Statistical theory of bimolecular reactions

The statistical theory can be applied to the calculation of rate constants of bimolecular reactions that occur through a long-lived complex. Probable reaction profiles of such reactions are shown in Fig. 2.8. 

The statistical theory of bimolecular reactions is based on two assumptions  

The latter assumption makes it possible to understand the term long-lived complex. The lifetime of the AB molecule before its decomposition should have such a value that the energy released at the step of AB formation had time to be statistically redistributed over all internal degrees of freedom of the AB molecule. 

The first statement of the statistical theory on the independent character of the steps of AB ( 8, J) formation and decomposition allow us to present the further reactions as microscopic reactions 

Let us present the rate of formation of the C and D products as A (T)[A][B]. The statistical theory relates i(T) to k (T) and microscopic decomposition rate constants M (e, T)[ and k t, T)[, which can he calculated using the statistical theory. 

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E.E.Nikitin and S.Ya. Umanski, Statistical theory of bimolecular reactions, in Khimiya Plasmy, ed. B.M.Smimov, Moscow, Atomizdat, 1974, p. 8... 

For a classical formulation that admits a host of alternate statistical theories of bimolecular reactions see A. F. Wagner and E K. Parks, J. Chem. Phys. 65,4343 (1976). 

An illustration of the relationship between the collision and statistical theory of bimolecular reactions will be given in the next section on the basis of approximate or accurate classical and quantum-mechanical calculations. Prom equation (50.Ill) it is evident that an evaluation of the ratio / ac particular interest in... 

Light, J. C. (1979). Complex mode chemical reactions statistical theories of bimolecular reactions. Atom-Molecular Collision Theory A Guide for the Experimentalist, R. B. Bernstein (ed.). New York, Plenum Press. 

J, C. Light, Complex-mode chemical reactions Statistical theories of bimolecular reactions, in reference 2, p. 647. 

E.E.Nikitin, On the statistical theory of endothermic reactions. I. Bimolecular reactions, Teor. Eksp. Khim. 1, 135 (1965)... 

The essential nature of this relationship is clear statistical theories are based on a number of simplifying assumptions consistent with chaotic behavior. Specifically,2 any such theory must satisfy microscopic reversibility and the condition of zero relevance. The latter condition requires that the final state be independent of all initial conditions other than conserved quantities, that is, from the viewpoint of classical mechanics, that the system display the relaxation characteristic of chaotic motion. We note, for reference, that this minimal set of requirements allows for the construction of a large number of theories,3 the most prominant of which are the RRK.M theory of uni-molecular dissociation4 and the phase space theory of bimolecular reactions.5 Such theories have analogues, and in some cases their origins are in other areas such as nuclear physics.6... 

White R Aand Light J C 1971 Statistical theory of bimolecular exchange reactions angular distribution J. Chem. Phys. 55 379-87... 

A rigorous relation between the collision and-statistical formulations of the theory of bimolecular reactions is obtained from equations (67.Ill) and (23.IV), which give, instead of (49.IV), the equation... 

J. C. Light, Statistical theory of bimolecular exchange reactions. Disc. Faraday Soc. 44, 14-29 (1967). 

E. E. Nikitin, Statistical theory of endothermic reactions. Part 1. Bimolecular reactions, Theor. Exp, Chem. 1, 83-89 (1965). 

The statistical adiabatic channel model (SACM) " is one realization of the laiger class of statistical theories of chemical reactions. Its goal is to describe, with feasible computational implementation, average reaction rate constants, cross sections, and transition probabilities and lifetimes at a detailed level, to a substantial extent with state selection , for bimolecular reactive or inelastic collisions with intermediate complex formation (symbolic sets of quantum numbers v, j, E,J. ..)... 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

F. A. Wolf and J. L. Haller, Statistical theory of four-body bimolecular resonant ion-molecule reactions, J. Chem. Phys. 52, 5910-5922 (1970). 

