Phenomenological rate constants

Depending on the particular experiment (see Fig. 6.17) we not only need to distinguish between the various diffusion coefficients (D, D, D ) but also the corresponding ic values (k, k, k ). In contrast to the diffusion case, in which the concentration gradients provide the driving force and the D s are defined as the proportionality constants between the fluxes and these gradients, the respective kinetic parameter of the reaction case, k, is understood to be the ratio of the flux to the concentration deviation from the equilibrium state, 6c = c — taken at the first 

This applies, in general and independent of mechanism, for small deviations from the equilibrium state, and corresponds to the linear approximation of the Taylor series of J in 6c. 

In this form, Eq. (6.126) applies for simple homogeneous kinetics. If, as is the case for oxygen incorporation, a reservoir of thickness L is coupled, a uniform concentration profile establishes in the sample under the condition of reaction control as directly confirmed by the space-resolved profiles in Fig. 6.51 (quasi-equilibrium). The deviation 6c then determines the flux into this volume and, hence, the amount 

There was an exponential solution too for the diffusion control (see Section 6.5) (but with L /D instead of L/k as time constant). However, the exponential solution only applied to the long-time range there, while here it is Amlid for small times as well. Hence, short time approximation 3delds linearity, and not a Vt approximation, as was obtained for diffusion control (compare Eq. 6.126 with Eq. 6.75a). It should be remembered that two significant presuppositions had to be made in the derivation of Eq. (6.126) and thus have to be considered in the evaluation of the measurements (see below) Namely small deviations from the equilibrimn state, and negligible time delay of the transfer from the gas phase to the locus, to which Eq. (6.126) refers . The detailed analysis can be found in subsection b, where we will also discuss the complexity of the surface reaction. 

It is of interest to pause here briefly and to consider a first-order reaction, just to demonstrate, on the one hand, that in this special case Eq. (6.126) apphes for any distance from equilibrium and, on the other hand, that the effective constant ka in Eq. (6.126) is correlated with the microscopic rate constants. In the case of the (closed) elementary reaction A =iB the rate is 71 — k[A]—k[B], whereby Tl = d[B]/dt. Conservation of mass requires that [A] (t) + [B] (t) = const, from which it follows that Tl = const — (k + k)[B], Equilibrium is reached when t— oo, ([B](t = oo) is termed BJ), Tl becomes zero and the constant is obviously (k + k) (B]. The result  

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From classical transition state theory, the phenomenological rate constant of ET can be expressed as... 

Table 20 Phenomenological rate constants (X10 ° cm molecule s ) and efficiencies (eff = feobs coii) (in parentheses) of the reaction of chiral 2-butylacetate ions with (5,5,5)-tri-sec-butylborate (Scheme 12)...

To distinguish between simultaneous or only consecutive epimerizations at CHD centers, kinetic experiments based on the four isomers of 1 -methyl-2,3-d2-cyclopropane were designed141. The phenomenological rate constants associated with the various reactions leading from one isomer to another employ subscripts to designate the carbon atom at which an epimerization occurs k, indicates a one-center epimerization at C(/), and /q is used for a two-center epimerization at C(i) and C(J) (Scheme 1). In this case, with all four isomers present in equal concentrations at equilibrium, the time dependence of each is governed by four rate constants kuk2- k2, kl2 = kl2 and /c23. But, at best, only three kinetic parameters may be found experimentally kt, k2i and (k2 + kl2). 

A different experimental approach to the relative importance of one-center and two-center epimerizations in cyclopropane itself was based on the isomeric l-13C-l,2,3-d3-cyclopropanes165"169. Here each carbon has the same substituents, one hydrogen and one deuterium, and should be equally involved in stereomutation events secondary carbon-13 kinetic isotope effects or diastereotopically distinct secondary deuterium kinetic isotope effects may be safely presumed to be inconsequential. Unlike the isomeric 1,2,3-d3-cyclo-propanes (two isomers, only one phenomenological rate constant, for approach to syn, anti equilibrium), the l-13C-l,2,3-d3-cyclopropanes provide four isomers and two distinct observables since there are two chiral forms as well as two meso structures (Scheme 4). Both chiral isomers were synthesized, and the phenomenological rate constants at 407 °C were found to be k, = (4 l2 + 8, ) = (4.63 0.19)x 10 5s l and ka = (4kl2 + 4, ) = (3.10 0.07) x 10 5 s 1. The ratio of rate constants k, kl2 is thus 1.0 0.2 both one-center and two-center... 

