The Isokinetic Relationship

It can be shown - p- that if an LFER is observed over a range of temperatures, and if the enthalpy and entropy changes are temperature independent, then the enthalpy changes must be directly proportional to the entropy changes for the reaction series. Let us start with the proposition that a real effect of this type has been demonstrated for a reaction series we write this as 

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, 

8AG = 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. 

Two extreme situations should be noted. If p =0, then 8AG = — 78A5, and the reaction series is entirely entropy controlled it is said to be isoenthalpic. If I/p = 0, then 8AG = 8A//, and the series is enthalpy controlled, or isoentropic. All of these relationships apply also to equilibria, but we will be concerned with kinetic quantities. 

Most of the isokinetic relationships in the literature have been established from plots of AH against AS collections of these have been published and several authors have discussed the mechanistic implications.9 6. csap. 12  

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. 

Transition state theory gives the Gibbs free energy of activation from either of the forms as 

To illustrate and use this equation, consider again reaction (7-10). The following are the values of AG at 77 °C, the middle temperature, calculated from the rate constant at this temperature 1 

Having AG values at other temperatures, perhaps at the reference temperature of 298 K, is sometimes useful. We then have 

For this calculation, the assumption was made that AH and A5 are temperature-independent, which is not unreasonable. The same cannot be said of the AG values, which will always vary nearly linearly with T. 

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It was not until the 1970s that the statistics of the isokinetic relationship was satisfactorily worked out.Exner first took this approach Let k, and 2 be the rate constants for a member of a reaction series at temperatures T, and T2, with T2 > T, and let k° and k° be the corresponding values for the reference member of the series. Then Eqs. (7-76) and (7-77) are easily derived for the reaction 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. 

The issues to be dealt with here have been touched upon earlier, and the reader should be reminded of selectivity (Section 5.5) and of the isokinetic relationship (Section 7.4). We saw that there is often a linear correlation between the values of A// and AS for a reaction series. This is usually written as... 

In addition to chemical reactions, the isokinetic relationship can be applied to various physical processes accompanied by enthalpy change. Correlations of this kind were found between enthalpies and entropies of solution (20, 83-92), vaporization (86, 91), sublimation (93, 94), desorption (95), and diffusion (96, 97) and between the two parameters characterizing the temperature dependence of thermochromic transitions (98). A kind of isokinetic relationship was claimed even for enthalpy and entropy of pure substances when relative values referred to those at 298° K are used (99). Enthalpies and entropies of intermolecular interaction were correlated for solutions, pure liquids, and crystals (6). Quite generally, for any temperature-dependent physical quantity, the activation parameters can be computed in a formal way, and correlations between them have been observed for dielectric absorption (100) and resistance of semiconductors (101-105) or fluidity (40, 106). On the other hand, the isokinetic relationship seems to hold in reactions of widely different kinds, starting from elementary processes in the gas phase (107) and including recombination reactions in the solid phase (108), polymerization reactions (109), and inorganic complex formation (110-112), up to such biochemical reactions as denaturation of proteins (113) and even such biological processes as hemolysis of erythrocytes (114). 

Equation (10) represents the simplest form of the isokinetic relationship however, several equivalent expressions are also possible and will now be discussed and shown in diagrams. It should be commented in advance that algebraic equivalence does not imply equivalence from the statistical point of view (see Section IV.). 

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. 

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 most general representation of the isokinetic relationship is the plot of logk against the reciprocal temperature. If the Arrhenius law is followed, each... 

The whole concept based on parameters, although used several times (3, 57, 155, 156, 201) and advocated particularly by Good and Stone (200), has a principal defect. The results are dependent not only on experimental rate constants, but also on the values of the parameter and on the form of the correlation equation used. Furthermore, the procedure does not give any idea of the possible error. Hence, it could be acceptable only in an unrealistic case, that in which the isokinetic relationship itself and the correlations with the parameter are very precise. 

Various algebraic expressions and various graphic representations of the isokinetic relationship offer the possibility of investigating each particular case from different sides and of stating the results and their consequences. A given kind of representation can be useful in a particular case, and no one of them can be considered to be erroneous in itself. 

