Kinetic case diagram

Equations 2.26 and 2.27 carmot be solved analytically except for a series of limiting cases considered by Bartlett and Pratt [147,192]. Since fine control of film thickness and organization can be achieved with LbL self-assembled enzyme polyelectrolyte multilayers, these different cases of the kinetic case-diagram for amperometric enzyme electrodes could be tested [147]. For the enzyme multilayer with entrapped mediator in the mediator-limited kinetics (enzyme-mediator reaction rate-determining step), two kinetic cases deserve consideration in this system in both cases I and II, there is no substrate dependence since the kinetics are mediator limited and the current is potential dependent, since the mediator concentration is potential dependent. Since diffusion is fast as compared to enzyme kinetics, mediator and substrate are both approximately at their bulk concentrations throughout the film in case I. The current is first order in both mediator and enzyme concentration and k, the enzyme reoxidation rate. It increases linearly with film thickness since there is no... 

FIGURE 1.16. (a) Kinetic case diagrams for dopant transport and reaction in electronically conducting polymer films according to the Bartlett-Gardner model. Thick lines separate different approximate solutions to the transport/kinetic problem. Six distinct cases are noted, (b) Computed concentration profiles u x) and site occupancy functions 0 for each of the six cases, (c) Schematic representation of the moving boundary problem (Case 6). 

FIGURE 1.62. Kinetic case diagram for linear sweep voltammetry applied to redox switching in a finite-diffusion space. Regions 1, 2, and 3 denote reversible, quasi-reversible, and irreversible charge percolation kinetics, respectively, whereas A, B, and C represent regions corresponding to infinite diffusion, finite diffusion, and surface behavior, respectively. (Adapted from Ref. 179.)... 

We now introduce the very important concept of a kinetic case diagram. This is a very useful device that enables us to present in a very... 

FIGURE 2.13. Three-dimensional kinetic case diagram according to the Albery-Hillman approach. The block diagram is similar to the one in Fig. 2.12, but now 0 = < 1-... 

FIGURE 2.14. Kinetic case diagram according to the Andrieux-Sav ant approach. The various kinetic possibilities are explicitly stated in the diagram. Note that denoting the Koutecky-Levich plot is linear, whereas implies that a second wave is observed. 

The important parameters defining system behavior are a and d>. We see that a defines the ratio of the substrate concentration in the layer to the Michaelis constant and O quantifies the extent of the reaction zone in the film. We show shortly that these two parameters can be used as axes defining a kinetic case diagram for the system. 

We now present the analysis in terms of a kinetic case diagram. As a natural set of axes we first choose log O as ordinate and log o as abscissa. Hence movement along the ordinate takes us from thin to the thick films, and the movement along the abscissa takes us from a condition of reactant unsaturation to saturation. The case diagram is presented in Fig. 2.32. We see that four cases must be presented. The main approximate expressions for the flux or current are given by Eqn. 118... 

FIGURE 2.32. Kinetic case diagram (log versus log a) for electrocatalysis in a polymer film exhibiting Michaelis-Menten kinetics. The four major cases discussed in the text are illustrated. Also included as insets are approximate analytical expressions for the normalized flux for various limiting values of 4> and a. 

This expression provides a link between Cases 3 and 4 in the kinetic case diagram. It is valid for all values of d> and for s K -... 

FIGURE 2.42. Kinetic case diagram for the Bartlett model, (a) The axes are log (kclka) and log s IKm). Also included as insets are the schematic concentration profiles of substrate and mediator for each of the approximate rate-limiting situations, (b) Case 3 for 0 1 and 4> 1. 

FIGURE 2.46. Kinetic case diagram illustrating the interrelation between the different rate-determining situations for catalytic microparticles immobilized in an ionomer or redox polymer film. The diagram is drawn for 4> = LIXj < 1. The system is characterized by two dimensionless parameters p = y =... 

FIGURE 2.48. A different perspective of the kinetic case diagram. Here the ratedetermining cases are indicated using log 4> and log jS as axes. The situations for y < 1 and y > 1 are shown separately. 

