Isokinetic temperature values

To illustrate this case, consider the parallel reactions of Eq. (7-30). Suppose Af/ = 60.0 kJmol-1 and A= 110.0 kJmol-1. There must exist a temperature (the isokinetic temperature) where the separate straight lines intersect. At this temperature, k(, = k i. Figure 7-2 shows the temperature profile for this case over the range 5-55 °C. The values of AS = -70.0 and ASf = 98.4 J mol-1 K l were chosen. The analysis of this situation yields an isokinetic temperature of 23.8 °C. The plot also shows k(, and k7 separately, to aid the appreciation of the nature of the summation. 

There is usually a compensation between values of rate constants along a series. That is, when AH increases, AS does so as well (and vice versa). As a consequence, the spread of the rate constant is less than if AH were varied at constant AS (or vice versa). This means that the isokinetic temperature is usually > 0 K, and often 0K. 

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. 

By repeating the calculation for various values of x, one can obtain y and Sx as functions of x and find the minimum of the latter by successive approximations. The value of x at this minimum (xo) gives the estimate of the isokinetic temperature Xo The corresponding values yo and So are obtained from eqs. (52) and (53) So has... 

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. 

Practically all values of 3 within the experimental interval claimed in the literature (1-5, 115-119, 153) have been shown to be artifacts (148, 149, 163) resulting from improper statistical treatment (see Sec. IV). Petersen thus believed (148) that actually no such value had been reported, and the meaning was offered that the isokinetic temperature probably is not accessible experimentally (149, 188). This view was supported by the existence of negative... 

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. 

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. 

One aspect of compensation behavior that would appear to have received less attention than perhaps it deserves is the use of the constants B and e, or the isokinetic temperature / and the isokinetic reaction rate constant lip, as quantitative measurements of reactivities between series of related reactions. In the literature, comparisons of relative reaction rates are often based on the values of k at a particular temperature, arbitrarily selected, though often within the range of measurements, or the temperature at which a specified value of k is attained (137). It can be argued, however, that where compensation exists, a more complete description of kinetic behavior is given by B and e. The magnitudes of these parameters define the temperature range within which reaction rates become significant and that at which these become comparable there is also the possibility that such behavior may be associated with the operation of a common reaction mechanism or intermediate. 

When E = 0, log A is equal to B, and this corresponds to the specific rate constant for reaction at the isokinetic temperature kti. It is also easily shown that e — (Rfi) . The variation of ft with e over the temperature range of interest, in the reactions considered in the present review, is given in Fig. 1, which includes dashed lines for values of = + 0.05c, 0.10c, and 0.20e, corresponding to the influence of various levels of inaccuracy of data upon the calculated precision with which )3 may be determined. For example, when ode = 0.05 and /I = 700 K the deviation is + 30 K and this reduces to 15 K. when fi = 300 K other values are very readily found. 

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 electronic nature of silylsilver intermediate was interrogated through inter-molecular competition experiments between substituted styrenes and the silylsilver intermediate (77).83 The product ratios from these experiments correlated well with the Hammett equation to provide a p value of —0.62 using op constants (Scheme 7.19). Woerpel and coworkers interpreted this p value to suggest that this silylsilver species is electrophilic. Smaller p values were obtained when the temperature of the intermolecular competition reactions was reduced [p = — 0.71 (8°C) and —0.79 (—8°C)]. From these experiments, the isokinetic temperature was estimated to be 129°C, which meant that the product-determining step of silver-catalyzed silylene transfer was under enthalpic control. In contrast, related intermolecular competition reactions under metal-free thermal conditions indicated the product-determining step of free silylene transfer to be under entropic control. The combination of the observed catalytically active silylsilver intermediate and the Hammett correlation data led Woerpel and colleagues to conclude that the silver functions to both decompose the sacrificial cyclohexene silacyclopropane as well as transfer the di-terf-butylsilylene to the olefin substrate. 

Since A° is constant for a given series satisfying equation (44), any structural change in the reactants must be reflected in the only parameter, E, determining the rate constant. If T = Tit the rate constants of all reactions in the given set will be identical. For that reason, Tt is called the isokinetic temperature . It is a mathematical consequence of equation (44) and has no physical meaning. (The only physically reasonable isokinetic temperature is the absolute zero.) Nevertheless, the value of as compared with a medium value of the temperature range of experiments (Texp) can help us to classify possible correlations of the Arrhenius parameters (Simonyi, 1967 Tiidos, 1969). 

Reactions conforming to Equation (31) have Arrhenius correlations (log versus 1/7) for rate constants of the reaction of each substituent which pass through a single point at the isokinetic temperature (T). Thus at 7" = Tj the value of p would be zero and the sign of p would reverse as the isokinetic temperature is traversed. 

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. 

In general, the constant L = 0, but on account of the complieating circumstances there appears an empirical correction, and then L may be equal to another (small) constant value [see further in Eq. (11.12), L = A], The larger B is, the less the height of the energy barrier and the higher the reaction rate at a temperature lower than the Schwab inversion temperature (or isokinetic temperature). 

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