Rate-equilibrium relationships

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

Figure 5-17. Rate-equilibrium relationship according to Eq. (5-69) for a hypothetical system with AG = 20.

There are several equations other than the Marcus equation that describe rate-equilibrium relationships. Murdoch writes all of these equations in the general form... 

These results suggest that the Marcus equations can be applied quite successfully to gas phase displacement reactions, as suggested by Professor Brauman. We are currently generating more cross reactions and intend to test other rate-equilibrium relationships using our data. 

When do rates and equilibria correlate in organic chemistry, and why do rate-equilibrium relationships break down ... 

Normal rate-equilibrium relationships are expected for methyl SN2 processes but the Bronsted parameter a is not a measure of TS charge development in these cases (Pross, 1984). 

The existence of rate-equilibrium relationships in the SN2 reactions of simple alkyl derivatives is well established (Arnett and Reich, 1980 Lewis and Kukes, 1979 Lewis et al., 1980 Bordwell and Hughes, 1982). A plot of the rate constants for a family of nucleophiles against the pKa for the nucleophiles generates linear Brensted plots whose slopes lie in the range 0.3 to 0.5. A typical example, taken from Bordwell s work (Bordwell and Hughes, 1982) is the reaction of a family of aryl thiolates with -butyl... 

An attempt to establish a rate-equilibrium relationship shows that a plot of E X -I- CH3X) as a function of the methyl cation affinity (MCA) of X-yields the linear correlation (44), where 7P(CH3) represents the ionization... 

As we have seen, the Hammett equation can, in principle, be applied to both equilibria and rate data. This implies that in certain cases, it is feasible to relate rate constants to equilibrium constants when both reflect the effects of a given structural moiety. In a general form a rate-equilibrium relationship can be written in terms of the corresponding changes in free energies of activation and of equilibration ... 

Rate and equilibrium constants for reactions of the trityl cation are summarized in the first two columns of Table 7 and clearly indicate that no simple rate-equilibrium relationship exists. The mild decrease in rate constants kx for... 

A similar picture holds for other nucleophiles. As a consequence, there might seem little hope for a nucleophile-based reactivity relationship. Indeed this has been implicitly recognized in the popularity of Pearson s concept of hard and soft acids and bases, which provides a qualitative rationalization of, for example, the similar orders of reactivities of halide ions as both nucleophiles and leaving groups in (Sn2) substitution reactions, without attempting a quantitative analysis. Surprisingly, however, despite the failure of rate-equilibrium relationships, correlations between reactivities of nucleophiles, that is, comparisons of rates of reactions for one carbocation with those of another, are strikingly successful. In other words, correlations exist between rate constants and rate constants where correlations between rate and equilibrium constants fail. 

Stefanidis, D. Bunting, J. W. Rate-equilibrium relationships for the deprotonation of 4-phenacylpyridines and 4-phenacylpyridi-nium cations./. Am. Chem. Soc. 1990, 112, 3163-3168. 

This final point signifies that the value of a in the rate-equilibrium relationship (2) is not constant but decreases as the reaction becomes increasingly exothermic. It should be noted however that since the Bell- Evans-Polanyi model and the Hammond postulate are couched in energy terms the assumption that free energy changes (AG°) are proportional to energy changes (A °) is inherent in eqns (1) and (2). 

It is important to note that the above presentation, justifying the reactivity-selectivity principle, is based on a number of fundamental assumptions. First, it is assumed that the Leffler-Hammond postulate is valid, which in turn implies that the reaction under consideration obeys a rate-equilibrium relationship [eqn (2)]. This assumption often cannot be verified since for reactions of highly active species such as carbenes, free radicals, carbonium ions, etc., equilibrium constants are generally not measurable. However it follows that for reactions which do not conform to a rate-equilibrium relationship, no reactivity-selectivity relationship is expected. Also, in Fig. 4, the difference in the free energy of the... 

Since (16) is a rate-equilibrium relationship [equivalent to the relationship (2) discussed earlier], a is considered to reflect the degree of proton transfer in the transition state and hence is a measure of selectivity. Values of a close to 0 are associated with exothermic reactions in which the degree of proton transfer in the transition state is as yet small. Similarly, values of a close to 1 are associated with endothermic reactions in which the degree of proton transfer in the transition state is almost complete. 

