Theory of Bimolecular Reactions

The rate of collision of gas molecules is given by gas kinetic theory. Molecules have an average kinetic energy given by the expression 

From the kinetic theory of gases we find that the frequency Vab of an A molecule colliding with a B molecule is 

Now putting all these together, we obtain a rate of collision of A with B 

This gives a collision rate constant for a second-order reaction as 

This is the mcvcimumpossible rate ojbimolecular reaction, the collision rate of the molecules that can react. We must multiply this by a probability of reaction in the collision so actual rates must be less than this. We know that the units of k should be Uters/mole time, and, since velocity is in lengthAime and cross section is in area/time, the units are correct if we make sure that we use volume in liters, and compute the area of a mole of molecules. If the molecular weight is 28 (air) and the temperature is 300 K, then we have 

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There is an inunediate coimection to the collision theory of bimolecular reactions. Introducing internal partition functions excluding the (separable) degrees of freedom for overall translation. 

The next more sophisticated theory of bimolecular reactions is called activated complex theory, which assumes that the collision of A and B forms a complex (AB) and that the rate of the reaction depends on the rate of decomposition of this complex. We write this as... 

Figure 4-14 Energy diagram illustrating the reactants A and B and the activated complex (ABT in the activated complex theory of bimolecular reactions.

Before discussing these points in detail, it is worthwhile to consider how the diffusion equation for relative motion of two species is developed from a reduction of the diffusion equation describing the motion of both species separately. It introduces some of the complexities to the many-body problem and, at the same time, shows an interesting parallel to the theory of bimolecular reaction rates in the gas phase [475]. 

The application of the previously developed theory of bimolecular reactions gives satisfactory results. 

The reaction rate in differential theories of bimolecular reactions is always the product of the reactant concentrations and the rate constant, does not matter whether the latter is truly the constant or the time-dependent quantity. In integral theories there are no such constants at all they give way to kernels (memory functions) of integral equations. However, there is a regular procedure that allows reduction the integral equations to differential equations under specificconditions [34,127]. This reduction can be carried out in full measure or partially, but the price for it should be well recognized. 

Mahan, B. H.. Activated complex theory of bimolecular reactions, J. Chem. Educ., 51, 709-711 (1974). 

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

Present, R.D. (1955) Note on the simple collision theory of bimolecular reactions Proc. Natl. Acad. Sci. USA 41, 415-417. 

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... 

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). 

E.E.Nikitin, The current status of the theory of bimolecular reactions, Uspekhi Khimii, 38, 1153 (1968)... 

In order to complete the set of equations describing polymerization reactions in dense and concentrated regimes, the rates kr and kd must be specified. In the stationary regime, in which the environmental responses to the microscopic motion of the polymer reactants can be assumed to follow the same regression throughout the reaction, these rates are time-independent. The theory of bimolecular reactions in liquids can then be applied at every reaction step. 

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... 

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... 

E. E. Nikitin, Present-day state of the theory of bimolecular reactions, Russ. Chem. Rev. 38, 505-512 (1969). 

Simple collision theory (SCT) is an early theory of bimolecular reactions that was developed in the hrst decades of the twentieth century. Although SCT oversimplihes collision dynamics, and is of limited predictive power, it provides a beginning point for the collision dynamics approach to bimolecular reactions, and the beginnings of insight into factors that affect chemical reactivity. SCT also permits estimates to be made of the upper limit expected for the value of the bimolecular gas phase reaction rate coefficients from the rate of gas phase collisions. For these reasons SCT is worthy of examination. 

Most theories of bimolecular reactions in solution have been based on Smoluchowski s treatment of the coagulation of colloid paurbicles (Smolu-chowski, 1917). We will briefly recall the development of this model, with its recent in rovements, and then examine how recent eiqperiments with a fast time resolution have permitted to test these theories. 

The empirical Arrhenius formula for the temperature dependence of elementary rate constants was presented. This empirical formula was based on an idea that activated molecules with high energy are necessary for the reaction to occur and that the population of molecules with a characteristic activation energy is given by the Boltzmann probability distribution. We presented the collision theory of bimolecular reaction rates, using first the assumption that all collisions with a relative kinetic energy greater than a critical value would lead to reaction. 

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. 

The statistical theory of bimolecular reactions is based on two assumptions ... 

Density in the Transition State Theory of Bimolecular Reactions. 

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

The basis of the collision theory of bimolecular reactions is the idea that molecular encounters that lead to reaction are associated with reactive cross sections. Consider a general reaction between species A and B moving with velocities and... 

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