Rate constant, dimensionless

Observe that aok has units of reciprocal time so that aokt is dimensionless. The grouping OQkt is the dimensionless rate constant for a second-order reaction, just as kt is the dimensionless rate constant for a first-order reaction. Equivalently, they can be considered as dimensionless reaction times. For reaction rates governed by Equation (1.20),... 

This contains the dimensionless rate constant, K = aokthatch, plus the initial and final densities. The comparable equation for reaction at constant density is... 

The second CSTR has the same rate constant and residence time, but the dimensionless rate constant is now based on (n, )2 = 0.618a rather than on Uin- Inserting A t2(am)2 = = (0-5)(0.618) = 0.309 into Equation... 

FIGURE 4.3 Effect of recycle rate on the performance of a loop reactor. The dimensionless rate constant is based on the system residence time, t = V/Q. The parameter is q/Q. 

FIGURE 8.1 Fraction unreacted versus dimensionless rate constant for a first-order reaction in various isothermal reactors. The case illustrated with diffusion is for = 0.1. 

For the formulation of a general solution, the dimensionless quantities defined by Eqs. (21)-(25) are used, but the dimensionless rate constant is defined by Eq. (33) rather than Eq. (26). 

In the case of 0-pipettes, the collection efficiency also decreases markedly with increasing separation. The situation becomes more complicated when the transferred ion participates in a homogeneous chemical reaction. For the pseudo-first-order reaction a semiquantita-tive description is given by the family of dimensionless working curves calculated for two disks (Fig. 6) [23]. Clearly, at any separation distance the collection efficiency approaches zero when the dimensionless rate constant (a = 2kr /D, where k is the first-order rate constant of the homogeneous ionic reaction) becomes 1. 

When the B/C conversion is fast, C is produced close to the electrode surface and is likely to diffuse back and be oxidized there. The situation is similar to the ECE case in the ECE-DISP problem discussed in Section 2.2.5. In the ECE case, the cyclic voltammetric responses depend essentially on the dimensionless rate constant, 2 = (7ZT/F)(k/v), of the B/C reaction in the framework of two subcases according to the order in which the two standard potentials, Z yBand c, lie (note that in the D/C couple, D is the oxidized form). Typical cyclic voltammograms are shown in Figure 2.25a and b for the two subcases. 

Because of the importance of points where the trace of the Jacobian matrix vanishes, we will denote such values of the parameters by a superscript asterisk. Equation (3.65) has two real roots, provided the dimensionless rate constant for the uncatalysed step has a value less than In terms of the original rate constants this requires k2 > 8ku as presented previously ( 2.5). For the data in Table 2.3, n = 0.9847 and /rf = 0.1021. As k u increases, these two bifurcation points move closer together for ku = 0.1, // = 0.790 and n f = 0.420, so the oscillatory range is smaller. 

There are two dimensionless rate constants which characterize this system. One is related to the reactant decay step and we use the symbol e as in the previous chapter ... 

For a system with fixed fi, 0SS is inversely proportional to the dimensionless rate constant k. However, if we look at this result in terms of the original dimensional quantities, the temperature rise does not depend on the value of... 

We may view eqns (4.71)—(4.73) in another way. Choose a system with y < Next choose the dimensionless rate constant k. If k is less than (1 — 4y)e-2, eqn (4.71) can be solved to yield two positive roots 9 and 9. From these values for the stationary-state temperature excess we calculate the reactant concentration required for Hopf bifurcation from eqn (4.72) whilst (4.73) gives the stationary-state concentration of the intermediate A. 

As the dimensionless rate constant K is of order unity, this time will be very short for small y. For our example, y = 0.1 and A7bc 1.6 x 10-4 K-1. 

This range decreases in size as k2 increases. In fact the form of the numerator in eqn (6.60) shows that there is an upper limit on the dimensionless rate constant, i.e. for an isola to exist at all we require that... 

For any given value of f 0, each root of this equation gives a quadratic for the residence time at which extinction (upper root) or ignition (lower root) occurs. Clearly we again require fi0 < , otherwise the right-hand side of (6.67) is not real. Whether or not the resulting quadratics for the extinction and ignition residence times have real roots then depends on the value of the dimensionless rate constant k2. 

If the flow line is to have a minimum gradient which coincides exactly with the ignition tangent Tv then we require 4k2 = 0.12096 so that k2 = 0.03024. For this value of the dimensionless rate constant, the stationary-state isola has grown so large that it just touches the lowest branch of solutions, as shown in Fig. 6.16(e). 

The third, and most interesting, case corresponds to the smallest values of the dimensionless rate constant ... 

If the dimensionless rate constant satisfies inequality (10.78), the well-stirred system again has two Hopf bifurcation points fi and n. However, within the range of reactant concentrations between these, the uniform state also changes character from unstable focus to unstable node at n and n 2, as shown in Fig. 10.10. 

The dispersion relation det(J) = 0 can also be viewed in the n-K plane, as in Fig. 10.14. This allows us to read off the maximum value of the dimensionless rate constant for which a given mode can be found. The left-hand ordinate corresponds to general values of the group nn/y112, the right-hand ordinate to our specific example y = 6n2. Thus, for the latter case, a pattern with n = 5 requires 0.01 < k < 0.0148. 

Thus one may obtain kt by multiplying the quantity previously referred to as the dimensionless time by k,tf, the dimensionless rate constant. This is particularly useful in constructing the single-potential-step working curve for the ECE mechanism mentioned earlier. This parametric substitution allows the experimental time to be rendered dimensionless by the inverse of the rate constant instead of by some known time tk. 

The shape and location of the current-potential curves are strongly dependent on the kinetics of the electrode reaction through the dimensionless rate constant, defined 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. 

It is also worth pointing out that a similar result to that shown in Fig. 3.3 is obtained if we analyze the effect of the time in the response of a quasi-reversible charge transfer process, i.e., for a given value of the rate constant k°, a decrease of the time leads to a decrease of the dimensionless rate constant Kplane and therefore to a higher irreversible character of the process. This fact can be used to ascertain at a glance if a particular electrode process behaves in a reversible or non-reversible way, since in the first case no influence of time on the normalized current is observed (see Eq. (2.36)). 

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.,... 

This mixed influence can be observed from the expression of (Eqs. 3.68 and 3.69). In order to analyze the influence of the electrode size, Fig. 3.10a shows the current-potential curves obtained for a charge transfer process with different values of the dimensionless rate constant K°phe for a fixed/ 0 = 10-4 cm s 1 in NPV with a time pulse t = 0.1 s (i.e., for different values of the electrode radius ranging from 100 to 1 pm). As a limiting case useful for comparison, the current-potential... 

The influence of the chemical kinetics is analyzed in Fig. 4.31 where ADDPV curves are plotted for different values of the dimensionless rate constant %2(= (k + ki)zi). For comparison, the curve corresponding to a simple, reversible charge transfer process (Er) of species C + B for the CE mechanism and of species A for the EC one has also been plotted (dashed line in Fig. 4.31a, b). As can be observed, the behavior of ADDPV curves with is very different depending on the reaction scheme. For the CE mechanism with K = (1 /Kepeak current increases and the peak potential shifts toward more negative values as the kinetics is faster, that is, as xi increases. For very fast chemical reactions, the ADDPV signal is equivalent to that of a reversible E mechanism (Er) with... 

Fig. 5.12 CV response corresponding to a charge transfer process of different reversibility degrees taking place at a planar electrode, calculated numerically by following the procedure given in [21, 22] The values of the voltammetric dimensionless rate constant plane aPPCar tllC g C...

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