Rate constants differences

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

These values of the reaction rate constants differ from those cited by Pannetier and Souchay (6) because these individuals erroneously treated the two reactions as if they were of the simple parallel type instead of as if there were a competition between the acrolein and butadiene molecules for other butadiene molecules. 

A comparison of equations 7.3.43 and 7.3.38 shows that they are of the same mathematical form. Both can be written in terms of four measurable kinetic constants in the manner of equation 7.3.40. Only the relationship between the kinetic constants and the individual rate constants differs. Thus, no distinction can be made between the two mechanisms using steady-state rate studies. In general, the introduction of unimolecular steps involving only isomerization between unstable intermediate complexes does not change the form of the rate expression. 

The second and third relaxation processes were coupled, where the observed rate constants differed by a factor of 3 to 7 and the rate constant for each relaxation process varied linearly with the DNA concentration.112 This dependence is consistent with the mechanism shown in Scheme 2, where 1 binds to 2 different sites in DNA and an interconversion between the sites is mediated in a bimolecular reaction with a second DNA molecule. For such coupled kinetics, the sum and the product of the two relaxation rate constants are related to the individual rate constants shown in Scheme 2. Such an analysis led to the values for the dissociation rate constants from each binding site, one of the interconversion rate constants and the association rate constant for the site with slowest binding dynamics (Table 2).112 The dissociation rate constant from one of the sites was similar to the values that were determined assuming a 1 1 binding stoichiometry (Table 1). 

The kinetics for all guests were fit to the sum of two exponentials. The recovered observed rate constants differed by a factor of 4 for the kinetics of these guests with ct-DNA and by a factor of 10 for the kinetics of the guests with the polydeoxynuc-leotides. For this reason, the kinetics were analyzed by determining an apparent observed rate constant defined by the fractional amplitudes (A,) and the individual rate constants ... 

AS -8 to +4 e.u. Even though the racemization rate constants differ slightly, their distinct dependence on the steric and to a lesser extent on the electronic effects of the substituents bonded to the sulfinyl sulfur atom was noted. It deserves adding that the activation volume for racemization of methyl p-tolyl sulfoxide 41, A F 0 ml/mol, is also consistent with the pyramidal inversion mechanism (249). 

Therefore, rather than varying the test temperature, a much simpler and more accurate procedure is to actuaily perform aii measurements at the standard temperature of 538°C (1000°F), as has been proposed previously [2], By an appropriate choice of F, W and conversion (e.g. 0.3-60%) rate constants differing by over four orders of magnitude can be readily measured. It has been found that even for the medium pore zeolite ZSM-5, of high activity, no diffusion limitations exist even at this relatively high temperature [8,12] except for very large crystals exceeding 40 pm [13]. At 538°C it is also easy to woh<... 

Therefore rate constant differs from that obtained in the atomic case by the factor (fy/fj ). Since fyis usually of the order of unity, whereas may be from 10 to 100 for a complex molecule, the ratio fylfji is 10-1 to 10-2, so that ifylf is 10- to lO-. ... 

One of the surprising features of Ritchie s studies was a constant selectivity. While the absolute rate constants differed considerably over the series of cations, the selectivities toward pairs of nucleophiles were constant and independent of the reactivities of R" ". Rate constants were correlated by a simple two parameter equation... 

As expected from Baldwin s rules, with suitable substrates, both regiospecific 5- or 6-exo addition could be observed. The electronically symmetrical alkene 10 underwent both 5-exo and 6-endo-trig addition in a ratio of 18 1. This is in accord with Baldwin s rule in that, whilst both processes are favourable, five-membered rings are formed more readily than six-membered rings. They also showed that 6-endo-trig and 6-exo-trig were both favourable processes with rate constants differing by a factor of less than 3. 

Hancock et al. [1989] used a version of the small curvature semiclassical adiabatic approach introduced by Truhlar et al. [1982] to calculate the temperature dependence of the rate constant, as shown in Figure 6.29. Variations in k(T) below the crossover point (25-30 K) are due to changes in the prefactor due to zero-point vibrations of the H atom in the crystal. Obviously, the gas-phase model does not take these into account. The absolute values of the rate constant differ by 1-2 orders of magnitude from the experimental ones for the same reason. 

The models rely on the derivation of rate constants through laboratory experiments or field experiments. Because rate constants differ among organisms and chemical substances, the development of kinetic models often requires a substantial effort. In some cases, empirically based correlations can be used to assess the rate constants. 

