Heterogeneous process kinetic curves

Fig. 12 Experimental approach curves of a 238 nm-radius pipet fitted with theoretical values. The tip potential was 0.45 V and the substrate potential was 0.20 ( ), 0.225 (x), 0.25 ( ), 0.275 (A), 0.30 (4 f), 0.325 (O), 0.35 ( ), 0.375 (A), 0.40 (O), and 0.425 V ( ). Curve 1 shows the theoretical curve for a diffusion-controlled process, and curves 2-6 are theoretical curves for kinetically controlled processes. Inset the dependence of the heterogeneous rate constants on Es. Reprinted from ref 59. Copyright 2009 with permission from WILEY-VCH Verlag GmbH Co. KGaA, Weinheim.

FIGURE 5.2 Simulated feedback SECM transients with various rate constants for an irreversible heterogeneous process at the substrate. The upper and lower dashed curves correspond, respectively, to the limits A ijs ° and A [,s=0. The solid curves (from top to bottom) represent log A s=3.0, 1.5, 1.0, 0.5, 0, -0.5, and -1.0. RG=10. (Reprinted with permission from Bard, A.J., Mirkin, M.V., Unwin, P.R., and Wipf, D.O., Scanning electrochemical microscopy. 12. Theory and experiment of the feedback mode with finite heterogeneous electron-transfer kinetics and arbitrary substrate size, J. Phys. Chem., 96, 1861-1868, 1992. Copyright 1992 American Chemical Society.)... 

The numerical values for ki. .. k4 vary with RG. For instance, for RG = 10, the following values provide the analytical function Jfei = 0.40472, k2 = 1.60185, k3 = 0.58819, and k4 = -2.37294 [12]. The analytical approximations for hindered diffusion provide a way to determine d from experimental approach curves. For this purpose, one can use an irreversible reaction at the UME (often 02 reduction). In such a case, Fig. 37.2, curve 1 is obtained irrespective of the nature of the sample. Besides the mediator flux from the solution bulk, there might be a heterogeneous reaction at the sample surface during which the UME-generated species O is recycled to the mediator R. The regeneration process of the mediator might be (i) an electrochemical reaction (if the sample is an electrode itself) [9], (ii) an oxidation of the sample surface (if the sample is an insulator or semiconductor) [14], or (iii) the consumption of O as an electron acceptor in a reaction catalyzed by enzymes or other catalysts immobilized at the sample surface [15]. All these processes will increase (t above the values in curve 1 of Fig. 37.2. How much iT increases, depends on the kinetics of the reaction at the sample. If the reaction of the sample occurs with a rate that is controlled by the diffusion of O towards the sample, Fig. 37.2, curve 2 is recorded. If the sample is an electrode itself, such a curve is experimentally obtained if the sample potential... 

In Fig. 4.18, the influence of the kinetic parameters (k°, a) on the ADDPV curves is modeled at a spherical microelectrode l /Dr /r, = 0.2). In general terms, the peak currents decrease and the crossing and peak potentials shift toward more negative values as the electrode processes are more sluggish (see Fig. 4.18a). For quasireversible systems (k° 10-2 — 10 4 cm s ). the peak currents are very sensitive to the value of the heterogeneous rate constant (k°) whereas the variation of the crossing potential is less apparent. On the other hand, for totally irreversible... 

The potentials Ex and E2 should be chosen in such way that at Ex no electrode process occurs and at E2 the electrode reaction of an electroactive species takes place. If the rate of the electrode process is controlled only by diffusion, the Cottrell equation [Eq. (3.6)] can be applied. Therefore, the observed current should be a linear function of t m with the intercept at the origin (a test for diffusion control). The diffusion coefficient of the electroactive species is directly proportional to the slope of the curve. The heterogeneous rate constant of a kinetically limited electrode reaction (kc or k3) also can be evaluated. 

Although Eq. (23) was derived for a one-step heterogeneous ET reaction, it was shown to be applicable to more complicated substrate kinetics (e.g., liquid-liquid interfacial charge transfer [38, 39, 67], ET through self-assembled monolayers [68, 69], and mediated ET in living cells [70-73]). The effective heterogeneous rate constant obtained by fitting experimental approach curves to Eq. (23) can be related to various parameters, which determine the rates of those processes, as discussed in the referred publications. 

As in heterogeneously catalyzed processes, the rate of enzyme reactions usually follow saturation kinetics with respect to the concentration of S as shown in Fig. 7.1. This curve is redrawn in Fig. 7.2 to show the relationship to and K]vi which is the substrate concentration when v = 1/2 V ax- At low values of [S]... 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Many different types of interfacial boundaries can be probed by SECM. The use of the SECM for studies of surface reactions and phase transfer processes is based on its abilities to perturb the local equilibrium and measure the resulting flux of species across the phase boundary. This may be a flux of electrons or ions across the liquid/liquid interface, a flux of species desorbing from the substrate surface, etc. Furthermore, as long as the mediator is regenerated by a first-order irreversible heterogeneous reaction at the substrate, the current-distance curves are described by the same Eqs. (34) regardless of the nature of the interfacial process. When the regeneration kinetics are more complicated, the theory has to be modified. A rather complete discussion of the theory of adsorption/desorption reactions, crystal dissolution by SECM, and a description of the liquid/liquid interface under SECM conditions can be found in other chapters of this book. In this section we consider only some basic ideas and list the key references. 

As a sufficiently negative tip potential both ET reactions are diffusion controlled, and the rate of the overall process is limited by heterogeneous reaction (33b). Its rate constant can be determined from the current-distance curves as discussed in Sec. II. The kinetic analysis of a more complicated ECE mechanism can be reduced to the measurement of an effective heterogeneous rate constant at the ITIES. 

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