Rate Constant Expressions

It is of interest to detennine when the linewidth F( ) associated with the RRKM rate constant lc(E) equals the average distance p( ) between the reactant energy levels. From equation (A3.12.54) F( ) = Dk( ) and from the RRKM rate constant expression equation (A3.12.15) p(Ef = hl% K( )/M( - q). Equating these two... 

The termination constants kt found previously (see Table XVII, p. 158) are of the order of 3 X10 1. mole sec. Conversion to the specific reaction rate constant expressed in units of cc. molecule" sec. yields A f=5X10". At the radical concentration calculated above, 10 per cc., the rate of termination should therefore be only 10 radicals cc. sec., which is many orders of magnitude less than the rate of generation of radicals. Hence termination in the aqueous phase is utterly negligible, and it may be assumed with confidence that virtually every primary radical enters a polymer particle (or micelle). Moreover the average lifetime of a chain radical in the aqueous phase (i.e., 10 sec.) is too short for an appreciable expectation of addition of a dissolved monomer molecule by the primary radical prior to its entrance into a polymer particle. 

Worked Example 8.10 What is the relationship between the values of rate constant expressed in units of s-1, and expressed in units of min-1 and year-1 ... 

When translational diffusion and chemical reactions are coupled, information can be obtained on the kinetic rate constants. Expressions for the autocorrelation function in the case of unimolecular and bimolecular reactions between states of different quantum yields have been obtained. In a general form, these expressions contain a large number of terms that reflect different combinations of diffusion and reaction mechanisms. 

In both TST and VTST, quantum mechanical tunneling is introduced into the rate constant expression as a correction factor usually referred to as k. A short discussion of k which is used largely with TST is presented in Section 6.3.1. Tunneling has been explored much more thoroughly in connection with VTST and this work will be discussed later. 

As mentioned earlier, practically all reactions are initiated by bimolecular collisions however, certain bimolecular reactions exhibit first-order kinetics. Whether a reaction is first- or second-order is particularly important in combustion because of the presence of large radicals that decompose into a stable species and a smaller radical (primarily the hydrogen atom). A prominent combustion example is the decay of a paraffinic radical to an olefin and an H atom. The order of such reactions, and hence the appropriate rate constant expression, can change with the pressure. Thus, the rate expression developed from one pressure and temperature range may not be applicable to another range. This question of order was first addressed by Lindemann [4], who proposed that first-order processes occur as a result of a two-step reaction sequence in which the reacting molecule is activated by collisional processes, after which the activated species decomposes to products. Similarly, the activated molecule could be deactivated by another collision before it decomposes. If A is considered the reactant molecule and M its nonreacting collision partner, the Lindemann scheme can be represented as follows ... 

For a quasireversible electrode reaction, the kinetic equation for reaction (2.204) can be attributed with a standard rate constant expressed in units of either cms (2.222), or s- (2.223) ... 

Following the determination of the geometry and the thermochemistry of transition states, the rate parameters for the two silane decomposition pathways can be obtained directly by the TST formulation presented earlier. These calculations have led to unimolecular rate constant expressions 10 exp(-91000/RT)s-S and 10 exp(-62000/l r)s" for Si-H bond scission and H2 elimination reactions, respectively. These results clearly... 

A quantitative measure of an enzyme s ability to lower the activation barrier for the reaction of a substrate in solution. Catalytic proficiency (a unitless parameter) equals the enzyme-catalyzed reaction rate constant (expressed as Acat/Xm) divided by the rate constant (Anon) for the noncatalyzed reference reaction. 

As seen in Tables 22—25, the Arrhenius preexponential factors Aa for the initiation step are very low, 10 in 7, 10 in 20, 10 " in 41 and 1in 44. These are very low values for bimolecular reactions for which values of about 10 ° are observed and also predicted by the Transition State Theory Thus step (a) belongs to a class of slow reactions , some of which might have ionic transition states . The activation entropies AS obtained from the Transition State Theory rate constant expression... 

Equation (C) gives the most commonly used form of the rate constant expression of Troe and co-workers ... 

