Measuring Reaction Rates

Note that the sums are restricted to the portion of the frill S matrix that describes reaction (or the specific reactive process that is of interest). It is clear from this definition that the CRP is a highly averaged property where there is no infomiation about individual quantum states, so it is of interest to develop methods that detemiine this probability directly from the Scln-ddinger equation rather than indirectly from the scattering matrix. In this section we first show how the CRP is related to the physically measurable rate constant, and then we discuss some rigorous and approximate methods for directly detennining the CRP. Much of this discussion is adapted from Miller and coworkers [44, 45]. 

Nitration at a rate independent of the concentration of the compound being nitrated had previously been observed in reactions in organic solvents ( 3.2.1). Such kinetics would be observed if the bulk reactivity of the aromatic towards the nitrating species exceeded that of water, and the measured rate would then be the rate of production of the nitrating species. The identification of the slow reaction with the formation of the nitronium ion followed from the fact that the initial rate under zeroth-order conditions was the same, to within experimental error, as the rate of 0-exchange in a similar solution. It was inferred that the exchange of oxygen occurred via heterolysis to the nitronium ion, and that it was the rate of this heterolysis which limited the rates of nitration of reactive aromatic compounds. 

It may seem, at first sight, paradoxical that a competition reaction carried out under conditions in which the measured rate is independent of the concentration of the aromatic can tell us about the relative reactivities of two aromatics. Obviously, the measured rate has nothing to do with the rate of the product-determining step, and what is important in determining relative reactivities is the ratio of the values of ( 3.2.4) for two compounds. The criteria to be met for a correct application of the competitive method are well understood. ... 

There are available from experiment, for such reactions, measurements of rates and the familiar Arrhenius parameters and, much more rarely, the temperature coefficients of the latter. The theories which we use, to relate structure to the ability to take part in reactions, provide static models of reactants or transition states which quite neglect thermal energy. Enthalpies of activation at zero temperature would evidently be the quantities in terms of which to discuss these descriptions, but they are unknown and we must enquire which of the experimentally available quantities is most appropriately used for this purpose. 

Sensitivity The sensitivity for a one-point fixed-time integral method of analysis is improved by making measurements under conditions in which the concentration of the monitored species is larger rather than smaller. When the analyte s concentration, or the concentration of any other reactant, is monitored, measurements are best made early in the reaction before its concentration has substantially decreased. On the other hand, when a product is used to monitor the reaction, measurements are more appropriately made at longer times. For a two-point fixed-time integral method, sensitivity is improved by increasing the difference between times t and f2. As discussed earlier, the sensitivity of a rate method improves when using the initial rate. 

Mechanisms. Mechanism is a technical term, referring to a detailed, microscopic description of a chemical transformation. Although it falls far short of a complete dynamical description of a reaction at the atomic level, a mechanism has been the most information available. In particular, a mechanism for a reaction is sufficient to predict the macroscopic rate law of the reaction. This deductive process is vaUd only in one direction, ie, an unlimited number of mechanisms are consistent with any measured rate law. A successful kinetic study, therefore, postulates a mechanism, derives the rate law, and demonstrates that the rate law is sufficient to explain experimental data over some range of conditions. New data may be discovered later that prove inconsistent with the assumed rate law and require that a new mechanism be postulated. Mechanisms state, in particular, what molecules actually react in an elementary step and what products these produce. An overall chemical equation may involve a variety of intermediates, and the mechanism specifies those intermediates. For the overall equation... 

The integral values of effective interfacial area can thus be obtained by measuring the reaction (extraction) rate and using physico-chemical properties of the reactants. A reaction satisfying the above conditions consists of hydrolysis of hexyl formate (11) ... 

Steric factors are important ia transesterification reactions. With a given alcohol, primary alkyl borates react at a rate too fast to measure, secondary alkyl borates react at measurable rates, and tert-huty borate reacts very slowly. 

Rates of Reaction. The rates of formation and dissociation of displacement reactions are important in the practical appHcations of chelation. Complexation of many metal ions, particulady the divalent ones, is almost instantaneous, but reaction rates of many higher valence ions are slow enough to measure by ordinary kinetic techniques. Rates with some ions, notably Cr(III) and Co (III), maybe very slow. Systems that equiUbrate rapidly are termed kinetically labile, and those that are slow are called kinetically inert. Inertness may give the appearance of stabiUty, but a complex that is apparentiy stable because of kinetic inertness maybe unstable in the thermodynamic equihbrium sense. 

Compounds of types (286) and (287) are in tautomeric equilibria with 4- or 5-hydroxyazoles. However, the non-aromatic form is sometimes by far the most stable. Thus oxazolinone derivatives of type (287) have been obtained as optically active forms they undergo racemization at measurable rates with nucleophiles (77AHC(21)175). Reactions of these derivatives are considered under the aromatic tautomer. 

