Interpretation of Rate Constants

The utility of rate constants for understanding reaction mechanisms depends largely on interpreting them in terms of energies. Energy information is ordinarily obtained from rate data by either of two methods, one empirical and the other more theoretical. 

The temperature dependence of observed rate constants follows the Arrhenius equation (2.50) with good accuracy for most reactions. A and Ea are parameters determined experimentally, R is the gas constant, 1.986 cal °K-1 mole-1, and 

T is the temperature in degrees Kelvin. The units of A, called the pre-exponential factor, are the same as those of kobs for a first-order rate constant, time 1 for a second-order rate constant, l mole-1 time-1. We use the notation A obs to emphasize that the equation applies to the observed rate constant, which may or may not be simply related to the microscopic k s characterizing the individual steps of a reaction sequence. 

If we write Equation 2.50 in the form of Equation 2.51, we see at once a resemblance to the familiar relation 2.52 between the equilibrium constant of a 

M Transition state theory is discussed in standard texts on physical chemistry, kinetics, and physical organic chemistry. See, for example, (a) W. J. Moore, Physical Chemistry, 3rd ed., Prentice-Hall, Englewood Cliffs, N.J., 1962, p. 296 (b) S. W. Benson, Thermochemical Kinetics, Wiley, New York, 1968 (c) K. J. Laidler, Chemical Kinetics, 2nd ed., McGraw-Hill, New York, 1965 (d) K. B. Wiberg, Physical Organic Chemistry, Wiley, New York, 1964 (e) L. P. Hammett, Physical Organic Chemistry, 2nd ed., McGraw-Hill, New York, 1970. For a different approach to chemical dynamics, see (f) D. L. Bunker, Accts. Chem. Res., 7, 195 (1974). 

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The formidable problems that are associated with the interpretation of LP kinetic data for nonstatistical IM reactions can be entirely avoided if the reactions can be studied in the HPL of kinetic behavior. In the HPL, the energy content of the initially formed species, X and Y, in reaction (2) would be very rapidly changed by collisions with the buffer gas so that the altered species, X and Y, would have normal Boltzmann distributions of energy. Furthermore, those Boltzmann energy distributions would be continuously refreshed as the most energetic X and Y within the distributions move forwards or backwards along the reaction coordinate. The interpretation of rate constants measured in the HPL is expected to be relatively straightforward because conventional transition-state theory can then be applied. 

It has been shown that, by adding the macrocycUc ligand 18-crown-6 or cr5q>t-[222] to complex the K" " ions (see Figure 10.8), the K -catalysed pathway is replaced by a cation-independent mechanism. The rate constant that is often quoted for the [Fe(CN)g] /[Fe(CN)5] self-exchange reaction is of the order of lO dm moP s , whereas the value of k determined for the cation-independent pathway is 2.4 X lO dm moP s , i.e. sslOO times smaller. This significant result indicates that caution is needed in the interpretation of rate constant data for electron-transfer reactions between complex anions. 

S. B. Woo and S. F. Wong, Interpretation of rate constants measured in drift tubes in terms of cross sections, J. Chem. Phys. 55, 3531-3541 (1971). 

Other possible problems which may arise in connection with the measurement and interpretation of rate constants, such as spin delocalization etc. are briefly dealt with in the introductory part of the following electron transfer rate constant compilation. 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

Gomes, W. (1961). "Definition of Rate Constant and Activation Energy in Solid State Reactions," Nature (London) 192, 965. An article discussing the difficulties associated with interpreting activation energies for reactions in solids. 

As a consequence of these various defined quantities, care must be taken in assigning values of rate constants and corresponding pre-exponential factors in the analysis and modeling of experimental data. This also applies to the interpretation of values given in the literature. On the other hand, the function [ [ c and the activation energy EA are characteristics only of the reaction, and are not specific to any one species. 

Since these electrochemical problems are of dominant importance for the interpretation of the kinetic results and the evaluation of the propagation rate-constants, we must explore them before we can discuss determination of rate-constants and their significance. 

In order to answer the questions we have posed, we need to accomplish a number of things. First, we need to measure a reaction rate in the gas phase. Second, we need a framework for interpreting the rate constant and relating the measurement to a potential surface. Third, we need to acquire sufficient information to allow us to extract the value of a barrier height. Then, we can apply theories for interpreting these barrier heights in terms of chemical structure. 

Quantitatively, many observed deviations from simple equilibrium processes can be interpreted as consequences of the various isotopic components having different rates of reaction. Isotope measurements taken during unidirectional chemical reactions always show a preferential emichment of the lighter isotope in the reaction products. The isotope fractionation introduced during the course of an unidirectional reaction may be considered in terms of the ratio of rate constants for the isotopic substances. Thus, for two competing isotopic reactions... 

Gas-phase reactions which result in nucleophilic displacement at a saturated, or an unsaturated, carbon centre have been observed in positive and negative ion chemistry. By far, the most widely occurring case is the formal analog of the Sn2 reaction initially reported by Bohme and Young (1970). The experimental determination of rate constants for SN2 reactions has received a great deal of attention as has the mechanistic point of view including the interpretation of the potential energy surface for the gas-phase reaction. 

In more than one respect, the small-amplitude sinuosoidal a.c. method can be superior to the large-amplitude step methods for the study of coupled homogeneous reactions. First, the wide range of frequencies at which meaningful data can be obtained will correspond to an equally wide range of rate constants on which, in principle, information can be obtained. Second, the a.c. perturbation can be superimposed on a large-amplitude d.c. or step perturbation so that information in the time scale of the latter is incorporated as well. Moreover, this affords an internal check on the reliability of data interpretations. Finally, it is important... 

The study of reaction rates has two purposes first, to compare the form of the rate equation with predictions of the various mechanisms under consideration, and second, to measure numerical values of rate constants and to interpret them in terms of elementary reaction steps. 

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