Isokinetic temperature relationships

The kinetics of nitration have been studied fairly intensively over a number of years and show that the reaction can be represented by the general steps 

The outstanding problem is to decide how much, if any, association exists between N02 and X in the generally rate-determining step of the reaction. Kinetic studies tend to indicate the presence of different electrophiles under different conditions whereas the derived partial rate factors are closely similar and therefore indicate one electrophile common to most, if not all, nitrating agents. The more electron-attracting is X , the more easily is N02 displaced from it and hence a reactivity sequence should be 

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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 ... 

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. 

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. 

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)... 

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. 

Figure 13. Isokinetic relationship for the reaction of substituted dinitromethanes with formaldehydes (57). The standard deviation is shown as function of the supposed isokinetic temperature (full curve).

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 physical meaning of the constant (3, connected with the reversal of reactivity at the temperature T = /3, is a puzzling corollary of the isokinetic relationship, noted already by older authors (26, 28) and discussed many times since (1-6, 148, 149, 151, 153, 163, 188, 212). Especially when the relative reactivity in a given series is explained in theoretically significant terms, it is hard to believe that the interpretation could lose its validity, when only temperature is changed. The question thus becomes important of whether the isokinetic temperature may in principle be experimentally accessible, or whether it is merely an extrapolation without any immediate physical meaning. 

In this equation, 6AH must equal the first and 6AS the second term on the right-hand side so that there is no simple relationship between them. However, the imaginary isokinetic temperatures Pi and P2, corresponding to the two interaction mechanisms, can be defined as J3i=-A/B and P2 -C/D. The resulting relation between AH and AS is scattered (2) as shown in Figure 19. 

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. 

The isokinetic temperature for nonvalid relationships, it is given in parentheses. The symbols 0 and isoentrppic series, respectively. 

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. 

Figure 9 demonstrates this compensation effect by the linear relationship between AS and AH. This indicates that both activation parameters depend equally on a and that the isokinetic temperature, i.e. the slope of the line, amounts to 256°K. Thus, at -17°C the rate would become independent of a, whereas it increases with a at higher temperatures. 

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. 

Isokinetic relationship, 261, 368 Isokinetic temperature, 368 Isolation technique, 26, 78 Isotope effects inverse, 299 kinetic, 292 primaiy, 293, 295 secondaiy, 298 solvent, 272, 300 Isotopic exchange, 300 Isotopic substitution, 6... 

When the temperature of the measurement (T) equals the isokinetic temperature (jS), AG is a constant. At the isokinetic temperature, a given acid will decompose at the same rate in all of the solvents for which eqn. (5) holds. In some instances the results for several acids will fall on the same line for the AH vs. AS plot. Table 52 lists the reported isokinetic temperatures for a number of systems that obey eqn. (5). The validity of the linear enthalpy-entropy of activation relationship has been questioned as an artifact due to experimental error in the enthalpy of activation. Error analysis was performed for some of the systems given in Table 52, and it was concluded that the linear enthalpy-entropy of activation relationships were valid . It has been reported that the isokinetic temperature for decarboxylation of several acids corresponds to the melting point of the acid. Our evaluation of the data, given later, does not support this conclusion. 

That means that the phenomenon isokinetic effect is indicated by the observation that the Arrhenius lines of the closely related reactions intersect in only one point, characterized by the isokinetic pre-exponential factor kao and by the isokinetic temperature Tiso, or, in other words, there is a linear relationship between In ko and Ea with the slope IRT so-... 

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. 

Frequently, the differences in entropy between two diastereomeric transition states are small, and the enthalpy term predominates. However, the entropy term cannot always be neglected, and sometimes it may even dominate according to the reaction conditions [2], In some cases, the observed selectivity can be inverted by changing the temperature of the reaction, and the temperature at which AAH = TAAS is called the isokinetic temperature (isoinversion). The occurrence of an isokinetic temperature in the normal operating range is not veiy frequent, but it is observed sometimes in multistep processes (see below). The ratio of the diastereo-isomeric products formed in kinetically controlled reactions is given by the relationship... 

The rates of epoxidation of cyclododecene with a series of aliphatic peroxy-acids have been correlated, using the Taft equation. The reaction constant (p ) was + 2.0 and the steric constant (6) was found to be essentially zero. A two-parameter correlation has been found for the effect of basicity and polarity of the solvent on the rate of epoxidation of propene with peracetic acid. Rate constants and activation parameters for the epoxidation of a number of cycloalkenes, including (11 R = H or COOMe), (12 R = H, Ph, or 2-furyl), (13), (14), and cyclo-octa-l,5-diene, have been measured. An isokinetic relationship was demonstrated, with the isokinetic temperature of 3 C. There was only a weak dependence of the rate on the structure of the alkene. 

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