Evans-Polanyi model

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... 

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). 

A First Approximation The Evans-Polanyi Model of the Bronsted Law and Other Linear Free-Lnergy Relationships... 

Figure 19.18 The Evans-Polanyi model for linear free-energy relationships. AG is the stability of the product relative to the reactant, E is the activation energy, T and r are the stable bond lengths of AB and BC, and r" is the A-B bond length in the activated state. 

This simple model shows how linear free energy relationships can arise from reactant and product energy surfaces with a transition state between them that is defined by the curve-crossings. Figure 19.19 shows how shifting the equilibrium to stabilize the products can speed up the reaction. It also illustrates that such stabilization can shift the transition state to the left along the reaction coordinate, to earlier in the reaction. If mil Im l, the transition state will be closer to the reactants than to the products, and if I mi correlate linearly with rates. 

G. The Evans-Polanyi model. Go to the classic text Glasstone et al. (1941) and see the derivation that leads to the diabatic potentials for the reactants. In the notation of Section 5.1.3, this diabatic surface has the functional form... 

Let us bring together several key ideas that we have discussed. We seek a imified approach where both the role of the solvent and the rearrangement of the reactants to form products are taken into consideration. We further want the approach to center attention on the correlation of reactivity with slmcture, a theme that we started in Chapter In essence, we generalize the one-coordinate discussion of solvation in Section 11.1 to a two-dimensional world that consists of a solvation coordinate and a reaction coordinate. Starting from the gas phase, what we do is generalize the one-coordinate Evans-Polanyi model to include the role of the solvent. 

The systematics of the energy profile along the reaction coordinate are obtained from the Evans-Polanyi model. Recall, Sections 5.1.4 and 7.0.2, that this model regards the adiabatic potential energy profile leading from reactants to prodncts... 

Fukuzumi and Kochi, 1981a). On the basis of a Bell-Evans-Polanyi diagram, an -value closer to 0.5 would have been expected. However we do not feel that the -value of 1 necessarily constitutes evidence for the ion-pair-like character of the transition state. On the basis of the CM model the transition state is described by (74). This would imply only c. 50% ion-pair character in... 

The predictions of the CM model are exactly the same. In line with a simple Bell-Evans-Polanyi diagram (e.g. Fig. 18), stabilization of the product configuration leads to an earlier transition state, while stabilization of an intermediate configuration leads it increasingly to mix into the transition-state wave-function. For example, stabilization of the carbocationic configuration [36] results in the transition state acquiring more of that character so that an E2 process becomes more El-like (Fig. 266). 

In the present chapter, we have attempted to illustrate how surface bonding and catalytic activity are closely related. One of the main conclusions is that adsorption energies of the main intermediates in a surface catalyzed reaction is often a very good descriptor of the catalytic activity. The underlying reason is that we find correlations, Brpnsted-Evans-Polanyi relations, between activation barriers and reaction energies for a number of surface reactions. When combined with simple kinetic models such correlations lead to volcano-shaped relationships between catalytic activity and adsorption energies. 

A study directed toward understanding when gas phase dynamics closely resembles the dynamics of the same reaction in solution was performed by Li and Wilson. io In this work, they used a model asymmetric A -t- BC reaction. By using an asymmetric reaction, Li and Wilson were able to test the validity in the solution phase of the Evans—Polanyi rule,3n which has proven to be quite useful in understanding gas phase reaction dynamics. The Evans-Polanyi rule states for a collinear A -t- BC reaction, that if the barrier to reaction is located early in the reaction coordinate, then translational excitation of the reactants is necessary to climb this barrier and vibrational excitation of the products will result. Conversely, a late barrier to reaction requires vibrational excitation of the reactants and results in translational excitation of the products. This rule has been validated numerous times in the gas phase and is an ideal example of how a simple rule can explain the dynamics of a large number of reaction systems. 

The objective of the study presented in this paper is to inspect the nature of the relation between the acidity and the activity of a given site towards the transformation of hydrocarbons over zeolites and to compare the relation derived from first-principles modelling of the catalytic mechanism with the type of correlations that are experimentally obtained and which can be viewed as an application of the Bell-Evans-Polanyi principle. 

Recently, semiempirical methods based on DFT calculations have been developed for catalyst screening. These methods include linear scaling relationships [41, 42] to transfer thermochemistry from one metal to another and Brpnsted-Evans-Polanyi (BEP) relationships [43 7]. Here, these methods and also methods for estimation of the surface entropy and heat capacity are briefly discussed. Because of their screening capabilities, semiempirical methods can be used to produce a first-pass microkinetic model. This first-pass model can then be refined using more detailed theory aided by analytical tools that identify key features of the model. The empirical bond-order conservation (BOC) method, which has shown good success in developing microkinetic models of small molecules, has recently been reviewed [11] and will not be covered here. 

Other methods for calculating the activation energies of the ORR have been reported in the literature. Based on the linear Bronsted-Evans-Polanyi relationship, Norskov et al. [85] proposed a method to estimate the least activation energy by calculating the stability of the reaction intermediate. With this simple model, the study reported a Tafel slope of 60 mV at 300 K (71 mV at 357 K). The value was consistent with experimental results. 

Beyond a number of assumptions and thermodynamic constraints a substantial reduction of the number of parameters to be determined from a set of experimental data is only possible by modeling the rate parameters. The modeling is based upon transition state theory, it makes use of the single event concept introduced by Froment and co-workers [Baltanas et al., 1989 Vynckier and Froment, 1991 Park and Froment, 2001 Feng et al., 1993 Svoboda et al., 1995 De Wachtere et al., 1999 Martinis and Froment, 2006 Kumar and Froment, 2007 Froment, 2005] and of the Evans-Polanyi relationship for the activation energy [1938]. 

The foregoing discussion deals with the stmcture effect on A (or entropy change). The stmcture effect on the activation energy (or reaction enthalpy change) is described by the Evans-Polanyi relation, with just two parameters Eo and a) for each single event type, which generally are obtained from model-compound experiments. 

---