Rate Constant Differences Among Classes of Surfactants... 

In a long series of papers on the master equation, Pritchard and his coworkers elucidated for the first time the effects of rotational and vibrational disequilibrium on the dissociation and recombination of a dilute diatomic gas. Ultrasonic dispersion in a diatomic gas was analyzed by similar computational experiments, and the first example of the breakdown of the linear mixture rule in chemical kinetics was demonstrated. A major difficulty in these calculations is that the eigenvalue of the reaction matrix (corresponding to the rate constant) differs from the zero eigenvalue (required by species conservation) by less than... 

Declustering seems to be also possible by the application of appropriate catalyst supports. For tris-allyl-neodymium Nd(r/3- C3H5P dioxane the rate constants for monomer consumption are determined for supported as well as for unsupported catalysts. Under comparable experimental conditions the respective rate constants differ by a factor of 100. This observation can be explained by differences in Nd efficiency or by differences in the concentration of active Nd sites [408]. 

Platination Step, i) Starting from the dichloro complex, the first aquation (1) (Scheme 2) is rate-determining, but the kinetic data do not tell which is/are the actual DNA-platinating species in the various conditions, ii) The kpi rate constants differ by one order of magnitude from 0.3 to 2.5 m-1 s-1 [48] [78] [82], but the value of 2.08 0.07 m-1 s-1 that we have also deter-... 

The previously discussed characters will influence the electron transfer rates implying anions. One of the simplest examples was given by the rate constant difference observed in reactions in which pyrene (Py) reacts with an electron or an electron-cation pair [44,45]. The same type of difference was measured in the exchange between the radical anion of biphenyl (B) and pyrene (Py) [46]. The reduced reactivity is the consequence of the cation proximity in the ion pair. 

Tight absorption, light emission, and nonradiative electron-transfer rate constants differ in the last term of equation (18) (as well as in /(veff) note that AG p is positive for light absorption and negative for emission). 

Under typical polymerization conditions, the total concentration of growing species is e.g. 10 mole F and the concentration of polymer is equal to e.g. 2.5 mole 1. Under these conditions, taking into account the value of Kg = 3 10 mole", we would have 10" mole 1" of oxonium ions (assuming that dimethoxymethane is a suitable model fmonomer molecules converted into polymer, only 10 wouU be added through carbenium ions and the rest throu oxonium ions. The actual data for 1,3-dioxolane are not known at pr nt and may differ from values found for dimethoxymethane. Nevertheless, of primary importance is the finding that the reactivities of alkoxycarbenium ions and tertiary oxonium ions toward linear acetals, expressed through the corresponding rate constants, differ for the discussed above conditions only by 10 times. 

The excess energy above the origin of the LE state required to reach these high rate constants differs somewhat among different systems. In the intermolecular systems, it is in the range 20-400 cm, whereas the linked systems, it may be as high as 1000 cm or more. The difference may be due to the more stringent restrictions on the relative motion of the donor and acceptor in the linked systems. 

The measured rate constants for cross-reactions and those calculated from Eq. (104) are listed in Tables 53-55. A comparison of the calculated and observed constants indicates that Eq. (104) is solved with an accuracy of an order of magnitude. An attempt to solve this equation by introducing several adjusted 2 values was made in Ref. 364. Eqs. (105-108) are justified at appreciable differences in the charges of the reacting species. But in this case, the correction makes no marked contribution to the improvement in data coincidence [106]. Even for the structurally very similar complexes with close properties, i.e., various isomers of sarcophaginate [Co(diAMHl,2pnsar)] + cation, the experimental rate constants differ from those calculated from Eq. (104) by a factor of 5-7. Sargeson attributed this difference to the... 

If we compare the rate equation above with the phenomenological one for desorption, we see that the transition probability is nothing but the reaction rate constant. This is not always the case. It is quite common that the transition probability and the reaction rate constant differ by a factor that depends on the number of neighbors of a site. It s very important to know that factor when one tries to calculate rate constants quantum chemically, because one can only calculate transition probabilities using quantum chemical methods, and one needs the derivations we present here to obtain the rate constants. 

Figure 4. Normalized peak potential versus the homogeneous rate constant. Different values of k, demonstrate the dependence of peak potential on the heterogeneous rate constant. Other parameters are the same as for Fig. 1.

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