Bell-Evans-Polanyi equation

In the mid-1930s of the twentieth century, Bell, Evans and Polanyi correlated energies of activation, E, for several reactions in the vapour phase with heats of reaction, A/I°, according to the following expression  

Both free-energy relationships expressed by eqs. (7.6) and (7.18) would imply that reactions which have a strong driving force thermodynamically will also proceed rapidly. However, if we consider this more deeply we will see that such an implication does not hold generally. For example, the energetically favourable oxidation of hydrocarbons in the presence of air may not take place for years, whereas the energetically unfavourable hydration of carbon dioxide takes place in seconds. For elementary reactions, the correct interpretation rests on the values of the intrinsic barriers. 

---

The Bell-Evans-Polanyi relationship and the Hammond postulate (see Section 3.3) provide a basic framework within which to discuss structure-reactivity relationships. The Bell-Evans-Polanyi equation implies that there will be a linear relationship between and the C-H BDE. 

Equation (82) predicts that for reactions with zero free energy change a = 0.5, while for exothermic reactions, a < 0.5 and for endothermic reactions, a > 0.5. Since according to both Marcus theory and the Bell— Evans-Polanyi model early transition states are related to exothermic reactions and late transition states to endothermic reactions, a may be interpreted as a relative measure of transition state geometry. However, in our view even this interpretation should be treated with a measure of healthy scepticism. Even if one accepts Marcus theory without reservation, the a... 

For five out of the six reactions investigated, Figure 1.10 shows a decrease in the activation enthalpy AH with increasingly negative reaction enthalpy AH. Only for the sixth reaction— drawn in red in Figure 1.10—is this not true. Accordingly, except for this one reaction AH and AHr are proportional for this series of radical-producing thermolyses. This proportionality is known as the Bell-Evans-Polanyi principle and is described by Equation 1.3. 

The thermolyses presented in this chapter are one example of a series of analogous reactions. The Bell-Evans-Polanyi relationship of Equation 1.3 also holds for many other series of analogous reactions. The general principle that can be extracted from Equation 1.3 is that, at least for a reaction series, the more exothermic the enthalpy of reaction, the faster it will be. But this doesn t mean that all reactions that are exothermic are fast, so be careful. 

The thermolyses presented in this chapter are one example of a series of analogous reactions. The Bell-Evans-Polanyi relationship of Equation 1.3 also holds for many other series of analogous reactions. 

The curve acts as a reference line, but it also shows that the slope, a, in Equation (1.30) has a value near one for AG° near zero, and approaches zero as AG ° becomes a large negative number. Such behavior is an expected consequence of the Bell-Evans-Polanyi-Leffler-Hammond principle.It corresponds to a late transition state for the more difficult reactions and a progressively earlier transition state for more exergonic reactions. However, the curve appears to level off at an n value near 8, which corresponds to a second-order rate constant of about 1 s at 25 °C, far from the diffusion-controlled limit. 

Many reactions exhibit effects of thermodynamics on reaction rates. Embodied in the Bell-Evans-Polanyi principle and extended and modified by many critical chemists in a variety of interesting ways, the idea can be expressed quantitatively in its simplest form as the Marcus theory (15-18). Murdoch (19) showed some time ago how the Marcus equation can be derived from simple concepts based on the Hammond-Leffler postulate (20-22). Further, in this context, the equation is expected to be applicable to a wide range of reactions rather than only the electron-transfer processes for which it was originally developed and is generally used. Other more elaborate theories may be more correct (for instance, in terms of the physical aspects of the assumptions involving continuity). For the present, our discussion is in terms of Marcus theory, in part because of its simplicity and clear presentation of concepts and in part because our data are not sufficiently reliable to choose anything else. We do have sufficient data to show that Marcus theory cannot explain all of the results, but we view these deviations as fairly minor. 

The Bell-Evans-Polanyi relationship, Hammond s postulate, and the Marcus equation are all approaches to analyzing, understanding, and predicting relationships between the thermodynamics and kinetics of a series of closely related reactions. This is an important issue in organic chemistry, where series of reactions differing only in peripheral substituents are common. Each of these approaches provides a sound basis for the intuitive expectation that substituents that favor a reaction in a thermodynamic... 

A closely related statement of the correlation of energy barriers with heats of reaction is known as the Bell-Evans-Polanyi (BEP) principle (equation 6.69). ° Note that the BEP principle is concerned with the activation energies, while the Hammond and Leffler postulates are concerned with the structures of transition states. Of course, bonding and energy are inherently related, so the Hammond-Leffler postulate and the Bell-Evans-Polanyi principle are complementary. 

FIGURE 5.7. Bell-Evans-Polanyi plot of energy vs. a reaction coordinate for the reaction of equation (5.70). 

Our initial interest in applying the cross relation to HAT grew out of the limitations of the Bell Evans Polanyi (BEP) equation diseussed above. This equation holds within a set of similar reactions, but with the expansion of HAT reactions to include transition metal reactions it was not clear what made reagents similar. It was not evident why different classes of reactions fall on different correlation lines (defined by the parameters a and p, see above). For example, it has long been known that, at the same driving force, H abstraction from O-H bonds is substantially faster than from C-H bonds. Transition metal... 

The equation of Bell-Evans—Polanyi (see eq. (7.18)) implies that exothermic reactions will have lower barriers than the endothermic ones. The view that a transition state has structural and energy features that are intermediate between those of starting materials and products is due to George Hammond [9], who resurrected the view implied by those three authors in 1955. The aim of Hanunond was that of mechanistic interpretations, postulating that the changes in structure of the TS are affected by the manner in which the substituents affect the energies of intermediates on alternate pathways from reactants to products. Since then this assumption has been known as the Hammond postulate. 

---