Reaction series isokinetic

To conclude this section, we can state that all of the theories presented hitherto, even when starting from general principles, inevitably embody several assumptions, which in fact represent the heart of the analysis. However, the physical meaning of these assumptions usually is not known, so that no theory is able to predict in which reaction series isokinetic behavior appears and in which it does not. Neither is the structural theory of organic chemistry able to make such a prediction and to define the terms reaction series or similar reactions or small structure changes it can only afford many examples. 

The second use of activation parameters is as criteria for mechanistic interpretation. In this application the activation parameters of a single reaction are, by themselves, of little use such quantities acquire meaning primarily by comparison with other values. Thus, the trend of activation parameters in a reaction series may be suggestive. For example, many linear correlations have been reported between AT/ and A5 within a reaction series such behavior is called an isokinetic relationship, and its significance is discussed in Chapter 7. In Section 5.3 we commented on the use of AS to determine the molecularity of a reaction. Carpenter has described examples of mechanistic deductions from activation parameters of organic reactions. 

It was not until the 1970s that the statistics of the isokinetic relationship was satisfactorily worked out.Exner first took this approach Let k, and 2 be the rate constants for a member of a reaction series at temperatures T, and T2, with T2 > T, and let k° and k° be the corresponding values for the reference member of the series. Then Eqs. (7-76) and (7-77) are easily derived for the reaction series. 

Thus, a linear plot of log 2 against log ki for a reaction series implies an isokinetic relationship for the series. The reason that this plot is a reliable test for such a relationship is that the errors in and 2 are independent (unlike the errors in A// and AS ). From the slope b of the straight line the isokinetic temperature p can be found ... 

Obviously for this method to work the ratio T1IT2 must be appreciably smaller than unity. Provided this condition is met, this method is a simple and reliable way to test for an isokinetic relationship or to detect deviations from such a relationship. Exner shows examples of systems plotted both as log 2 vs. log and as AH vs. A5, demonstrating the inadequacy of the latter plot. Exner has also developed a statistical analysis of the Petersen method this analysis yields p and an uncertainty estimate of p. Exner has applied his statistical methods to 100 reaction series, finding that 78 of them follow approximately valid isokinetic relationships. 

Suppose it is known that an isokinetic relationship holds for a reaction series. Then give the slope of a plot of log k against for the series. 

The issues to be dealt with here have been touched upon earlier, and the reader should be reminded of selectivity (Section 5.5) and of the isokinetic relationship (Section 7.4). We saw that there is often a linear correlation between the values of A// and AS for a reaction series. This is usually written as... 

Figure 2. Example of the isokinetic relationship in the coordinates AS versus AG (the same reaction series as in Figure 1).

Having in mind the various forms of the isokinetic relationship, we can also show its physical meaning in kinetics more clearly. Let us consider a reaction series with a variable substituent, solvent, or other factor. The term reaction series was discussed by Bunnett (14), with the conclusion that the common mechanism of all reactions is a necessary condition (12). However, this condition can seldom be ascertained, and best after finishing the whole analysis. At the beginning, it may be sufficient that the reaction products are invariable and the kinetic order equal. In addition, the structural changes should not be too large of course, this condition cannot be defined precisely. 

Figure 8. Real (full lines) and apparent (broken heavy lines) isokinetic relations and experimental Arrhenius lines in the graph logk versus T" the same reaction series as in Figures 6 and 7.

Figure 12. Isokinetic relationship for the same reaction series as in Figure 11, in the coordinates E versus log A.

These new statistical procedures permit reexamination of a number of reaction series to reach more definite conclusions than formerly concerning the occurrence, accuracy, and significance of isokinetic relationships and possible values of the isokinetic temperatures. In this section, the consequences of these findings will be discussed and confronted with theoretical postulates or predictions. 