The simple collision theory for bimolecular gas phase reactions is usually introduced to students in the early stages of their courses in chemical kinetics. They learn that the discrepancy between the rate constants calculated by use of this model and the experimentally determined values may be interpreted in terms of a steric factor, which is defined to be the ratio of the experimental to the calculated rate constants Despite its inherent limitations, the collision theory introduces the idea that molecular orientation (molecular shape) may play a role in chemical reactivity. We now have experimental evidence that molecular orientation plays a crucial role in many collision processes ranging from photoionization to thermal energy chemical reactions. Usually, processes involve a statistical distribution of orientations, and information about orientation requirements must be inferred from indirect experiments. Over the last 25 years, two methods have been developed for orienting molecules prior to collision (1) orientation by state selection in inhomogeneous electric fields, which will be discussed in this chapter, and (2) bmte force orientation of polar molecules in extremely strong electric fields. Several chemical reactions have been studied with one of the reagents oriented prior to collision. ... 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

These well-known results of the physics of phase transitions permit us to stress useful analogy of the critical phenomena and the kinetics of bimolec-ular reactions under study. Indeed, even the simplest linear approximation (Chapter 4) reveals the correlation length 0 - see (4.1.45) and (4.1.47), or 0 = /d for the diffusion-controlled processes. At t = 0 reactants are randomly distributed and thus there is no spatial correlation between them. These correlations arise in a course of the reaction, the correlation length 0 increases monotonously in time but 0 — 00 at t —> 00 only. Consequently, a formal difference from statistical physics is that an approach to the critical point is one-side, t0 —> 00, and the ordered phase is absent here. There is also evident correspondence between the parameter t in the theory of equilibrium phase transitions and time t in the kinetics of the bimolecular... 

For example, the standard synergetic approach [52-54] denies the possibility of any self-organization in a system with with two intermediate products if only the mono- and bimolecular reaction stages occur [49] it is known as the Hanusse, Tyson and Light theorem. We will question this conclusion, which in fact comes from the qualitative theory of non-linear differential equations where coefficients (reaction rates) are considered as constant values and show that these simplest reactions turn out to be complex enough to serve as a basic models for future studies of non-equilibrium processes, similar to the famous Ising model in statistical physics. Different kinds of auto-wave processes in the Lotka and Lotka-Volterra models which serve as the two simplest examples of chemical reactions will be analyzed in detail. We demonstrate the universal character of cooperative phenomena in the bimolecular reactions under study and show that it is reaction itself which produces all these effects. 

The last decade has witnessed an intense interest in the theory of radiative association rate coefficients because of the possible importance of the reactions in the interstellar medium and because of the difficulty of measuring these reactions in the laboratory. Several theories have been proposed these are all directed toward systems of at least three or four atoms and utilize statistical approximations to the exact quantum mechanical treatment. The utility of these treatments can be partially gauged by using them to calculate three body rate coefficients which can be compared with laboratory measurements. In order to explain these theories briefly, it would be helpful to write down equations for the mechanism of association reactions. Consider two species A+ and B that come together with bimolecular rate coefficient kj to form a complex AB+ which can then be stabilized radiatively with rate coefficient kr, be stabilized collisionally with helium with rate coefficient kcoll, or redissociate with rate coefficient k j ... 

In this Section we continue studies of particle dynamical interactions. For this purpose the formalism of many-particle densities is applied to the study of the cooperative effects in the kinetics of bimolecular A -f B —> 0 reaction between oppositely charged particles (reactants) interacting via the Coulomb forces. We show that unlike the Debye-Hiickel theory in statistical physics, here charge screening has essentially a non-equilibrium character. For the asymmetric mobility of reactants (Da = 0, 0) the joint spatial distri-... 

The goal of the present study lies in the establishment of realistic kinetic database from reliable ab initio MO and statistical-theory calculations as alluded to above beginning from the gas phase. The results from a recent series of studies on ClOx radical reactions, including their formation from the decomposition of HClOx (x = 3, 4) and their unimolecular decomposition and bimolecular reactions with OH and HO2, are summarized in this chapter of review. 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

The situation is quite different in bimolecular reactions with an activation energy (E >0). In particular, the "diatomic" model is certainly a bad approximation for radical-radical rebinding along a double bond in which the maximum of the effective potential (35 IV) lies near the saddle-point of the potential energy surface /141/, In this case no central forces govern the nuclear motion hence, the total angular momentum is not a constant, which means that the reaction cannot be rotationally adiabatic. Therefore, in this situation the statistical theory cannot correctly reproduce the results of the simple collision theory. 

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