Temperature isokinetic

The proportionality constant p has the dimension of absolute temperature, and it is called the isokinetic temperature. It has the significance that when T = P,... 

AG = 0 that is, all substituent (or medium) effects on the free energy change vanish at the isokinetic temperature. At this temperature the AH and TAS terms exactly offset each other, giving rise to the term compensation effect for isokinetic behavior. 

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 ... 

Thus, y is the slope of the plot of A// against AG at the harmonic mean temperature, and from y the isokinetic temperature P is calculated. Tomlinson has shown many examples of this type of analysis. 

In a series of reactions for which an acceUrative decrease in the activation energy is accompanied by a decelerative decrease in the entropy of activation (Compensation Law ), or the two increase together, there wiU be an isokinetic temperature (between 0-200° C for three-fourths of the 79 reactions tabulated by Leffler ). The rate vs. temperature curves for all the reactions in the series pass through this single point. Comparisons are affected since the isokinetic temperature is a point of inversion of relative reactivity in the series. 

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. 

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. 

A corollary of this pattern for parallel reactions is that the one with the larger activation enthalpy grows relatively more important at higher temperatures. Also, the reaction that is slower below the isokinetic temperature is the faster one above it. One can also show that the logarithm of product yield, ln([P]]/tP2]), is a linear function of l/T (see Problem 7-13). 

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. 

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. 

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)... 

Figure 4.35. Effect of catalyst work function on the activation energy EA, preexponential factor k° and catalytic rate enhancement ratio r/r0 for C2H4 oxidation on Pt/YSZ 4 p02=4.8 kPa, Pc2H4=0-4 kPa,4,54 kg is the open-circuit preexponential factor, T is the mean temperature of the kinetic investigation, 375°C.4 T0 is the (experimentally inaccessible) isokinetic temperature, 886°C.4 25,50...

Figure 4.36. Effect of catalyst potential UWR and work function on the activation energy E (squares) and preexponential factor r° (circles) of C2H4 oxidation on Rh/YSZ. open symbols open-circuit conditions. Te is the isokinetic temperature 372°C and r is the open-circuit preexponential factor. Conditions po2=l.3 kPa, pc2n =7.4 kPa.50 Reprinted with permission from Academic Press.

It is quite interesting that ln(r°/r0°), where r0° is the open-circuit preexponential factor, also varies linearly with Uwr and O (Figs. 4.35 to 4.37) and in fact, when plotted as kbT0ln(r°/ro°) where T is the isokinetic temperature discussed below, with the same slope as EA. This decrease in apparent preexponential factor with increasing <3> can be attributed to the reduced binding strength and thus enhanced mobility of chemisorbed oxygen on the catalyst surface. 

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).

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. 

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. 

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... 

---