The idea that /3 continuously shifts with the temperature employed and thus remains experimentally inaccessible would be plausible and could remove many theoretical problems. However, there are few reaction series where the reversal of reactivity has been observed directly. Unambiguous examples are known, particularly in heterogeneous catalysis (4, 5, 189), as in Figure 5, and also from solution kinetics, even when in restricted reaction series (187, 190). There is the principal difficulty that reactions in solution cannot be followed in a sufficiently broad range of temperature, of course. It also seems that near the isokinetic temperature, even the Arrhenius law is fulfilled less accurately, making the determination of difficult. Nevertheless, we probably have to accept that reversal of reactivity is a possible, even though rare, phenomenon. The mechanism of such reaction series may be more complex than anticipated and a straightforward discussion in terms of, say, substituent effects may not be admissible. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

The crucial test of all of the theories based on solvation would be the absence of the isokinetic relationship in the gas phase, but the experimental evidence is ambiguous. Rudakov found no relationship for atomization of simple molecules (6), whereas Riietschi claimed it for thermal decomposition of alky] chlorides (96) and Denisov for several radical reactions (107) however, the first series may be too inhomogeneous and the latter ones should be tested with use of better statistics. A comparison of the same reaction series in the gas phase on the one hand and in solution on the other hand would be most desirable, but such data seem not to be available. 

Ritchie and Sager (124) distinguish three types of reaction series according to whether the Hammett equation or the isokinetic relationship is obeyed, or both. The result that the former can be commonly valid without the latter seems to be based on previous incorrect statistical methods and contradicts the theoretical conclusions. Probably both equations are much more frequently valid together than was anticipated. The last case, when the isokinetic relationship holds and the Hammett equation does not, may be quite common, of course, and has a clear meaning. Such a series meets the condition for an extrathermo-dynamic treatment when enough experimental material accumulates, it is only necessary to define a new kind of substituent constant. 

In the isoenthalpic and isoentropic reaction series, the isoenthalpic and isoentropic activation parameters, respectively, may be defined as a special case of isokinetic values. Values of this kind are certainly more reliable than the usual ones in all cases when the isokinetic behavior has been proved, or, better, when it cannot be rejected. In dubious cases, when it cannot be disproved with any reliability but yet would cause a reduction of the overall accuracy, we recommend the calculation of both isokinetic and unconstrained parameters. In... 

Figure 22. Representation of isokinetic (o) and unconstrained ( ) activation parameters, the same reaction series (57) as in Figure 13.

The necessity of the statistical approach has to be stressed once more. Any statement in this topic has a definitely statistical character and is valid only with a certain probability and in certain range of validity, limited as to the structural conditions and as to the temperature region. In fact, all chemical conceptions can break dovra when the temperature is changed too much. The isokinetic relationship, when significantly proved, can help in defining the term reaction series it can be considered a necessary but not sufficient condition of a common reaction mechanism and in any case is a necessary presumption for any linear free energy relationship. Hence, it does not at all detract from kinetic measurements at different temperatures on the contrary, it gives them still more importance. 

Some Reaction Series Tested as to Their Isokinetic Behavior... 

Up to now (1971) only a limited number of reaction series have been completely worked out in our laboratories along the lines outlined in Sec. IV. In fact, there are rather few examples in the literature with a sufficient number of data, accuracy, and temperature range to be worth a thorough statistical treatment. Hence, the examples collected in Table III are mostly from recent experimental work and the previous ones (1) have been reexamined. When evaluating the results, the main attention should be paid to the question as to whether or not the isokinetic relationship holds i.e., to the comparison of standard deviations of So and Sqo The isokinetic temperature /J is viewed as a mere formal quantity and is given no confidence interval. Comparison with previous treatments is mostly restricted to this value, which has generally and improperly been given too much atention. 

In a series of reactions for which an acceUrative decrease in the activation energy is accompanied by a decelerative decrease in the entropy of activation (Compensation Law ), or the two increase together, there wiU be an isokinetic temperature (between 0-200° C for three-fourths of the 79 reactions tabulated by Leffler ). The rate vs. temperature curves for all the reactions in the series pass through this single point. Comparisons are affected since the isokinetic temperature is a point of inversion of relative reactivity in the series. 

---