Relation isokinetic

Jacobsen EN, Zhang W, Giiler ML (1991) J Am Chem Soc 113 6703 Isaacs N (1995) Physical Organic Chemistry. Wiley, New York, p 296 For a discussion of related isokinetic effects, see Isaacs N (1995) Physical Organic Chemistry. Wiley, New York, p 116... 

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. 

Figure 7-3 shows this treatment for reaction (7-35).8 Its slope defines the isokinetic temperature as 489 K or 216 °C. Also shown are the data for reaction (7-36), forward and reverse.9 Here, too, there is a valid isokinetic relation, with an isokinetic temperature of 331 K or 58 °C. 

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

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... 

Figure 8. Real (full lines) and apparent (broken heavy lines) isokinetic relations and experimental Arrhenius lines in the graph logk versus T" the same reaction series as in Figures 6 and 7.

Figure 11. Isokinetic relationship for the Lossen rearrangement of dihydroxamic acids (208), in the log kj versus log k, plot, real (full line) and apparent (broken line) relations.

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. 

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. 

Relations of another kind between LFER and the isokinetic relationship were sought by Lee (165-167), who tried to incorporate both in one extended... 

A special case of the isokinetic temperature is still to be mentioned, confined to a single reaction only, not strictly obeying the Arrhenius law (53). Temperature itself thus represents the variable factor, and the relation of AH and AS may be written... 

Related to the isokinetic temperature given in the appropriate column. 

For simplicity we assumed that the transition states are charged. However, it is not necessary to do so because the only requirement is that the difference in entropy of forming the transition states be offset by the difference in enthalpy of activation. The transition states could have different polarities and the same result be obtained. In fact, the transition states need not have high polarity. Forming a transition state in which there is a reduction in charge separation could result in more favorable solvation when the solvent is nonpolar. For there to be an isokinetic relationship for a series of reactions, it is required only that AH and AS be related in such a way that AG be approximately constant. 

A linear relationship between the standard enthalpies and entropies of a series of structurally related molecular entities undergoing the same reaction thus, AH° -I3AS° = constant or AAH° = (3AS°. When P > 0, this relationship is referred to as an isoequilibrium relationship. When the absolute temperature equals the factor P (often referred to as the isoequilibrium temperature), then all substituent effects on the reaction disappear (i e., AAG° = 0). In other words, a reaction studied at T = p will exhibit no substituent effects. This would suggest that, when one studies substituent effects on a reaction rate, the reaction should be studied at more than one temperature. Note also that the p factor in the Hammett equation changes sign at the isoequilibrium temperature. See Isokinetic Relationship... 

Other factors related to sample injection can also be important. It has been shown by Kirkland et al. (27) that the way of injectin the sample, especially by using a syringe, is critical. It seems that isokinetic" sampling is necessary to establish a uniform profile of the sa nple band at the column inlet. Otherwise eddies can form at the needle tip and the resulting multimodal injection yields to band nonuniformities ready at the column inlet. The time during which the sample is injected ia also important and it should be less than a maximum v ue, ts,u> isi eii by the relationship... 

The ETR proceeds through 11 in which the 0—0 bond is cleaved. The process of the bond-breaking follows isokinetic relations, indicating that fi = —173 °C, which is far below our reaction temperatures. This emphasizes the importance of entropy for the rates. Comparison of the magnitude of AA7/ [ Y H and A AS [ y h shows that the relative rates are controlled by T AA.S y h- The entropic dominance is derived from the translational degree of freedom occurring in the cleavage of the O- -O bond. 

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. 

Fig. 1. The slope of the compensation line e is related to the isokinetic temperature /i by the full line e = (Rfl)1. The influence of scatter of data on the accuracy with which P is determined is shown by the dashed lines, which correspond to uncertainty in aje = 0.05, 0.10, and 0.20.

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. 

If Eapp and In Aapp indeed obey to the Constable-Cremer relation, then there must be an isokinetic relation between all catalysts. At the isokinetic temperature, the activities for all catalysts are the same. However, it is very difficult to establish this isokinetic relation with statistical certainty. Therefore, a second, more reliable way to establish the presence of an isokinetic behavior is to plot all activity plots in one graph, and to check if there is an isokinetic temperature 7] where all activity plots intersect. Since the parameters in such a plot... 

It is evident that a calculation of the rate constant neglecting AS will only be correct for a limited number of cases. Series of reactions are known in which AS really remains approximately constant. Other series obey the isokinetic relation in others the change in AS is independent of E [14]. Nevertheless, for the time being, our only choice is to consider only the energy, and compare the result with experiment. A procedure for calculating E at variable AS in a generally applicable form is not yet available. 

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