Diffusive transport/reaction regime

Laminar flame speed is one of the fundamental properties characterizing the global combustion rate of a fuel/ oxidizer mixture. Therefore, it frequently serves as the reference quantity in the study of the phenomena involving premixed flames, such as flammability limits, flame stabilization, blowoff, blowout, extinction, and turbulent combustion. Furthermore, it contains the information on the reaction mechanism in the high-temperature regime, in the presence of diffusive transport. Hence, at the global level, laminar flame-speed data have been widely used to validate a proposed chemical reaction mechanism. 

Figure 26b shows the impedance predicted by eqs 8 and 9. As previously discussed, this function is known as the Gerischer impedance, derived earlier in section 3.4 for a situation involving co-limited adsorption and surface diffusion (in the context of Pt). As with the surface-mediated case, the present result corresponds to a co-limited reaction regime where both kinetics and transport determine the electrode characteristics (as reflected in the dependency of 7 chem and Qs on both fq and T eff)- The essential difference between this and the Pt case is that here the kinetics and diffusion parameters refer to a bulk-mediated rather than surface-mediated process. 

Available reaction-transport models describe the second regime (reactant transport), which only requires material balances for CO and H2. Recently, we reported preliminary results on a transport-reaction model of hydrocarbon synthesis selectivity that describes intraparticle (diffusion) and interparticle (convection) transport processes (4, 5). The model clearly demonstrates how diffusive and convective restrictions dramatically affect the rate of primary and secondary reactions during Fischer-Tropsch synthesis. Here, we use an extended version of this model to illustrate its use in the design of catalyst pellets for the synthesis of various desired products and for the tailoring of product functionality and molecular weight distribution. 

To appreciate the impact of SECM on the study of phase transfer kinetics, it is useful to briefly review the basic steps in reactions at solid/liquid interfaces. Processes of dissolution (growth) or desorption (adsorption), which are of interest herein, may be described in terms of some, or all, of the series of events shown in Figure 1. Although somewhat simplistic, this schematic identifies the essential elements in addressing the kinetics of interfacial processes. In one limit, when any of the surface processes in Figure 1 (e.g., the detachment of ions or molecules from an active site, surface diffusion of a species across the surface, or desorption) are slow compared to the mass transport step between the bulk solution and the interface, the reaction is kinetically surface-controlled. In the other limit, if the surface events are fast compared to mass transport, the overall process is in a mass transport-controlled regime. 

It is also essential that the period of the ac stimulus not be so long that convection becomes a factor within a few cycles. The lower frequency limit was set here at 1 Hz because convection would become a problem in the range of several seconds in most liquid systems with water-like viscosity. Current equipment for EIS can operate at much lower frequencies (as low as 10 jU,Hz) and can be usefully applied in the low-frequency (long-time) regime when the processes being examined are not controlled by convection. Examples include transport or reaction at a solid-solid interfaces or diffusion and reaction in extremely viscous media, such as glasses or polymers. 

Perhaps one reason for the non-competitiveness of liquid films as gas separators - besides the difficulties of fabricating ultra-thin porous membranes and preventing their dessication - is that the theoretical aspects of CO transport in alkaline media have not been fully explored. Solutions to the differential equations governing steady-state CO2 diffusion with non-equilibrium chemical reaction are available, and the appreciable effects of catalysts and buffers have been elucidated. However, noteworthy aspects of the equilibrium (fast reaction) regime in simple alkaline solutions have not been fully examined. 

The pyrolytic regime is distinguished from the (diffusion-limited) dialytic regime by the effect of the input gas flow rate on the deposition rate. Both transport-controlled regimes occur at high temperatures, where the slopes of the Arrhenius plot decrease (Figure 6.12). In the case of small equilibrium constants, the slope of the Arrhenius plot in the pyrolytic regime equals the reaction heat, as will be shown below. 

