Chemical Kinetics Collision Theory

TRANSITION-STATE THEORY CHEMICAL KINETICS COLLISION THEORY TEMPERATURE DEPENDENCE... 

There are two major theories of chemical kinetics, collision theory (CT) and transition-state theory (TST). Both theories lead to rate equations that obey Generalization I, i.e., the effects of temperature and concentration are separable. Unfortunately, both CT and TST apply to a very limited category of reactions known as elementary reactions. An elementary reaction is one that occurs in a single step on the molecular level exactly as written in the balanced stoichiometric equation. The reactions that chemists and chemical engineers deal with on a practical level almost never are elementary. However, elementary reactions provide the link between molecular-level chemistry and reaction kinetics on a macroscopic level. Elementary reactions will be discussed in some depth in Chapter 5. For now, we must look at Eqn. (2-1) as an empirical attempt to extrapolate a key result of CT and TST to complex reactions that are outside the scope of the two theories. 

The simplest version of the theory of chemical reactions rates is the kinetic collision theory of gas reactions /1/ which has been developed several decades ago by LEWIS (1918), HERZPBLD (1919), POLA-NYI (1920), HINSHELWOOD (1937) a.o./2/. For a simple bimolecular reaction of the type... 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

In chemical kinetics, it is often important to know the proportion of particles with a velocity that exceeds a selected velocity v. According to collision theories of chemical kinetics, particles with a speed in excess of v are energetic enough to react and those with a speed less than v are not. The probability of finding a particle with a speed from 0 to v is the integral of the distribution function over that interval... 

FIGURE 13.25 (a) In the collision theory of chemical reactions, reaction may take place only when two molecules collide with a kinetic energy at least equal to a minimum value, /rmn (which later we identify with the activation energy), (b) Otherwise, they simply bounce apart. 

The rate is thus the number of collisions between A and B - a very large number - multiplied by the reaction probability, which may be a very small number. For example, if the energy barrier corresponds to 100 kj mol , the reaction probability is only 3.5 x lO l at 500 K. Hence, only a very small fraction of all collisions leads to product formation. In a way, a reaction is a rare event For examples of the application of collision theory see K.J. Laidler, Chemical Kinetics 3 Ed. (1987), Harper Row, New York. 

The science of reaction kinetics between molecular species in a homogeneous gas phase was one of the earliest fields to be developed, and a quantitative calculation of the rates of chemical reactions was considerably advanced by the development of the collision theory of gases. According to this approach the rate at which the classic reaction... 

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. ... 

An interesting, but probably incorrect, application of the probabilistic master equation is the description of chemical kinetics in a dilute gas.5 Instead of using the classical deterministic theory, several investigators have introduced single time functions of the form P(n1,n2,t) where P(nu n2, t) is the probability that there are nl particles of type 1 and n2 particles of type 2 in the system at time t. They use the transition rate A(nt, n2 n2, n2, t) from the state with particles of type 1 and n2 particles of type 2 to the state with nt and n2 particles of types 1 and 2, respectively, at time t. The rates that are used are obtained by assuming that only uncorrelated binary collisions occur in the system. These rates, however, are only correct in the thermodynamic limit for a low density system. In this limit, the Boltzmann equation is valid from which the deterministic theory follows. Thus, there is no reason to attach any physical significance to the differences between the results of the stochastic theory and the deterministic theory.6... 

There are several theories to account for chemical kinetics. The simplest is the collision theory, which will be used in Chapter 4 to calculate the rate constants for the collision of molecules in solution. A more sophisticated theory, one that... 

A simple way of analyzing the rate constants of chemical reactions is the collision theory of reaction kinetics. The rate constant for a bimolecular reaction is considered to be composed of the product of three terms the frequency of collisions, Z a steric factor, p, to allow for the fraction of the molecules that are in the correct orientation and an activation energy term to allow for the fraction of the molecules that are sufficiently thermally activated to react. That is,... 

According to the molecular collision theory of kinetics, only collisions between molecules with high enough energy can result in reaction(s). In other words, a chemical reaction occurs only when the two conditions are met (1) the molecules to take part in the reaction collide with each other and (2) the colliding molecules have a sufficiently high energy level. 

