate is an essential task in order to gain a realistic mathematic expression for the overall reaction rate. Since the volume hold-ups of the liquid phases are the same and water exhibits a higher surface tension, it is obvious that the organic and gas phases are dispersed in the aqueous phase. In terms of the film model there are laminar boundary layers on both sides of an interphase where transport of the substrates takes place due to concentration gradients by diffusion. The overall transport coefficient /cl can then be calculated based on the resistances on both sides of the interphase (Eq. 1) ... 

In FPTRMS, transport of the reactive species of interest from the reactor to the detector can make a contribution to the observed time dependence such that the chemical kinetics becomes convoluted with mass transport rates. This will have to be accounted for in data analysis if reliable rate coefficients are to be obtained. If the physical rate processes are sufficiently fast they will make a negligible contribution to the kinetics. In this section we examine the above four factors to see when they influence the chemical kinetics. The first, third, and fourth items put an upper limit on the rate at which decays and growths can be reliably determined, and the second one sets a lower limit on the decay rate. 

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 ... 

It should be noted that the local mass transfer coefficient can only be obtained experimentally and is case specific. An analytical relationship for the local mass transfer rate coefficient can be obtained if a mathematical expression describing the gradient of the dissolved concentration at the NAPL-water interface is known. Unfortunately, the local mass transfer coefficient usually is not an easy parameter to determine with precision. Thus, in mathematical modeling of contaminant transport originating from NAPL pool dissolution, k(t, x,y) is often replaced by the average mass transfer coefficient, k(t), applicable to the entire pool, expressed as [41]... 

Reaction of dissolved gases in clouds occurs by the sequence gas-phase diffusion, interfacial mass transport, and concurrent aqueous-phase diffusion and reaction. Information required for evaluation of rates of such reactions includes fundamental data such as equilibrium constants, gas solubilities, kinetic rate laws, including dependence on pH and catalysts or inhibitors, diffusion coefficients, and mass-accommodation coefficients, and situational data such as pH and concentrations of reagents and other species influencing reaction rates, liquid-water content, drop size distribution, insolation, temperature, etc. Rate evaluations indicate that aqueous-phase oxidation of S(IV) by H2O2 and O3 can be important for representative conditions. No important aqueous-phase reactions of nitrogen species have been identified. Examination of microscale mass-transport rates indicates that mass transport only rarely limits the rate of in-cloud reaction for representative conditions. Field measurements and studies of reaction kinetics in authentic precipitation samples are consistent with rate evaluations. 

In the frontal analysis experiment described in Section 5.3.2, the transport model of chromatography was used to fit the experimental data [40]. Neglecting axial and eddy diffusion, band broadening was accounted for by one single mass transfer rate coefficient. The mass transfer rate coefficients estimated were small and strongly dependent on the temperature and solute concentration, particularly the rate coefficients corresponding to the imprinted L-enantiomer (Fig. 5.12). Above a concentration of ca. 0.1 g/L the mass transfer rate coefficients of the two enantiomers are similar. 

In Eq. (13) km is the local mass transport coefficient D/S, that is, diffusion coefficient divided by the thickness of diffusion layer. High mass transport rates can be achieved by electrode movement,... 

In the simple examples considered here, the mass transport rate constant is the ratio of the diffusion coefficient of the species in question to the diffusion layer thickness of the latter ... 

The concept of fast or slow couples is therefore independent of the potential applied, since it is intrinsic to the system. However it does depend on other experimental parameters through the mass transport rate constant. The latter parameter is in fact a function of the quantities specific to the mass transport of the species in question (diffusion coefficient or electrochemical mobility), but it also depends on other characteristics in the system which vary according to each type of experiment, as illustrated in the examples below. 

If the diffusion coefficients of the two reactive species are very close, then their mass transport rate constants are also very close and the half-wave potential is equal to the standard potential of the redox couple % =E°... 

Here / h is the Chilton-Colburn factor for mass transport, i th mass transport coefficient, u the linear flow velocity. Sc = V/D is the Schmidt number and / is the friction coefficient. As a consequence, a higher shear stress also means a higher mass transport rate and vice versa. 

Figure 10.27 Effect of an expansion on local mass transport rate the dimensionless mass transport coefficient ki lki Q is plotted as a function of the dimensionless distance from the expansion x/Dq. Here q is the mass transport coefficient far away from the expansion and Dq is the pipe diameter down stream from the expansion. Reynolds number (o) 2.1 x 10 ( ) 4.2 X 10, ( ) 8.4 X 10 and (+ ) 13 X 10. Ratio of pipe diameters = 0.625, adapted...

The friction coefficient/ for a given flow velocity, increases as the wall rugosity increases and according to (10.53) so does the mass transport rate. As a result, rough surfaces corrode at a higher rate. 

Scanning electrochemical microscopy (SECM) (see Chapter 12) combines useful features of UMEs and thin-layer cells. The mass transfer rate in SECM is a function of the tip-substrate distance d. Eor an UME far from a substrate, the mass transfer coefficient, m D/a, while for the tip near a conductive substrate (d mass transport rate can be increased sufficiently for quantitative characterization of the ET kinetics, preserving the advantages of steady-state methods, i.e., the absence of problems associated with ohmic drop, adsorption, and charging current. Eor example, with... 

Both diffusion and convection will determine the net mass transport rate of a reactant to the surface of the electrode. The convention controls the thickness of the diffusion layer and the diffusion controls the transport rate of the reactant through the diffusion layer. The flux normal to the electrode surface due to diffusion is given by D(5 C/5x ), and that due to convection is given by v (dC/dx), where D is the diffusion coefficient of the species, C is the bulk concentration of the species, and is the solution velocity in the x direction, which is normal to the electrode surface. 

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