All this is valid only in the catalytic or reaction-limited regime (see Chapter 6). There is another effect of the fractal dimension of the surface on the growth rate and on the morphology of the resulting product. This is the ease of transport of the reactants to the surface before they adsorb and react. In the dialytic regime, e.g., in the case of an Eley-Rideal mechanism with slow diffusion, the reaction occurs mainly on the top of a fractal surface and less on the less accessible parts. The Eley-Rideal expression of the rate is then raised to a power dependent on the dimension of the surface.Higher surface dimensions mean relatively higher reaction rates because the reactants have better accessibility to the surface. 

If microelectrodes (hemispherical or inlaid-disc types) are used, the mass transport rate due to radial diffusion to the electrode is enhanced so that the current contribution of chemical steps is decreased relative to the diffusive transport. This statement can be easily verified on any EQrrE sequence at a macroelectrode of radius a 100 /zni the diffusion-controlled two-electron wave is observed when two one-electron electrode processes occurs at the same potential. At microelectrodes in steady-state regime the time scale is too short for the overall reaction and the number of electrons measured on the limiting current gradually declines to unity with decreasing electrode radius. On the other hand, the change in the current as a function of the electrode radius can be used to determine the rate constants of involved chemical steps. 

Cyclic voltammetric inspection of the permeation of electroactive species in thin layer immobilized on an electrode illustrated by the permeation of FcMeOH in PGMA brushes. Typical voltammograms of 3 mM FcMeOH solution at v = 20 mV s" on 0.03 cm Au surfaces showing (a) permeation in a 12 nm thick brush controlled by diffusive transport (here diffusion in the solution phase) or (b) permeation in a 110 nm thick brush steady-state transport controlled by the partition reaction at the brush-solution interface, (c) Interpretation of FcMeOH permeation into (from top to bottom) e = 12, 70, 110, and 170 nm thick PGMA brushes. Comparison with theoretical prediction of the different kinetic regimes observed for permeation in thin layers (transition from control by (i) solution diffusion to (ii) partition to (iii) diffusion in the layer). (Adapted from Matrab, T., et al., ChemPhysChem, 11, 670-682,2010.)... 

Equations 6.6 and 6.7 show that in both unentangled and entangled regimes, when the reaction is diffusion controlled, the interfacial rate constant is insensitive to interfacial structure (e.g. interfacial thickness) and dynamics. This is because the rate limiting step is the slow diffusive transport of a reactive chain to within a critical distance of the interface. In the entangled regime, this critical distance is comparable to the radius of gyration of the chain, Rg. Sub-diffusive transport is assumed to be very rapid. 

At low temperatures, the speed is very slow, the transport of the species to the boundary layer being fast the limiting stage then corresponds to the reaction of the species adsorbed at the substrate surface (reaction regime). At high temperatures, as the reactions become instantaneous, the slow stage then corresponds to the diffusion of the reactive species (diffusion regime). 

These observations are consistent with the proposed mechanism of the reaction being diffusion controlled in the laminar flow regime. The mass transport is aided by the velocity gradient and thus the reaction rate increases as the Reynolds number is increased. 

Regime of transport limitation, here

diffusion through the hydrodynamic boundary layer. The apparent activation energy under these conditions gets close to zero. Every educt molecule reacts instantaneously on the outer catalyst surface, no educt diffusion inside the catalyst particle takes place. 

It can be seen in the plot in Figure 11 that EA . shows a clear temperature dependence. For rising temperatures the mass transport limitation can be observed, which leads to a lowering of EAs by a factor of V2 in the pore diffusion regime down to 0, owing to the shift of the reaction from the interior of the pore system of the catalytic particle to the outer surface. In the final state, the diffusion through the boundary layer becomes the rate-limiting step of the reaction. 

For all reactions, the mass transport regime is controlled by the diffusion of the reacting ligand only, as the mercury electrode serves as an inexhaustible source for mercury ions. Hence, with respect to the mathematical modeling, reactions (2.205) and (2.206) are identical. This also holds true for reactions (2.210) and (2.211). Furthermore, it is assumed that the electrode surface is covered by a sub-monomolecular film without interactions between the deposited particles. For reactions (2.207) and (2.209) the ligand adsorption obeys a linear adsorption isotherm. Assuming semi-infinite diffusion at a planar electrode, the general mathematical model is defined as follows ... 

---