Having in mind the various forms of the isokinetic relationship, we can also show its physical meaning in kinetics more clearly. Let us consider a reaction series with a variable substituent, solvent, or other factor. The term reaction series was discussed by Bunnett (14), with the conclusion that the common mechanism of all reactions is a necessary condition (12). However, this condition can seldom be ascertained, and best after finishing the whole analysis. At the beginning, it may be sufficient that the reaction products are invariable and the kinetic order equal. In addition, the structural changes should not be too large of course, this condition cannot be defined precisely. 

The isokinetic relationship can further yield a preliminary test of a common mechanism i.e., when one reaction deviates from the others, it follows to a high probability that its mechanism is different. The deviations are best seen in a plot like Figure 4, while in Figure 5, it is difficult to decide which of the straight lines does go through the common point of intersection. 

Of course, Sqo Sq if the difference is significant, the hypothesis of a common point of intersection is to be rejected. Quite rigourously, the F test must not be used to judge this significance, but a semiquantitative comparison may be sufficient when the estimated experimental error 6 is taken into consideration. We can then decide whether the Arrhenius law holds within experimental error by comparing Soo/(mi-21) with 6 and whether the isokinetic relationship holds by comparing So/(ml — i— 2) with 5. ... 

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... 

The natural and correct form of the isokinetic relationship is eq. (13) or (13a). The plot, AH versus AG , has slope Pf(P - T), from which j3 is easily obtained. If a statistical treatment is needed, the common regression analysis can usually be recommended, with AG (or logK) as the independent and AH as the dependent variable, since errors in the former can be neglected. Then the overall fit is estimated by means of the correlation coefficient, and the standard deviation from the regression line reveals whether the correlation is fulfilled within the experimental errors. 

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. 

Figure 18. Schematic representation of the isokinetic relationship, a, in an isoentropic series, b, in an isoenthalpic series, c, with compensation Texp d, with compensation < Tgxp.

Since its discovery (23), the isokinetic relationship has attracted the attention of theoretical chemists who attempted to give reasons for its existence or to predict its range of validity. They used quite different approaches in the framework of various theoretical disciplines. For this reason, the individual arguments cannot be discussed here, and an attempt is made only to confront the main results with the experimental evidence now available. 

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. 

Many workers have offered the opinion that the isokinetic relationship is confined to reactions in condensed phase (6, 122) or, more specially, may be attributed to solvation effects (13, 21, 37, 43, 56, 112, 116, 124, 126-130) which affect both enthalpy and entropy in the same direction. The most developed theories are based on a model of the half-specific quasi-crystalline solvation (129, 130), or of the nonideal conformal solutions (126). Other explanations have been given in terms of vibrational frequencies involving solute and solvent (13, 124), temperature dependence of solvent fluidity in the quasi-crystalline model (40), or changes of enthalpy and entropy to produce a hole in the solvent (87). 

The crucial test of all of the theories based on solvation would be the absence of the isokinetic relationship in the gas phase, but the experimental evidence is ambiguous. Rudakov found no relationship for atomization of simple molecules (6), whereas Riietschi claimed it for thermal decomposition of alky] chlorides (96) and Denisov for several radical reactions (107) however, the first series may be too inhomogeneous and the latter ones should be tested with use of better statistics. A comparison of the same reaction series in the gas phase on the one hand and in solution on the other hand would be most desirable, but such data seem not to be available. 

Ritchie and Sager (124) distinguish three types of reaction series according to whether the Hammett equation or the isokinetic relationship is obeyed, or both. The result that the former can be commonly valid without the latter seems to be based on previous incorrect statistical methods and contradicts the theoretical conclusions. Probably both equations are much more frequently valid together than was anticipated. The last case, when the isokinetic relationship holds and the Hammett equation does not, may be quite common, of course, and has a clear meaning. Such a series meets the condition for an extrathermo-dynamic treatment when enough experimental material accumulates, it is only necessary to define a new kind of substituent constant. 

An attempt to derive the isokinetic relationship still more generally considering a temperature-dependent a (2) is not quite correct. Equation (72), corresponding to eq. (4), p. 317, of (2), then has a solution... 

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

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