Isokinetic relationship 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. 

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... 

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. 

Example 2. Equilibrium constants of the reaction of twenty substituted dinitromethanes with formaldehyde have been measured (57) in the range 10-50°C. The isokinetic relationship is valid for only nine of them, as revealed in a preliminary graphical treatment using the plot of log Kjo versus log Kio( 163) the pertinent values of logK are reproduced in Table I. The values of x = T" were transformed according to eq. (36a) with... 

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. 

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. 

Cationic vesicles, for example those formed from di-n-hexadecyldimelhylammonium bromide (DHAB) accelerate the decarboxylation by a factor of about 1000 relative to pure water. Dehydration of the carboxylate group at the binding sites is most likely the main factor behind the catalysis. Different isokinetic temperatures (obtained from linear plots of enthalpies v.y. entropies of activation) have been observed above and below the main phase transition temperature. These excellent isokinetic relationships indicate that the catalytic effects are caused by a single important interaction mechanism. ... 

A significant amount of kinetic data exists for the decarboxylation and oxidation of carboxylic acids. However, a relatively small fraction of these results deals with n-C2 to n-C4 aliphatic mono- and dicarboxylic acids under conditions pertinent to geological interests. For example, the early studies of the decarboxylation kinetics of acetic acid utilized flow-though silica tubes in which the anhydrous gas was exposed to very high temperatures for only seconds (Bamford and Dewar 1949 Blake and Jackson 1968, 1969). Nevertheless, it is useful to consider all of these results because it reveals trends common for structural classes of carboxylic acids. In this background discussion, a brief introduction to the subject of isokinetic relationships is given, as well as an overview of the decarboxylation and oxidation of carboxylic acids in which isokinetic relationships are used to establish trends and gross variations in reaction mechanisms between structural classes of acids. 

It follows that for a special value of one parameter, the observed value of y is independent of the second parameter. This happens at Ii= a2/ai2 or I2 = -ai/ai2 any of these values determines y= a -aia2/ai2, the so called isoparametrical point. The argument can evidently be extended to more than two independently variable parameters. Experimental evidence is scarce. In the field of extrathermodynamic relationships, i.e., when j and 2 are kinds of a constants, eq. (84) was derived by Miller (237) and the isoparametrical point was called the isokinetic point (170). Most of the available examples originate from this area (9), but it is difficult to attribute to the isoparametrical point a definite value and even to obtain a significant proof that a is different from zero (9, 170). It can happen—probably still more frequently than with the isokinetic temperature—that it is merely a product of extrapolation without any immediate physical meaning. 

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. 

The dependences, such as Eq. 2.47, are known as compensation effect, and coefficient (3C is denoted as isokinetic temperature at which all reactions of given series have the same rate constant. An example of compensation effect for for catalytic rate constant of the Sulfolobus solfataricus p-glycosidase reaction with different substrates is shown in Fig. 2.19. Similar relationships were reported for many other prosesses, involving the binding ligands to hemoglobin, the oxidation of alcohols by catalase, the hydroxylation of substrates by cytochrome c, etc. 

The estimation of the optimal pressure was previously discussed by taking into account the possible pressure dependence of [2] as well as the interrelation of pressure and temperature defined under isokinetic conditions [3], The relationship (Eq. (10.1)) underlines that the rate constant increases exponentially with pressure. The logarithmic behavior is illustrated in Fig. 10.1 which shows the variation of the rate constant ratio fep/ko with pressure at 25 °C. As an example, let us consider a pressure of 300 MPa which is usually an upper limit for large commercial pressure vessels. At that pressure the value of fep/fco approaches 10-40 for pressure-... 

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