The theory of absolute reaction rates

The theory of absolute reaction rates, which i s based on statistical mechanics, was developed in full generality by H. Eyring in 1935, although it was foreshadowed in kinetic theory investigations as early as 1915. A simplified development of the equations will be given here. In this theory, we have a postulate of equilibrium away from equilibrium, applied more broadly here than in the irreversible thermodynamics. 

The fundamental postulate of the theory of absolute reaction rates is that the reactants are always in equilibrium with activated complexes. The activated complex is that configuration of the atoms which corresponds energetically to the top of the energy barrier separating the reactants from the products (Fig. 33.2). We write the equilibrium 

If we know the concentration of activated complexes, the problem resolves into the calculation of the rate at which these complexes decompose into products that is, we must calculate the rate of the reaction 

The activated complex is an aggregate of atoms, which may be thought of as being similar to an ordinary molecule except that it has one special vibration with respect to which it is unstable. This vibration leads to dissociation of the complex into products. If the frequency 

Comparing Eqs. (33.27) and (33.28), we find that the rate constant is given by 

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The transition state theory provides a useful framework for correlating kinetic data and for codifying useful generalizations about the dynamic behavior of chemical systems. This theory is also known as the activated complex theory, the theory of absolute reaction rates, and Eyring s theory. This section introduces chemical engineers to the terminology, the basic aspects, and the limitations of the theory. 

An understanding of the mechanism of creep failure of polymer fibres is required for the prediction of lifetimes in technical applications. Coleman has formulated a model yielding a relationship similar to Eq. 104. It is based on the theory of absolute reaction rates as developed by Eyring, which has been applied to a rupture process of intermolecular bonds [54]. Zhurkov has formulated a different version of this theory, which is based on chain fracture [55]. In the preceding sections it has been shown that chain fracture is an unlikely cause for breakage of polymer fibres. 

The Arrhenius theory (above) was wholly empirical in terms of it derivation. A more rigorous, but related, form of the theory is that of Eyring (also called the theory of absolute reaction rates). The Eyring equation is... 

The equation (3) generates the famous BO potential energy hypersurface. The practical power of this concept is well documented and it remains at the foundation of important domains in computational quantum chemistry. The theory of absolute reaction rates is entirely based upon it [32-34, 63] as well as all modem quantum theories of reaction rates [36, 39, 64-80],... 

For the activated transfer of ions, the transfer current can be derived from the theory of absolute reaction rates as shown in Eqn. 7- 3 [Horiuti-Nakamura, 1967] ... 

THL.9. I. Prigogine et A. Mertens, Sur I hypothese d equilibre dans la theorie des vitesses absolues de reaction (On the equilibrium hypothesis in the theory of absolute reaction rates), in Contribution a VEtude de la Structure Moleculaire, volume commemoratif Victor Henri, Desoer, Liege, 1948. 

The reaction between two complex molecules proceeds more slowly compared to the reaction between two less complex species. Explained in short using the theory of absolute reaction rates ... 

When there is a strong interaction between the vapor species and the substrate, a can be assumed to be unity. For weak interactions the theory of absolute reaction rates may provide order-of-magnitude estimates. For many cases, this would probably be sufficient. 

To investigate the quantitative variation of enthalpy of activation and entropy of activation with x, a series of rate constants with different temperatures for each value of x were taken, and by using the theory of absolute reaction rates (6) ... 

The theory of absolute reaction rates has been applied to heterogeneous processes through equations of the form (3,121)... 

We would, therefore, agree with Bond s conclusion (3) that application of the transition state theory to heterogeneous reactions has not so far provided insight into the mechanisms of surface reactions and that the failures of the theory are generally more significant than the successes. We do not accept that the use of the theory of absolute reaction rates in the interpretation of kinetic data provides a general and reliable method for the estimation of the concentration of surface active sites but conclude that results should always be considered with reference to appropriate quantitative supporting evidence (133). 

From the theory of absolute reaction rates it follows4 6,50-561 for a simple barrier model (Fig. 1 a) that the rate constant of the jumps of a particle (motional unit) from site 1 to site 2 is given by... 

The partial rate factor is equal to a ratio of rate constants. It may be used as a measure of the substituent effect on the activation energy for the reaction, provided it is assumed that the pre-exponential factors in the kinetic equation for the reaction with benzene and its derivatives are the same. On the basis of the Arrhenius equation and the theory of absolute reaction rates the rate constant can be expressed as... 

To obtain information about both relaxations, the Eyring equation, from the theory of absolute reaction rates, for the dependence of the frequency of an absorption peak on temperature has been used ... 

