Dimensionless rate constant, heterogeneous

Matsuda and Ayabe [15] defined a dimensionless heterogeneous rate constant A as... 

This dimensionless rate constant contains typical parameters of the process (i.e., the heterogeneous rate constant k°, the diffusion coefficient, and the experiment time), thus reflecting that the behavior of the process is the result of a combination of intrinsic (kinetics and diffusion) and extrinsic (time window) effects. The effect of Kplane in the voltammograms obtained when both species (a) or only oxidized species O (b) are initially present can be seen in Fig. 3.3. 

By assuming that a reversible process corresponds to R() > 10 and a fully irreversible one to R° < 0.05 (i.e., the heterogeneous rate constant is ten times higher or 20 times smaller than the mass transport coefficient, respectively), in the interval 0.05 < R° < 10 the process can be considered as quasi-reversible. The variation of log(/ °) with log(/°) for / = 1 has been plotted in Fig. 3.5, and the three regions have been delimited. From the above criterion, a totally irreversible process is characterized by a value of/0 < 0.17 (which corresponds to a dimensionless rate constant /c°lane < 0.042), and a reversible behavior is attained for,/0 > 23.6 (i.e.,... 

Theoretical i scv E ancl Gscvc/2f — E curves, calculated from Eqs. (6.202) and (6.205), respectively, for different values of the dimensionless catalytic rate constant /cv. have been plotted in Fig. 6.31. These curves correspond to a cyclic staircase potential with AE = 5mV and t = 10ms (v = 0.5 V s-1). Two values of the heterogeneous rate constant °(s-1) = 200 and 2, which refer to reversible (laand lb) and totally irreversible (lc and Id) electrochemical behavior, have been considered. 

Eqs. (7.143) and (7.148), for different values of the catalytic rate constant kc and a fixed square wave pulse amplitude Esw = 100 mV are plotted in Fig. 7.57. Two values of the dimensionless heterogeneous rate constant have been considered... 

Read in starting potential pstart and reversal potential prev (both in dimensionless potential units), nrper, the number of time intervals per potential unit swept, and A, the simulation parameter, and Ko, the dimensionless heterogeneous rate constant. 

In the example program EX CV, as mentioned, two kinds of boundary conditions are accommodated those for a quasireversible reaction, and for a fully reversible reaction. The division is made on the basis of the dimensionless heterogeneous rate constant Ko if it exceeds 1000, the reaction is considered reversible. 

For the irreversible case with dimensionless heterogeneous rate constant K, G is given as... 

When the reaction being studied is relatively slow, the HE relationship does not follow Eq. 17D. The problem was treated by Koulecky, who was able to express the activation-controlled current i in terms of a dimensionless parameter %, which is linearly related to the heterogeneous rate constant of the reaction... 

Figure 7. Simulated cyclic voltam-mograms pertaining to O + e R for three different values of the dimensionless heterogeneous rate constant...

In the preceding chapter, the charge transfer was assumed to be so fast that it was Nernstian. However, this condition cannot always be fulfilled, either because the charge transfer is intrinsically slow (i.e., k° is small), or because the follow-up reaction causes the voltammetric wave to shift, thereby lowering the apparent heterogeneous rate constant kapp by the potential factor (Eq. 59). Let us consider once more the DIM 1 (EC2) mechanism, but now open for the possibility that the heterogeneous step is not necessarily reversible. It is then convenient to define two dimensionless parameters related to k° and /cdimi... 

Figure 19. (a) Steady-state voltammograms of a simple redox system as a function of its intrinsic heterogeneous rate constant (number on the curves k° in cms ) for 5"/D = 0.1 s. (b) Variations in the half-wave potential E1/2 as a function of the dimensionless rate constant A = k°5/D for different values of the transfer coefficient a. 

Dimensionless heterogeneous rate constant Chemical potential Standard chemical potential Sweep rate in CV or LSV (Vs )... 

Lastly, the system described by Reaction (5) might be quasireversible or even irreversible, in which case the boundary condition is given by the Butler-Volmer equation. It is preferable to express it in dimensionless form, using potential p, and the heterogeneous rate constant ko... 

In LSV and CV, a redox system may show a Nernstian, quasireversible or totally irreversible behavior depending on the scan rate employed, since V determine the time available for the electrodesolution interphase to attain the equilibrium condition dictated by the Nernst equation. Such a dependence is usually rationalized by the following dimensionless parameter, comparing the standard heterogeneous rate constant with the scan rate v ... 

Fig. 25. Dependence of the dimensionless peak-potential shift Cp on the logarithm of the kinetic parameter A p for different values of the electron transfer coefficients. Cp is defined in the Eq. (51), = k° is the standard heterogenous rate constant, tp the pulse...

In the present work, the above continuum will be described by means of dimensionless boundary condition parameters, r and r, for positive and negative mobile charge species, respeclively. These parameters may be related to thermally activated heterogeneous rate constants or to surface recombination rates. Although they may sometimes be complex and frequency-dependent in ac situations, such possibilities will not be further considered herein. When r = 0 for a given positive species at a given electrode, the contact is completely blocking for this species. 

The heterogeneous rate constant is converted to dimensionless form by multiplying through by the appropriate simulation units. 

For the quasireversible case, two species A and B must again be considered and the two boundary conditions are the flux condition (2.49) and the dimensionless form of the Butler-Volmer equation. The forward and backward heterogeneous rate constants kf and kb are normalised ... 

Here, cp = (E —E ) is a dimensionless potential and rs = 1 cm is an auxiliary constant. Recall that in units of cm s is heterogeneous standard rate constant typical for all electrode processes of dissolved redox couples (Sect. 2.2 to 2.4), whereas the standard rate constant ur in units of s is typical for surface electrode processes (Sect. 2.5). This results from the inherent nature of reaction (2.204) in which the reactant HgL(g) is present only immobilized on the electrode surface, whereas the product is dissolved in the solution. For these reasons the cathodic stripping reaction (2.204) is considered as an intermediate form between the electrode reaction of a dissolved redox couple and the genuine surface electrode reaction [135]. The same holds true for the cathodic stripping reaction of a second order (2.205). Using the standard rate constant in units of cms , the kinetic equation for reaction (2.205) has the following form ... 

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