Isokinetic relationship statistics

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. 

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. 

The problem of relationship between the activation parameters-the so called isokinetic relationship or compensation law—is of fundamental importance in structural chemistry, organic or inorganic. However, there are few topics in which so many misunderstandings and controversies have arisen as in connection with this problem. A critical review thus seems appropriate at present, in order to help in clarifying ideas and to draw attention to this treatment of kinetic or equilibrium data. The subject has already been reviewed (1-6), but sufficient attention has not been given to the statistical treatment which represents the heaviest problems. In this review, the statistical problems are given the first place. Theoretical corollaries are also dealt with, but no attempt was made to collect all examples from the literature. It is hoped that most of the important... 

Several doubts about the correctness of the usual statistical treatment were expressed already in the older literature (31), and later, attention was called to large experimental errors (142) in AH and AS and their mutual dependence (143-145). The possibility of an apparent correlation due only to experimental error also was recognized and discussed (1, 2, 4, 6, 115, 116, 119, 146). However, the full danger of an improper statistical treatment was shown only by this reviewer (147) and by Petersen (148). The first correct statistical treatment of a special case followed (149) and provoked a brisk discussion in which Malawski (150, 151), Leffler (152, 153), Palm (3, 154, 155) and others (156-161) took part. Recently, the necessary formulas for a statistical treatment in common cases have been derived (162-164). The heart of the problem lies not in experimental errors, but in the a priori dependence of the correlated quantities, AH and AS. It is to be stressed in advance that in most cases, the correct statistical treatment has not invalidated the existence of an approximate isokinetic relationship however, the slopes and especially the correlation coefficients reported previously are almost always wrong. 

Equation (10) represents the simplest form of the isokinetic relationship however, several equivalent expressions are also possible and will now be discussed and shown in diagrams. It should be commented in advance that algebraic equivalence does not imply equivalence from the statistical point of view (see Section IV.). 

The natural and correct form of the isokinetic relationship is eq. (13) or (13a). The plot, AH versus AG , has slope Pf(P - T), from which j3 is easily obtained. If a statistical treatment is needed, the common regression analysis can usually be recommended, with AG (or logK) as the independent and AH as the dependent variable, since errors in the former can be neglected. Then the overall fit is estimated by means of the correlation coefficient, and the standard deviation from the regression line reveals whether the correlation is fulfilled within the experimental errors. 

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. 

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. 

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. 

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. 

C. P. Brink and A. L. Crumbliss, Inorg. Chem. 23, 4708 (1984). A number of LFER are included with a short discussion of the strict statistical criteria for isokinetic relationships. 

This is called the isokinetic relationship, and 0 is the isokinetic temperature, where all k values for related series of reactions are the same. Although 0 can be obtained as the slope of plot of AS versus AH, significant statistical problems may be encountered (Exner, 1972). At T< 0, reactions with smaller Eacl are faster. At T>6, reactions with larger ac, are faster. 

From the data listed in Tables I-V, we conclude that most authors would probably accept that there is evidence for the existence of a compensation relation when ae < O.le in measurements extending over AE 100 and when isokinetic temperature / , would appear to be the most useful criterion for assessing the excellence of fit of Arrhenius values to Eq. (2). The value of oL, a measure of the scatter of data about the line, must always be considered with reference to the distribution of data about that line and the range AE. As the scatter of results is reduced and the range AE is extended, the values of a dimin i, and for the most satisfactory examples of compensation behavior that we have found ae < 0.03e. There remains, however, the basic requirement for the advancement of the subject that a more rigorous method of statistical analysis must be developed for treatment of kinetic data. In addition, uniform and accepted criteria are required to judge quantitatively the accuracy of obedience of results to Eq. (2) or, indeed, any other relationship. 

Exner O (1970) Determination of the isokinetic temperature. Nature 227 366-367 Exner O (1972) Statistics of the enthalpy-entropy relationship. I. The special case. Collect Czech Chem Commun 37 1425-1444... 

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