In the theory of absolute reaction rates of Polanyi and especially Eyring (transition state theory) the frequency factor A contains instead of the collision number the frequency kt h, and the probability factor is replaced by eAslRT where AS is the entropy of activation. This entropy includes the concept of steric hindrance which maintains that the probability that partners collide in the correct way is small in certain cases. 

Arrhenius classical theory of reaction kinetics is based on the assumption that the starting materials (reactants) have to overcome an energy barrier, the activation energy, in order to be transformed into the products. This picture has been developed and made more explicit in the theory of absolute reaction rates [2-5, 7, 8, 11, 24, 464-466, 770, 771]. The influence of solvent on reaction rates is best treated by means of this theory -also known as transition-state theory, developed almost simultaneously in 1935 by Eyr-ing as well as Evans and Polanyi [464]. 

Eyring and Steam made the first study of the probability that the proton would have enough energy to climb over the potential-energy barrier. The basic application of the theory of absolute reaction rates made by Eyring begins with the eqiwtion for the frequency at which the proton crosses the barrier, i.e., the rate constant k... 

According to the theory of absolute reaction rates [4-9], the rate is the product of a universal frequency factor and the concentration of the activated complex or transition state, M (the system in the transient state of highest potential energy), crossing the energy barrier in the direction toward the products. The activated complex, in turn, is postulated to be in equilibrium with the reactants. Say, for a single-step reaction A + B — P ... 

In terms of statistical thermodynamics and the theory of absolute reaction rates, the equilibrium and rate constants of gas phase reactions... 

Once one accepts molecular fracture as an activated process, as outlined above, the theory of absolute reaction rates immediately leads to the following expression for the net rate v of the forward reaction A -> B, ie net bond fractiue. 

Protein monolayers can attain very high interfacial viscosities and elasticities. Moore and Eyring (1938) have developed a theory of interfacial viscosity, based on the theory of absolute reaction rates. In this theory, the flow of a molecule in a monolayer is treated as a movement of flow units, normally molecules, from one equilibrium position to another, passing over an intermediate activation energy barrier. The equation for the interfacial viscosity, tjs, which is derived is... 

From the transition-state model, or the theory of absolute reaction rates (Ref. 15, p. 89), the forward rate constant, k, for a reaction, the rate of... 

Curtiss, C. F., The Equilibrium Assumption in the Theory of Absolute Reaction Rates/ University of Wisconsin, Report CM-476, June 1948. 

Eyring, H. (1938) The theory of absolute reaction rates, Trans. Faraday Soc. 34, 41-48. 

His endless energy and glowing personality have carried worldwide his scientific genius to perceive and formulate the most reasonable and simple physical models with which to interpret important characteristics of complex systems and processes. The theory of absolute reaction rates and the significant structure model of liquids are classic examples. 

It is also commonly known as absolute-rate theory, the theory of absolute reaction rates, and transition-state theory. Our preference is for activated-complex theory. ... 

The theory of absolute reaction rates has not and probably will not, in the foreseeable future, permit the satisfactory a priori calculation of most reaction rates, particularly in condensed phases. What it does do, instead, is to identify an attainable goal toward which speculation, semiempirical calculation, and experimental work may be directed. That is the elucidation of transition state structure. If transition state structure could be determined, by whatever means, as well as the structure of many stable molecules is now known, it would be possible to calculate reaction rates. Further, even well short of enough information to calculate rates, even very approximate and incomplete transition state structures may often suffice for the calculation of such rate-related quantities as isotope effects,Bronsted a values, and Hammett p values. When successful, such partial calculations tend to confirm the partial transition state structures on which they are based and, thus, to permit further progress on a more secure base. We cannot be certain how well pleased Henry Eyring is with this hybrid of theory and empiricism, but it doesn t seem likely that he is too disturbed. The pragmatist, pleased with what works and ready to take information from anywhere, has always seemed to be as well represented in his work as the rigorous theoretician. 

At the time when the theory of absolute reaction rates appeared in 1935, a vast literature had already accumulated in the above three, seemingly independent areas of research. The seeming independence was emphasized by the contemporary reviews or monographs which were published in each field, with little if any mention of the other two fields (Belehradek, Ref. 16, on temperature Cattell, Ref. 17 on pressure Henderson, Ref. 14, and Clark, Ref. 15, on narcosis). Soon thereafter, however, the erstwhile separate fields were brought together in the application of the Eyring theory to bacterial luminescence, as alluded to above. 

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