Structure-reactivity relationships can be probed by measurements of rates and equiUbria, as was diseussed in Chapter 4. Direct comparison of reaction rates is used relatively less often in the study of radical reactions than for heterolytic reactions. Instead, competition methods have frequently been used. The basis of competition methods lies in the rate expression for a reaction, and the results can be just as valid a comparison of relative reactivity as directly measured rates, provided the two competing processes are of the same kinetic order. Suppose that it is desired to compare the reactivity of two related compounds, B—X and B—Y, in a hypothetical sequence ... 

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

Kinetic studies involving enzymes can principally be classified into steady and transient state kinetics. In tlie former, tlie enzyme concentration is much lower tlian that of tlie substrate in tlie latter much higher enzyme concentration is used to allow detection of reaction intennediates. In steady state kinetics, the high efficiency of enzymes as a catalyst implies that very low concentrations are adequate to enable reactions to proceed at measurable rates (i.e., reaction times of a few seconds or more). Typical enzyme concentrations are in the range of 10 M to 10 ], while substrate concentrations usually exceed lO M. Consequently, tlie concentrations of enzyme-substrate intermediates are low witli respect to tlie total substrate (reactant) concentrations, even when tlie enzyme is fully saturated. The reaction is considered to be in a steady state after a very short induction period, which greatly simplifies the rate laws. 

From these considerations we conclude that diffusion-limited bimolecular rate constants are of the order 10 -10 M s . If an experimentally measured rate constant is of this magnitude, the usual conclusion is, therefore, that it is diffusion limited. For example, this extremely important reaction (in water)... 

The electrode current depends on the rates of the coupled reactions, but by suitable adjustment of the electrode potential (into the diffusion current region for the electrode reaction) the rate of the reduction reaction can be made so fast that the current depends only on the rate of the prior chemical reaction. The dependence of the observed current on the presence of the chemical reaction is a measure of the rate. 

Other measures of nucleophilicity have been proposed. Brauman et al. studied Sn2 reactions in the gas phase and applied Marcus theory to obtain the intrinsic barriers of identity reactions. These quantities were interpreted as intrinsic nucleo-philicities. Streitwieser has shown that the reactivity of anionic nucleophiles toward methyl iodide in dimethylformamide (DMF) is correlated with the overall heat of reaction in the gas phase he concludes that bond strength and electron affinity are the important factors controlling nucleophilicity. The dominant role of the solvent in controlling nucleophilicity was shown by Parker, who found solvent effects on nucleophilic reactivity of many orders of magnitude. For example, most anions are more nucleophilic in DMF than in methanol by factors as large as 10, because they are less effectively shielded by solvation in the aprotic solvent. Liotta et al. have measured rates of substitution by anionic nucleophiles in acetonitrile solution containing a crown ether, which forms an inclusion complex with the cation (K ) of the nucleophile. These rates correlate with gas phase rates of the same nucleophiles, which, in this crown ether-acetonitrile system, are considered to be naked anions. The solvation of anionic nucleophiles is treated in Section 8.3. 

Bimolecular rate constants determined at temperatures giving conveniently measurable rates and calculated for the temperature given in parentheses, except for some of the catalyzed reactions (lines 1-4 and 14—19) which are third-order. 

The equation indicates that one MnO ion, five Fe+2 ions, and eight H+ ions (a total of fourteen ions) must react with each other. If this reaction were to take place in a single step, these fourteen ions would have to collide with each other simultaneously. The probability of such an event occurring is extremely small—so small that a reaction which depended upon such a collision would proceed at a rate immeasurably slow. Since the reaction occurs at an easily measured rate, it must proceed by some sequence of steps, none of which involves such an improbable collision. 

Since high temperatures and a nitrogen atmosphere are necessary to obtain measurable rates of polyesterification and to remove the reaction water, a loss of volatile reactants can hardly be avoided, especially in early stages of polyesterification. In the last stages, the decrease of the concentration of the volatile reactants can be of the same order of magnitude as their concentration. Consequently, the ultimate points of the kinetic plot have possibly no significance. 

Assuming that reactions 1 and 2 are reversible and are fast in comparison with reaction 5, the measured rate constants ( obs) at pH 8.2-9.8 are related to the intrinsic rate constant k5 according to Scheme 5-16, as shown by Ritchie and Wright (1971a). ... 

Quinone diazides (12.9) and their 1,2-isomers (Secs. 1.2, 2.4, and 4.2) simultaneously display the properties of both aliphatic and aromatic diazo components. They can be considered as analogues of conjugated diazoketones. On the other hand, a specific feature of many of their reactions is their conversion to hydroxyarenediazo-nium ions (12.8) in the presence of acids (Scheme 12-7). The p Ta-value of the 4-hydroxybenzenediazonium ion is 3.19 (Kazitsyna and Klyueva, 1972), so the reactivity of compounds of this type will depend considerably on the acidity of the reaction medium. Compound 12.8 is much more electrophilic than 12.9, and therefore the measured rate depends on the position of the equilibrium in Scheme 12-7. 

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