The question immediately arises as to how far this independence of collision will continue as the pressure of nitrogen pentoxide is indefinitely decreased. The matter is of great importance for checking theories of chemical kinetics and accordingly many other investigators undertook the study of the decomposition of nitrogen pentoxide at very low pressures. 

Rate constants for quenching can be compared with those predicted by the collision theory of chemical kinetics. According to this theory, a rate constant, k, is given by... 

The chemical reaction is characterized on the one hand by the kinetic mechanism, that is to say the dependence on the concentrations of the participants in the reaction, on the other hand by the reaction (velocity) constant. This latter in the simplest form is k — Ae EIRT in which E is the energy of activation and A the frequency factor. The latter is in the classical collision theory equal to where Z the collision number ( io11) and P the probability factor or steric factor. The latter can be much larger than unity if the activation energy is divided over several internal degrees of freedom (mono-molecular reactions) but it can also be as low as io 8, e.g., in cases where steric hindrance plays a role. 

If the collisions of molecules produce a chemical reaction, the Boltzmann equation is modified in obtaining the equations of change these problems are addressed and analyzed in the context of quantum theory, reaction paths, saddle points, and chemical kinetics. Mass, momentum, and energy are conserved even in collisions, which produce a chemical reaction. 

Besides aetive research he very much enjoys teaching. In the Physical-Teehnieal Institute of Moseow he taught (1966-1992) general courses on Molecular Dynamics and Chemical Kinetics. In the Teehnion (since 1992) he has taught and still teaehes graduate eourses on different subjeets Advanced Quantum Chemistry, Theory of Moleeular Collisions, Kinetie Proeesses in Gases and Plasma, Theory of Fluetuations, Density Matrix Formalisms in Chemical Physics etc. 

Four factors have marked effects on the rates of chemical reactions. They are (1) nature of the reactants, (2) concentrations of the reactants, (3) temperature, and (4) the presence of a catalyst. Understanding their effects can help us control the rates of reactions in desirable ways. The study of these factors gives important insight into the details of the processes by which a reaction occurs. This kind of study is the basis for developing theories of chemical kinetics. Now we study these factors and the related theories—collision theory and transition state theory. 

S. W. Benson, The Foundations of Chemical Kinetics, McGraw-Hill Book Company, New York, 1960. The stationary-state hypothesis mentioned in Secs. 2-6 and 2-11 is defined and illustrated on pp. 50-53. In chap. XII the collision theory is considered in detail, complications related to the energy distribution of molecules and the steric factor are discussed, and the results are compared in depth with those from the transition-state theory. 

As the fundamental concepts of chemical kinetics developed, there was a strong interest in studying chemical reactions in the gas phase. At low pressures the reacting molecules in a gaseous solution are far from one another, and the theoretical description of equilibrium thermodynamic properties was well developed. Thus, the kinetic theory of gases and collision processes was applied first to construct a model for chemical reaction kinetics. This was followed by transition state theory and a more detailed understanding of elementary reactions on the basis of quantum mechanics. Eventually, these concepts were applied to reactions in liquid solutions with consideration of the role of the non-reacting medium, that is, the solvent. 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

As Fowler and Guggenheim8 realized, the main difficulty with the theories of chemical kinetics lay in the so-called equilibrium Hypothesis, on which the treatments are grounded. As a matter of act, In both the collision and activated state approaches, one uses... 

We may say that the work which has been done after the war has somewhat changed our views about the physical grounds of chemical kinetics. We have learned that the old a posteriori experimental evidence of the collision theory has proved to be unsafe, being grounded upon ill-defined quantities, such as the diameter to be used in the rate expressions. This has been particu-lary evident for thermal dissociation reactions, where the knowledge of the activation energy makes possible a real test. 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann energy (or momentum or internal coordinate) distribution of reactants is postulated to persist during a reaction. In the collision theory, mainly due to Hinshelwood,7 the number of energetic, reaction producing collisions is calculated under the assumption that the molecular velocity distribution always remains Maxwellian. In the absolute... 

According to the collision theory" of chemical kinetics, the rate... 

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