In this chapter we consider bimolecular reactions from both a microscopic and a macroscopic point of view and thereby derive a theoretical expression for the macroscopic phenomenological rate constant. That is, a relation between molecular reaction dynamics and chemical kinetics is established. 

Fig. 8.13 (a) Potential dependence of the phenomenological rate constant k,r derived for mechanism I. Note that for a two step mechanism, this rate constant contains terms associated with surface recombination, so that it is not the true rate constant for electron transfer. The influence of the modulation of potential due to surface charging is shown, (b) Potential dependence of the phenomenological rate constant k,t for case /. The influence of the dynamic modulation of surface potential by accumulated reaction intermediates is... 

Here, /r is a phenomenological rate constant. The great advantage of this model is its computational simplicity, allowing rapid calculations even for real RPs with multiple nondegenerate HFCs. A more realistic model of diffusion in free solution developed by... 

The symbols k and k in equation 1 represent respectively the phenomenological rate constants for cis-trans isomerization and racemization. From this equation one can see that a pure Smith mechanism k = k 2 = ) would give kjk = 2. A pure Benson mechanism [ki = ki2 =0) would give kjk = 1. Finally, a pure Hoffmann mechanism (kj = = 0) would give kjk = 0. 

The mechanistic and phenomenological rate constants in equation 2 have the same meanings as those in equation 1 and Z12 are the mechanistic secondary isotope effects for formation of the biradical and correlated double rotation respectively. 

In fact the results are not quite as dramatically different as they appear when presented this way. The mechanistic rate constants are derived from phenomenological rate constants that can be described as linear combinations of the mechanistic ones. The discrepancy between the two results can be traced to a difference in just one of the phenomenological rate constants—the one corresponding to the first order loss of optical activity for a 1 1 mixture of (1R,25)- and (lR,2R)-phenylcyclopropane-2-d. The peculiar feature of this problem is that the Berson group obtained an internally consistent set of results from two independent experiments with one value for this rate constant and the Baldwin group also obtained an internally consistent set from two independent experiments but with a value that differed by a factor of 2.7 from that found by the Berson group. It is hard to know how to reconcile such results and so, to be objective, one must probably say that despite the immense amount of effort put into the problem the mechanism of stereomutation of phenylcyclopropane remains something of a mystery. 

Three phenomenological rate constants were obtained in three separate experiments (see Figure 18). The rate constant for racemization, was obtained from optically active l-methylspiro[2.4]hepta-4,6-diene. The rate constant for cis-trans isomerization, was... 

The three phenomenological rate constants, having the indicated experimental values, can be equated with combinations of mechanistic rate constants as follows ... 

The observed (phenomenological) rate constant, will be a function of... 

A detailed analysis of mechanisms I and II by Peter et al. [89] that takes all these factors into account has shown that the apparent or phenomenological rate constants k,r and krec derived from IMPS data are functions of the rate constants for all the steps in the reaction. For both mechanisms, the analysis shows that k,r is no longer potential independent because it contains a term associated with recombination via... 

Figs. 17a and 17b illustrate the quite complicated potential dependence of the apparent or phenomenological rate constants k,r and krec predicted for mechanism I when dynamic surface charging is taken into account. More work is required to develop the analysis of experimental data to this level of sophistication. 

Fig. 17b. Potential dependence of the phenomenological rate constant k for case I [89]. The influence of the dynamic modulation of surface potential by accumulated reaction intermediates is shown.

In the remainder of this subsection, the goal is to relate these phenomenological rate constants to more fundamental parameters of the growing oxide layer. Table 7.1 lists the parabolic rate constants for a number of metals oxidized in pure oxygen at 1000°C. 

Cross sections can be converted to phenomenological rate constants by using the formula... 

Dynamic parameters such as diffusion coefficients and the phenomenological rate constant of ion transfer can also be evaluated. 

Analogous expressions for the phenomenological rate constants can be derived that relax the assumptions of steady-state kinetics for the I minima and of local equilibrium in the A and B states (kf ... 

In essence, the DPS approach reduces the problem of global kinetics to a discrete space of stationary points. Phenomenological rate constants can then be extracted under the assumption of Markovian dynamics within this space, which requires that the system has time to equilibrate between transitions and lose any memory of how it reached the current minimum. The Markovian assumption is therefore an essential part of the framework. However, we can regroup the stationary points into states whose members are separated by low barriers so that the Markov property is likely to be better obeyed between the groups (Section 14.2.3). 

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