Rigorous quantitative treatments lead to kinetic zone diagrams that distinguish between different extreme and borderline cases depending on which parameters controll the overall reaction (Fig. 2). A realistic picture is obtained from cross... 

Fig. 2a-c. Kinetic zone diagram for the catalysis at redox modified electrodes a. The kinetic zones are characterized by capital letters R control by rate of mediation reaction, S control by rate of subtrate diffusion, E control by electron diffusion rate, combinations are mixed and borderline cases b. The kinetic parameters on the axes are given in the form of characteristic currents i, current due to exchange reaction, ig current due to electron diffusion, iji current due to substrate diffusion c. The signpost on the left indicates how a position in the diagram will move on changing experimental parameters c% bulk concentration of substrate c, Cq catalyst concentration in the film Dj, Dg diffusion coefficients of substrate and electrons k, rate constant of exchange reaction k distribution coefficient of substrate between film and solution d> film thickness (from ref. 

FIGURE 2.1 7. Homogeneous catalysis electrochemical reactions. Kinetic zone diagram in the case where the homogeneous electron transfer step is rate limiting. 

The plateau currents are thus a function of two dimensionless parameters, Jis/ik and 4/4(1 — k/i )- On this basis, a kinetic zone diagram may be established (Figure 4.19) as well as the expressions of the plateau currents pertaining to each kinetic zone (Table 4.1).17 Derivation of these expressions is described in Section 6.4.4. There are in most cases two successive waves, and the expressions of both limiting currents are given in Table 4.1. The general case corresponds to a situation where none of the rate-limiting factors... 

Another case of interest is the transition between no catalysis and the pure kinetic conditions leading to plateau-shaped responses. In the kinetic zone diagram of Figure 2.17, it corresponds to the extreme right-hand side of the diagram, where the cyclic voltammogram passes from the Nernstian reversible wave of the cosubstrate to the plateau-shaped wave, under conditions where the consumption of the substrate is negligible. The peak... 

This is a somewhat opaque and over-complicated result. Thus, to facilitate the extraction of useful kinetic information from experimental data, we return to equation (9.98) in order to identify simpler, asymptotic solutions. These are illustrated in the case diagram of Fig. 9.11. At the bottom of the diagram, where is small and r is therefore large, we obtain the asymptotic steady state solution ... 

The left-hand side (Ihs) of the case diagram (regions C1-C3) locates surface voltammograms for any degree of kinetic reversibility, whereas the situation corresponding to semiinfinite diffusion is located on the rhs of the case diagram (A1-A3). For > 10 the kinetic classification depends only on the kinetic parameter A, whereas when (o < 0.2, the case assignment is determined by the coupled parameter In the... 

The effect of layer thickness L on kinetic behavior in the system is also transparent from this analysis. Starting from the R case we note that increasing L leads to a proportional increase in the current in. As we see from the case diagram, the concentration profiles for both S and B are flat then throughout the film. The catalytic plateau current I l increases with increasing L, as illustrated in Fig. 2.15. However the latter quantity levels off as approaches ,. We note therefore that increasing the layer thickness produces a corresponding increase in the catalytic current but only up to a certain value. We eventually meet a limitation... 

Of course other limitations can occur before passing to rate control via substrate transport in solution. We note that ip and is both decrease as layer thickness increases. This means that they start to affect the kinetics of the mediation, and the phase point representing the system passes from the R zone in the case diagram to S E zone or eventually to the S or the E regions. In the latter cases substrate and electrons transport completely determines the kinetic response. In such a situation as we have previously noted, the catalytic reaction is so rapid that it takes place only in a relatively thin reaction layer whose location in the film depends on the relative rates of the S and E processes. If ip is, then... 

Thus the case diagram very conveniently summarizes the kinetics. The diagnostic criteria listed in Table 2.10 enables us to identify any case by varying the experimental quantities L, and c. ... 

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