The theory of activated complexes

In this chapter, we discuss the modeling of elementary steps in condensed liquids and solids as well as reactions located with the interphases. The aim is always to analyze the elementary step and deduce an expression of the reactivity. 

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The theory of activated complex formation can be applied to mineral... 

Despite difficulties of interpretation, the application of theoretical concepts permits conclusions, or at least hints at conclusions, concerning the mechanism. As a result of the application of the theory of activated complexes, Equ. 5.3a can be recast as... 

A more complete analysis based on the theory of activated complexes and on statistical mechanics has been given by Zener (1951,1952). He considered the system or an atom in its initial equilibrium condition and in the activated state at the top of the potential barrier which separates the initial position from its neighbouring equilibrium position. The rate of transition from one equilibrium site to another is given by... 

For direct reactions, the reaction path profile has one potential barrier. In the frameworic of the theory of activated complex, the deviations of the temperature dependence of the rate constant from the Arrhenius equation for direct reactions can be explained by the temperature dependence of statistical sums of the reactants and activated complex. After inserting rotation, vibration, and translational statistical sums into expression (4.76), the temperature dependence of the rate constant is presented by the expression... 

Following collision theory, a model of the bimolecular reaction, called the theory of activated complexs or the theory of absolute speeds, was estabhshed in 1938 by Eyring and Polanyi respectively. 

Figure 10.11. Eyring and Polanyi developed the theory of activated complexes...

The theory of equilibrium is treated on the basis of thermodynamics considering only the initial and final states. Time or intermediate states have no concern. However, there is a close relationship between the theory of rates and the theory of equilibria, in spite of there being no general relation between equilibrium and rate of reaction. A good approximation of equilibrium can be regarded between the reactants and activated state and the concentration of activated complex can, therefore, be calculated by ordinary equilibrium theory and probability of decomposition of activated complex and hence the rate of reaction can be known. 

In theory, one can use statistical thermod3mamics to calculate the partition functions of all the species from first principles, AS, AH, and hence k. For simple systems, the calculation results are in good agreement with experimental data (e.g.. Chapter 3 in Laidler, 1987). For complicated geological systems, however, it is not possible to calculate k from first principles, but the concept of activated complexes is very useful for a microscopic understanding of the reaction... 

The first assumption, that phase space is populated statistically prior to reaction, implies that the ratio of activated complexes to reactants is obtained by the evaluation of the ratio between the respective volumes in phase space. If this assumption is not fulfilled, then the rate constant k(E, t) may depend on time and it will be different from rrkm(E). If, for example, the initial excitation is localized in the reaction coordinate, k(E,t) will be larger than A rrkm(A). However, when the initially prepared state has relaxed via IVR, the rate constant will coincide with the predictions of RRKM theory (provided the other assumptions of the theory are fulfilled). 

According to transition state theory, the rate of a reaction is the number of activated complexes passing per second over the top of potential energy barrier. This rate is equal to the concentration of activated complex times the average velocity with which a complex moves across to the product side. The activated complex is not in a state of stable equilibrium, since it lies at a maximum potential energy. 

The transition state theory gives us a framework to relate the kinetics of a reaction with the thermodynamic properties of the activated complex (Brezonik, 1990). In kinetics, one attempts to interpret the stoichiometric reaction in terms of elementary reaction steps and their free energies, to assess breaking and formation of new bonds, and to evaluate the characteristics of activated complexes. If, in a series of related reactions, we know the rate-determining ele-mentaiy reaction steps, a relationship between the rate constant of the reaction, k (or of the free energies of activation, AG ), and the equilibrium constant of the reaction step, K (or the free energy, AG°), can often be obtained. For two related reactions. 

However, a final word of caution should be added. The above conclusions have been drawn for H + H., and similar reactions for which the vibrational-energy separations are large. Deviations from the predictions of activated complex theory are more likely for very low energy barriers, or when the energy barrier is large in comparison with the vibrational quanta and is situated in the product valley. There is indeed theoretical evidence, and evidence from molecular beam and chemiluminescence studies, that such is the case for a number of reactions. 

The total number of ways of distributing the species is independent of the order in which the species are attached. In transition state theory applied to heterogeneous catalysis the coverage of activated complexes is neglected. Consequently, interactions between a particular type of adsorbed species and activated complexe are also neglected. Equilibrium condition for adsorption again implies equality of the chemical potential in the gas-phase and the adsorbed phase ... 

