Statistical thermodynamics activated complex theory

While the collision theory of reactions is intuitive, and the calculation of encounter rates is relatively straightforward, the calculation of the cross-sections, especially the steric requirements, from such a dynamic model is difficult. A very different and less detailed approach was begun in the 1930s that sidesteps some of the difficulties. Variously known as absolute rate theory, activated complex theory, and transition state theory (TST), this class of model ignores the rates at which molecules encounter each other, and instead lets thermodynamic/statistical considerations predict how many combinations of reactants are in the transition-state configuration under reaction conditions. 

Nowadays, the basic framework of our understanding of elementary processes is the transition state or activated complex theory. Formulations of this theory may be found in refs. 1—13. Recent achievements have been the Rice—Ramsperger—Kassel—Marcus (RRKM) theory of unimol-ecular reactions (see, for example, ref. 14 and Chap. 4 of this volume) and the so-called thermochemical kinetics developed by Benson and co-workers [15] for estimating thermodynamic and kinetic parameters of gas phase reactions. Computers are used in the theory of elementary processes for quantum mechanical and statistical mechanical computations. However, this theme will not be discussed further here. 

The aim of this appendix is to gather together a few statistical thermodynamics notions and, in particular, to establish the expression of equilibrium constants using partition functions, which is useful in the activated complex theory. 

In transition-state theory, the absolute rate of a reaction is directly proportional to the concentration of the activated complex at a given temperature and pressure. The rate of the reaction is equal to the concentration of the activated complex times the average frequency with which a complex moves across the potential energy surface to the product side. If one assumes that the activated complex is in equilibrium with the unactivated reactants, the calculation of the concentration of this complex is greatly simplified. Except in the cases of extremely fast reactions, this equilibrium can be treated with standard thermodynamics or statistical mechanics . The case of... 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

At the same time transition state theory requires information about the activated complexes, assumes equilibrium only for reactants, but not products and requires introduction of a special partition function (minus one degree of freedom). Another question which remains is the applicability of statistical thermodynamics, if the life time of activated complexes is ca. 10 13 s. For instance the application of transmission coefficient contradicts the basic principles of TST, namely statistical equilibrium between reactants and activated complexes. 

The transition State Theory (TST) applies the principle of statistical mechanics and thermodynamic to a system in which activated complexes are effectively in equilibrium with... 

Parsons made an attempt to derive some theoretical values for the preexponential term for the above processes, based on statistical thermodynamics and classical absolute rate theory. He pointed out the effect of possible surface mobility of the loosely bonded activated complex on the rate of reaction, or rather that of the mobility of the adsorbate, since the latter implies a mobile activated complex. Low bonding strength of H in the electrochemical environment may well allow mobility, provided that the effective free energy of the adsorbed species is less than A further activation entropy effect... 

In addition to theoretical estimates of the rate constant based on postulated structures for the activated complex, the statistical theory of rate constants can be used to correlate kinetic trends in homologous series of reactions. For this purpose we construct a thermodynamic interpretation of the rate constant. 

In order to take account of the fact that the solvent is made up of discrete molecules, one must abandon the simple hydrodynamically-based model and treat the solvent as a many-body system. The simplest theoretical approach is to focus on the encounters of a specific pair of molecules. Their interactions may be handled by calculating the radial distribution function, whose variations with time and distance describe the behaviour of a pair of molecules which are initially separated but eventually collide. Such a treatment leads (as has long been known) to the same limiting equations for the rate constant as the hydro-dynamically based treatments, including the term fco through which an activation requirement can be expressed, and the time-dependent term in (Equation (2.13)) [17]. The procedure can be developed, but the mathematics is somewhat complex. Non-equilibrium statistical thermodynamics provides an alternative approach [16]. The kinetic theory of liquids provides another model that readily permits the inclusion of a variety of interactions the mathematics is again fairly complex [37,a]. In the computer age, however, mathematical complexity is no bar to progress. Refinement of the model is considered further below (Section (2.6)). 

Interactions between soluble polymer and colloidal particles control the behavior of a large number of chemical products and processes and, hence, their technological viability. These dispersions have also attracted considerable scientific interest because of their complex thermodynamic and dynamical behavior—stimulated by the synthesis of novel polymers, improved optical and scattering techniques for characterization, and a predictive capability emerging from sophisticated statistical mechanical theories. Thus, the area is active both industrially and academically as evidenced by the patent literature and the frequency of technical conferences. 

Kinetic treatment based on the theory of complex reactions introduced the necessity to calculate quite many parameters (pre-exponential factors, activation energies of elementary reactions, etc.). Therefore a need to estimate independently the rates and surface coverage called for the application of theoretical approaches, based on thermodynamics and transition state theory, as well as other tools (ultra-high vacuum studies, spectroscopy) to get necessary data and reduce the number of parameters in statistical data fitting. 

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