The clearest example of the danger in using a as a measure of transition state structure is illustrated in the work of Bordwell et al. (1969, 1970, 1975). In the rate-equilibrium relationship for the deprotonation of a series of nitroalkanes the unprecedented Br nsted slopes of 1 61 for l-aryl-2-nitropropanes and 1-37 for 1-arylnitro-ethanes were obtained. The simple exposition of the mechanistic significance of a disallows values greater than 1. This, coupled with the fact that the transition state for the proton transfer is not product-like (as established by alternative criteria) indicates at best that, in at least some cases, a does not reflect the selectivity of a particular reaction. Several attempts to rationalize these anomalous results have been made. 

In conclusion, it is apparent that the use of the Br nsted coefficient as a measure of selectivity and hence of transition state structure appears to be based on extensive experimental data. However, the many cases where this use of the Br nsted coefficient is invalid suggest that considerable caution be used in drawing mechanistic conclusions from such data. The limitations on the mechanistic significance of a require further clarification, but the first steps in defining them appear to have been taken. The influence of change in the intrinsic barrier and variable intermolecular interactions in the transition state, both of which will result in a breakdown of the rate-equilibrium relationship, as well as internal return appear to be some of the key parameters which determine the magnitude of the Br nsted coefficient in addition to the degree of proton transfer. 

Through lack of an unambiguous method for direct determination, the acidity constants of carbon acids have, for many years, been estimated from the rate of proton abstraction by means of rate-equilibrium relationships. Thus, Bell (1943) (see also Hibbert, 1977) estimated the acetaldehyde, acetone and acetophenone acidity constants (19.7,20.0 and 19.2, respectively) by assuming that the rate constants for proton abstraction from several mono- and dicarbonyl compounds to a single base (A-) with pAHA = 4.0 obey a Bransted equation in its differential form (47). By taking curvature into account, the... 

Imagine two molecules combining with each other in a simple, one-step, exothermic reaction leading to two possible products A and B (Fig. 3.1a). Chemists have long appreciated that the more exothermic reaction, that leading to the product B, is usually the faster—it has been called the rate-equilibrium relationship, and is related to the reactivity-selectivity principle. The explanation is easy enough—whatever features lead the product B to be lower in energy than the product A will have developed in the transition structure to some extent. Thermodynamics does affect kinetics—a source of endless confusion. 

Although the value of the coefficient 1.16 in (20) does not have as direct a physical significance as the a-exponent in the extended Brpnsted equation (19) because the reaction, solvents and temperature are different, there is still a good linear rate-equilibrium relationship for benzhydryl carbocation formation the overall correlation embraces clearly concave partial correlations with varying slopes for the respective Y series. The whole pattern of substituent effects, pXr vs should be essentially identical (with only the ordinate scale being slightly different) to that of log (/ xy/Z hh) vs 2 a for the solvolyses shown in Fig. 8. 

The progress of the reaction at the transition state, a, is usually obtained from the coefficients of the extended Brpnsted equation (19) or of other rate-equilibrium relationships which compare substituent effects on kinetics and thermodynamics. Using (24) the p values can also express this position if Pk for rates and pe for equilibria of the same elementary step are available. 

The identity of r values for solvolysis reactivities and the gas-phase stabilities of the corresponding carbocations implies the generality of the extended Brpnsted relationship or Hammond-Leffler rate-equilibrium relationship for benzylic solvolyses, i.e. (37a,b),... 

We have discussed the significant dependence of the substituent effect upon the coplanarity of aryl rings in [3C (X,Y,Z)]. As the excellent rate-equilibrium relationship shown in Fig. 34 indicates, the effect of the aryl conformation on and Xr+ must be similar for both terms. For the equilibria. 

Ritchie, 1986 McClelland et al., 1989, 1991) for a wide set of triarylmethyl cations, and that there is a reasonably linear correlation encompassing the entire set of triarylmethyl carbocations over 16 p/Cr+ units with a small amount of scatter (Fig. 34). The implication of this behaviour is that despite a change in the cation stability, there is a little change in the apparent position of the transition state, at least as revealed in this rate-equilibrium relationship. 

Scheme 14 is related to the potential energy profile in the vicinity of the transition state in the dissociation/recombination equilibria of [3C (X = Y = Z)] given by the Br0nsted-type analysis, the rate-equilibrium relationship in terms of a fixed Y-T a scale for the equilibrium. The potential energy surface should be essentially non-crossing, and consistently, the assumption that the Hammond shift in the transition-state coordinate is reflected exclusively in the p value but not in the r value can be applied to systems involving an appreciable shift in the structure of the solvolytic transition states. 

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