A simple diagram depicting the differences between these two complementary theories is shown in Fig. 1, which represents reactions at zero driving force. Thus, the activation energy corresponds to the intrinsic barrier. Marcus theory assumes a harmonic potential for reactants and products and, in its simplest form, assumes that the reactant and product surfaces have the same curvature (Fig. la). In his derivation of the dissociative ET theory, Saveant assumed that the reactants should be described by a Morse potential and that the products should simply be the dissociative part of this potential (Fig. Ib). Some concerns about the latter condition have been raised. " On the other hand, comparison of experimental data pertaining to alkyl halides and peroxides (Section 3) with equations (7) and (8) seems to indicate that the simple model proposed by Saveant for the nuclear factor of the ET rate constant expression satisfactorily describes concerted dissociative reductions in the condensed phase. A similar treatment was used by Wentworth and coworkers to describe dissociative electron attachment to aromatic and alkyl halides in the gas phase. ... 

Rate Constant. The logarithms of second-order rate constants (expressed as M sec.-1) are given for 25°C. unless otherwise indicated. These are rounded off to the nearest 0.1 unit. The ionic strengths are often 0.1-0.2 M. 

In solution the over-all process of excimer dissociation includes escape of the dissociation products from the solvent cage, and the corresponding rate constant expressed as... 

After substitution of these expressions for the partition functions and much simplification, the CTST rate constant expression becomes... 

The rate constant expression in equation (18) derived by Marcus is complete in the sense that it includes a pre-association between reactants, a time dependence arising from the frequency with which the reactants collide, and the thermal activation required for electron transfer to occur. On the other hand, the quantum mechanically derived expressions for fcet are dependent upon the interreactant separation, and the dependence on V must be included explicitly. 

The derivation of the cross-reaction equation follows by (1) solving equation (63) for k for each individual self-exchange reaction, e.g. u=4RT(ln(vetKA)u/ku) (2) inserting the expressions for Au and k22 into equation (63) for A12 (3) incorporating this expression for kl2 into the rate constant expression in equation (59), assuming that = [(vet/fA)j j(vetAlA)22]1/1 - The final... 

When a larger fraction of HOC was associated with the particulate phase, numerical solutions of Equation (6.132) and Equation (6.133) were required to predict PCB transformation rates. The instantaneous concentration of dissolved PCB was estimated by incorporating the terms S and OH into the rate constant expressions for Equation (6.133) and Equation (6.134). 

Table Vm. Competition Between Dehydrohalogenation and Substitution Pathways for Reactions Between Halogenated Aliphatic Substrates and Sulfur Nucleophiles (Rate Constants Expressed in Units of M V1.)...

What rate law is consistent with the above data and what is the best average value for the rate constant expressed in seconds and molar concentration units ... 

Figure 3. Elementary steps in polyhexene polymerization with rate constant expressions...

Earlier pulse radiolysis investigations were devoted to the radiation effects of the polymers dissolved in water. The rate constants of the reactions of OH radicals with polyethylene oxide) [75, 76], polyvinylpyrolidone [75], dextran [75], and sodium polyacrylate [77] have been measured as functions of the chainlength and the polymer concentrations. The rate constant expressed in polymer unit (mol -1 dm3 s- M increased more slowly with increasing chainlength than would be expected if the reactivity of the CH2 and CH groups were additive. The reaction between small molecules and the polymer radicals produced by the reaction of OH radical with polyethylene oxide) has also been demonstrated [78]. 

DNOSBP, was varied from 1.1 to 96.9 /unoles/liter. Relative rate constants, expressed in units of (/xmoles/gram)2/hr., derived from the present experiments are plotted in Figure 4 vs. initial concentration of solute. The linearity of the trace so obtained indicates good agreement with the concentration dependence of adsorption rate noted previously for sul-fonated alkylbenzenes (8). Thus, the previous and present experiments suggest that the nature of the effect of concentration may well be general and quantitatively as well as qualitatively predictable. 

We see in Table XII. 1 that we cannot separately identify the terms in the rate-constant expression for the thermodynamics equation or the collision theories without special assumptions. A complete identification of all the terms, frequencies, energies of activation and entropies of activation from experimental data is possible only for the Arrhenius equation and the transition-state theory. 

A major objective of a kinetic study is to deduce the simplest kinetic mechanism which is capable of satisfying all the kinetic measurements. It then becomes possible to state the rate constant expression which represents each of the steady state coefficients (the < s). 

The transfer distance with time, D, and the value of the tunneling decay constant, /3 were determined in the rate constant expression for tunneling, k = kge °t. Table 1 shows the results for these studies for both hole and electron transfer with MX as both the electron and the hole acceptor. 

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