There are various concepts about the aluminum silicates dissolution mechanism. Relatively recently a low rate of their dissolution was explained by inner diffuse regime. Currently more substantiated appears hydrolysis with the formation of activated complexes. According to this theory, the dissolution begins with the exchange of alkaline, alkaline-earth and other metals on the mineral surface of H+ ions from the solution (see Figure 2.26). At that, metals in any conditions are removed in certain sequence. In case of the presence of iron and other metals with variable oxidation degree the process may be accompanied with redox reaction. Hydrolysis is a critical reaction in the dissolution of aluminum silicates. It results in the formation on the surface of a very thin layer of activated complexes in Na, K, Ca, Mg, Al and enriched with H+, H O or H O. The composition and thickness of this weakened layer depend on the solution pH. These activated complexes at disruption of weakened bonds with mineral are torn away and pass into solution. For some minerals (quartz, olivine, etc.) the disruption of one inner bond is sufficient, for some others, two and more. The very formation of activated complexes is reversible but their destruction and removal from the mineral are irreversible. 

HIRSCHPELDER and WIGRER /6/ first discussed the validity of activated complex theory from the viewpoint of quantum mechanics. They showed that the notion of an activated complex is compatible with Heisenberg s uncertainty principle only when the potential V(x) along the reaction path in the saddle-point region is sufficiently flat that the condition... 

It should be noted that a very good correlation of the experimental data for the same isotopic reactions was previously achieved by SHAVITT /32b/ on the basis of activated complex theory and making use of the SSMK potential energy surface. In this work an adjustable parameter is introduced for scaling the surface, and, moreover, a one-dimensional potential along the reaction coordinate is adjusted in or-... 

D24.3 The Eyring equation (eqn 24.53) results from activated complex theory, which is an attempt to account for the rate constants of bimolecular reactions of the form A + B iC -vPin terms of the formation of an activated complex. In the formulation of the theory, it is assumed that the activated complex and the reactants are in equilibrium, and the concentration of activated complex is calculated in terms of an equilibrium constant, which in turn is calculated from the partition functions of the reactants and a postulated form of the activated complex. It is further supposed that one normal mode of the activated complex, the one corresponding to displaconent along the reaction coordinate, has a very low force constant and displacement along this normal mode leads to products provided that the complex enters a certain configuration of its atoms, which is known as the transition stale. The derivation of the equilibrium constant from the partition functions leads to eqn 24.51 and in turn to eqn 24.53, the Eyring equation. See Section 24.4 for a more complete discussion of a complicated subject. 

An extension of the theory of ES complex formation is used to explain the high specificity of enzyme activity. According to the lock-and-key theory, enzyme surfaces will accommodate only those substrates having specific shapes and sizes. Thus, only specific substrates fit a given enzyme and can form complexes with it (see I Figure 10.4A). 

To find evidence for tunneling, one must first account for kinetic Isotope effects that do not depend on tunneling. The most direct method of doing this Is by means of activated complex theory, which leads to a formulation of the rate constant In terms of partition functions of the reactants and the activated complex. Arguments concerning the validity of activated complex theory are easy to provoke and difficult to settle, and I shall not consider this question here. It can then be shown (15,16) that the ratio of rate constants for two Isotopic forms of the same reactant (AX] and AX2) Is given by... 

In the absolute reaction rate theory, the rate of the reaction is proportional to the concentration of activated complex, which is in equilibrium with reactants. Thus, the activated complex concentration can be calculated using the corresponding equilibrium constant and, consequently, the activities. This is the reason why the activity coefficients appear in kinetic equations. 

Transition state theory argues that the rate of the reaction (modified by some probability function that the activated complex will go on to product rather than back to starting material) is equal to the concentration of activated complex, that is, [AB ] times the rate at which the complex passes over the barrier. In detail, it is presumed that a bond that was, for example, stretching is now stretched just to the point of being broken, or that a bond that was inhibited from twisting has now overcome that inhibition, and that the event occurs with some frequency v. This frequency, v, is simply Eilh (i.e., since E = hv), where h is Planck s constant (6.62 X 10 ergs or 1.58 x KT cals). Furthermore, since the energy, Ei, at which the process occurs is simply k T, the frequency (v) with which the barrier is overcome is k TIh, that is. 

Similar calculations have been carried out to estimate the -factor for various reactions. The results, for some of the examples of Table 8.1, are given in Table 9.2. The calculations did not account for bond-length increases in the activated complex. Since expansion affects both the moments of inertia and the vibrational frequencies in the same way, recalculation would increase the predictions of the theory by factors of 3-5. Even without this correction the predictions of activated complex theory are generally within a factor of 10 of the experimental value. Only when the experimental... 

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