Diffusive-kinetic steady state

Kinetic Steady-State Conditions by Assuming a Purely Diffusive Behavior for Species f and D (Diffusive-Kinetic Steady State, dkss)... 

In this section, the different behavior of processes with coupled noncatalytic homogeneous reactions (CE and EC mechanisms) is discussed in comparison with a catalytic process. We will consider that the chemical kinetics is fast enough and in the case of CE and EC mechanisms K (- c /cf) fulfills K 1 so that the kinetic steady-state and even diffusive-kinetic steady-state approximation can be applied. 

Finding rigorous analytical expressions for the single potential step voltammograms of these reaction mechanisms in a spherical diffusion field is not easy. However, they can be found in reference [63, 64, 71-73] for the complete current-potential curve of CE and EC mechanisms. The solutions of CE and EC processes under kinetic steady state can be found in references [63, 64] and the expression of the limiting current in reference [74], Both rigorous and kinetic steady state solutions are too complex to be treated within the scope of this book. Thus, the analysis of these processes in spherical diffusion will be restricted to the application of diffusive-kinetic steady-state treatment. 

For high values of the chemical rate constant, i.e., under conditions of a diffusive-kinetic steady state (dcross potentials of the ADDPV curves,... 

In this section, new assumptions are introduced which will be fundamental for the general definition and understanding of reaction and diffusion layers. We will consider that variable ss retains the form given by Eq. (3.203b) deduced under kinetic steady-state approximation (i.e., by supposing that d(pss/dt = 3(cb — Kcq)/ dt = 0). In relation to the variables f and cD, it is assumed that their profiles have the same form as that for species that would only suffer diffusion and would keep time-independent values at the electrode surface, i.e., [63] ... 

This section deals with the solution corresponding to an EC mechanism (see reaction scheme 4.IVc) in Reverse Pulse Voltammetry technique under conditions of kinetic steady state (i.e., the perturbation of the chemical equilibrium is independent of time see Sect. 3.4.3). In this technique, the product is electrogenerated under diffusion-limited conditions in the first period (0 < t < ) and then exam-... 

Similarly to the response at hydrodynamic electrodes, linear and cyclic potential sweeps for simple electrode reactions will yield steady-state voltammograms with forward and reverse scans retracing one another, provided the scan rate is slow enough to maintain the steady state [28, 35, 36, 37 and 38]. The limiting current will be detemiined by the slowest step in the overall process, but if the kinetics are fast, then the current will be under diffusion control and hence obey the above equation for a disc. The slope of the wave in the absence of IR drop will, once again, depend on the degree of reversibility of the electrode process. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Given that, under the defined conditions, there is no interfacial kinetic barrier to transfer from phase 2 to phase 1, the concentrations immediately adjacent to each side of the interface may be considered to be in dynamic equilibrium throughout the course of a chronoamperometric measurement. For high values of Kg the target species in phase 2 is in considerable excess, so that the concentration in phase 1 at the target interface is maintained at a value close to the initial bulk value, with minimal depletion of Red in phase 2. Under these conditions, the response of the tip (Fig. 11, case (a)] is in agreement with that predicted for other SECM diffusion-controlled processes with no interfacial kinetic barrier, such as induced dissolution [12,14—16] and positive feedback [42,43]. A feature of this response is that the current rapidly attains a steady state, the value of which increases... 

As might be expected, similar trends to those identified above are observed as y is varied, while maintaining constant and K high and nonlimiting. The transient and steady-state current responses, shown respectively in Figs. 14 and 15 for = 1 and K = 10, vary between a lower limit which is close to the response for an inert interface when y < 0.01, and an upper limit (when y > 1000) which is characteristic of SECM diffusion-control in phase 1 with no resistance from interfacial kinetics or transport in phase 2. 

Under conditions of nonlimiting interfacial kinetics the normalized steady-state current is governed primarily by the value of K y, which is the relative permeability of the solute in phase 2 compared to phase 1, rather than the actual value of or y. In contrast, the current time characteristics are found to be highly dependent on the individual K. and y values. Figure 16 illustrates the chronoamperometric behavior for K = 10, log(L) = —0.8 and for a fixed value of Kf.y = 2. It can be seen clearly from this plot that whereas the current-time behavior is sensitive to the value of Kg and y, in all cases the curves tend to be the same steady-state current in the long-time limit. This difference between the steady-state and chronoamperometric characteristics could, in principle, be exploited in determining the concentration and diffusion coefficient of a solute in a phase that is not in direct contact with the UME probe. 

The normalized steady-state current vs. tip-interface distance characteristics (Fig. 18) can be explained by a similar rationale. For large K, the steady-state current is controlled by diffusion of the solute in the two phases, and for the specific and y values considered is thus independent of the separation between the tip and the interface. For K = 0, the current-time relationship is identical to that predicted for the approach to an inert substrate. Within these two limits, the steady-state current increases as K increases, and is therefore diagnostic of the interfacial kinetics. 

For comparable diffusion coefficients of the target solute in the two phases and nonlimiting transfer kinetics, systems characterized by different should be resolvable on the basis of transient and steady-state current responses to a value of up to 50 at practical tip-interface separations. If the diffusion coefficient in phase 2 becomes lower than that in phase 1, diffusion in phase 2 will be partly limiting at even higher values of K. On the other hand, as the value of y increases or interfacial kinetics become increasingly limiting, lower values of suffice for the constant-composition assumption for phase 2 to be valid. 

FIG. 24 Steady-state diffusion-limited current for the reduction of oxygen in water at an UME approaching a water-DCE (O) and a water-NB (A) interface. The solid lines are the characteristics predicted theoretically for no interfacial kinetic barrier to transfer and for y = 1.2, Aj = 5.5 (top solid curve) or y = 0.58, = 3.8 (bottom solid curve). The lower and upper dashed lines denote the... 

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