Isokinetic relationships

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. 

Most of the isokinetic relationships in the literature have been established from plots of AH against AS collections of these have been published and several authors have discussed the mechanistic implications.9 6. csap. 12 ... 

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. 

Equation (7-76) shows that, in general, a plot of log 2 against log k, is not expected to be linear. Linearity in such a plot can be assured, however, if an isokinetic relationship, 8rAW = P8rA5, is followed by the system. By incorporating this relationship into Eqs. (7-76) and (7-77), we obtain... 

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. 

Earlier analyses making use of AH vs. AS plots generated many p values in the experimentally accessible range, and at least some of these are probably artifacts resulting from the error correlation in this type of plot. Exner s treatment yields p values that may be positive or negative and that are often experimentally inaccessible. Some authors have associated isokinetic relationships and p values with specific chemical phenomena, particularly solvation effects and solvent structure, but skepticism seems justified in view of the treatments of Exner and Krug et al. At the present time an isokinetic relationship should not be claimed solely on the basis of a plot of AH vs. A5, but should be examined by the Exner or Krug methods. 

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. 

Comparisons of reactivity at different temperatures may be misleading if the Compensation Law or isokinetic relationship applies. i 152c... 

It is also a point of change in control of the reaction rate by the energy of activation below it to control by the entropy of activation above it. The effect of changes in structure, solvent, etc., will depend on the relation of the experimental temperature to the isokinetic temperature. A practical consequence of knowing the isokinetic temperature is the possibility of cleaning up a reaction by adjusting the experimental temperature. Reactions are cleaner at lower temperatures (as often observed) if the decrease in the experimental temperature makes it farther from the isokinetic temperature. The isokinetic relationship or Compensation Law does not seem to apply widely to the data herein, and, in any case, comparisons are realistic if made far enough from the isokinetic temperature. 

The existence of an isokinetic relationship supports (but does not prove) a claim that a single mechanism operates along the series. If certain members deviate from the correlation to a significant extent, this observation is a suggestion (that bears further checking) that a new mechanism has come into play. The deviant point(s) will likely lie above the isokinetic line, since a new mechanism wins out because it is faster than the normal one (and of course the rate constant records the sum of the contributions from all of the operative pathways). 

A plot depicting isokinetic relationships, (a) The thermal rearrangement of triarylmethyl azides, reaction (7-35) is shown with different substituents and solvent mixtures. The slope of the line gives an isokinetic temperature of 489 K. Data are from Ref. 8. (b) The complexation of Nr by the pentaammineoxalatocobalt(III) ion in water-methanol solvent mixtures follows an isokinetic relationship with an isokinetic temperature of 331 K. The results for forward (upper) and reverse reactions are shown with the reported standard deviations. Data are from Ref. 9. 

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... 

Isokinetic relationship. Show and discuss whether the existence of an isokinetic relationship (Section 7.4) is a necessary condition for a Hammett LFER. 

Induced reactions, 102 Induction period, 72 Inhibitor competitive, 92 noncompetitive, 93 Initial rates, method of, 8, 32 from power series, 8 Initiation step, 182 Inverse dependences, 130-131 Isokinetic relationship, 164—165 Isokinetic temperature, 163, 238 Isolation, method of (see Flooding, method of)... 

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... 

In addition to chemical reactions, the isokinetic relationship can be applied to various physical processes accompanied by enthalpy change. Correlations of this kind were found between enthalpies and entropies of solution (20, 83-92), vaporization (86, 91), sublimation (93, 94), desorption (95), and diffusion (96, 97) and between the two parameters characterizing the temperature dependence of thermochromic transitions (98). A kind of isokinetic relationship was claimed even for enthalpy and entropy of pure substances when relative values referred to those at 298° K are used (99). Enthalpies and entropies of intermolecular interaction were correlated for solutions, pure liquids, and crystals (6). Quite generally, for any temperature-dependent physical quantity, the activation parameters can be computed in a formal way, and correlations between them have been observed for dielectric absorption (100) and resistance of semiconductors (101-105) or fluidity (40, 106). On the other hand, the isokinetic relationship seems to hold in reactions of widely different kinds, starting from elementary processes in the gas phase (107) and including recombination reactions in the solid phase (108), polymerization reactions (109), and inorganic complex formation (110-112), up to such biochemical reactions as denaturation of proteins (113) and even such biological processes as hemolysis of erythrocytes (114). 

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.). 

Figure 1. Example of the isokinetic relationship in the coordinates AH versus AS isoequilibrium relationship in the ionization of anilinium ions (69, 71).

Instead of its reciprocal value, denoted 7, is used sometimes (3, 124, 156) in eqs. (10) and (11) however, the symbol 7 can also stand for 1/(2.303 Rj3) (154, 155). For this reason, it will not be used in this paper. Alternatively, these equations can be modified by taking TAS as a variable, and the proportionality constant is then j3/T and is called the compensation factor (173). As an example of the graphical representation of the isokinetic relationship in the coordinates AH and AS, see Figure 1, ionization of meta- and para-substituted anilinium ions in water. This example is based on recent exact measurements (69, 71) and clearly shows deviations that exceed experimental error, but the overall linear correlation cannot be doubted. 

Another expression for the isokinetic relationship relates two rate or equilibrium constants (kj, k ) measured at two temperatures (T2 > Tj) The linear relationship holds... 

Figure 4. Isokinetic relationship in the coordinates log versus log k, with k, at three different temperatures alkaline decomposition of Malachite-Green-like dyestuffs (186, 187).

The most general representation of the isokinetic relationship is the plot of logk against the reciprocal temperature. If the Arrhenius law is followed, each... 

Figure 5. Isokinetic relationship in the coordinates log k versus T" decomposition of formic acid on various catalysts (189).

The last method for illustrating an isokinetic relationship is based on the dependence on a parameter. If both AH and AS are related to the parameter then by its elimination from the two equations, the relation between AH and... 

The whole concept based on parameters, although used several times (3, 57, 155, 156, 201) and advocated particularly by Good and Stone (200), has a principal defect. The results are dependent not only on experimental rate constants, but also on the values of the parameter and on the form of the correlation equation used. Furthermore, the procedure does not give any idea of the possible error. Hence, it could be acceptable only in an unrealistic case, that in which the isokinetic relationship itself and the correlations with the parameter are very precise. 

Various algebraic expressions and various graphic representations of the isokinetic relationship offer the possibility of investigating each particular case from different sides and of stating the results and their consequences. A given kind of representation can be useful in a particular case, and no one of them can be considered to be erroneous in itself. 

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. 

The isokinetic relationship can further yield a preliminary test of a common mechanism i.e., when one reaction deviates from the others, it follows to a high probability that its mechanism is different. The deviations are best seen in a plot like Figure 4, while in Figure 5, it is difficult to decide which of the straight lines does go through the common point of intersection. 

In the graph of AH versus AS, large deviations in the direction of T are thus admissible, while much smaller ones in the perpendicular direction are not. Hence, sequences of points with the slope T can easily result from experimental errors only this is why the value of T is called error slope (1-3,115, 116, 118, 119). Isokinetic relationships with slopes close to T should be viewed with suspicion, but they have been reported frequently. However, we shall see later that even correlations with other slopes are only apparent, or at least the isokinetic temperature is determined erroneously from the plot of AH versus AS. 

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