Bimolecular reactions, collision model

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]. 

This section considers the cross section for reactive collisions ar. Bimolecular reactions will be treated explicitly. The rate (frequency) of collisions depends on the collision cross section. The larger the cross section, the more often molecules run into one another. In a similar way the reactive cross section determines how often molecules run into one another and react. This section introduces the simple line-of-centers model for scaling of the reactive cross section with energy. 

Table 4.2 Comparison of collision model and experimental data. Pre-exponential factors are given in units of dm3 mol-1 s 1. The experimental data are from J. Chem. Phys. 92, 4811 (1980) J. Phys. Chem. Ref. Data 15, 1087 (1986) and J. Phys. Chem. A 106, 6060 (2002), respectively. Note that the third reaction is a bimolecular association reaction. For this reaction, the experimental data are derived in the high-pressure limit.

For the first time the reaction of CO oxidation was considered in Refs. [137,138], However, the equations for two-site processes differ from corrected Eq. (62b) due to using in these works a different definition for the bimolecular reaction activation energy the term with Ey was absent in the reaction activation energy. Actually calculations were performed with simplified assumption about s = 0 (the collision model). The theoretical curves have given a qualitative agreement with experiment data. 

Based on the molecular collision model, that describes successfully experimental data for a large number of bimolecular reactions, the rate of the reaction can be calculated as the number of collisions of molecules having energy higher than the required value E [7] ... 

Figure 11 shows a typical example of the temperature-dependent behavior for the reactions of OH radical with aromatic compounds. The measured bimolecular rate constants of OH radical with nitrobenzene showed distinctly non-Arrhenius behavior below 350°C, but increased in the slightly subcritical and supercritical region. Feng a succeeded in modeling these data with a three-step reaction mechanism originally proposed by Ashton et while Ghandi etal. claimed to have developed a so-called multiple collisions model to predict the rates for the reactions of OH radical in sub- and super-critical water. 

In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, u, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j. Once the reactive cross-sections are known the state specific rate coefficient, can be determined from. 

Miklavc, A., Perdih, M. and Smith, I.W.M. (1995) The role of kinematic mass in simple collision models of activated bimolecular reactions. Chem. Phys. Lett. 241, 415-422. 

A transition-state theory (Safron et al, 1972) has been developed within the context of scattering theory, to provide suitable models for crossed molecular beam processes. As in the case of RRKM theory, it is based on the premise that the probability of complex decomposition is a product of a probability of break-up and a probability Of departure from the collision region. But it adds restrictions peculiar to bimolecular reactions, such as a limit on the maximum angular momentum that allows formation of the complex from reactants. Let p(E t) indicate the probability density for finding a product pair with kinetic energy E. This may be written as... 

Solvent effects enter through the potential of mean force and the activation energy they may cancel or nearly cancel in the expression for kj (cf. Northrup and Hynes ). The collision frequency per unit density of B, ab(8 b /Mab) 1 expression for k° for a bimolecular reaction, takes the place of the frequency Wq in (3.23) for an isomerization reaction. This analysis shows that the transition state expression for the rate coefficient appears in this theory as a singular contribution to the rate kernel for the hard-sphere model of the reaction. 

In Section 4.1 we discussed a simple collision model to describe an elementary reaction in the gas phase between two reactant species A and B, in other words a bimolecular reaction. 

According to Equation 6.3, this factor is equivalent to the Arrhenius A-factor. In the collision model it is a measure of the standard rate at which reactant species collide that is it is a measure of the number of collisions per second when the concentrations of the reactant species are both 1 mol dm"-. It is necessary to specify standard conditions since, in general, the collision rate depends on the concentrations of the species present (cf. Section 4.1). The value of Atheory a given bimolecular reaction depends on the hard-sphere radii and masses of the reactant species. Calculations show that it does not vary significantly from reaction to reaction with values usually of the order of 10 dm mol s . Table 7.1 compares the calculated values of Atheory for gas-phase bimolecular reactions with those derived from experiment. 

It is possible to combine these results on the frequency of binary collisions with a simple model of the dynamic mechanism of bimolecular reactions to obtain a formal result which agrees well with experiment in a number of cases. The model has been developed by Present [8], whose presentation we follow here. 

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. 

ET reactions are typically bimolecular in the RDS and therefore display second-order kinetics. The reaction approximates a simple collision model, where the free energy of activation (AG ) involves three terms, as shown in Equation (17.43). 

The development of the collision dynamics approach to bimolecular reactions has for the most part departed from models that seek analytical expressions for rate coefficients, and has centered on trajectory calculations, a method made possible by the development of high speed computers. 

Parameter Calculation and Establishment of Relationships. The use of molecular modeling tools not being evident for nonexperts in the field, alternative tools can be applied for the assessment of values for rate coefficients, preexponential factors, and/or activation energies (22). Collision rate theory provides a simple description of a kinetic process. It counts the number of collisions, Zab, between the reacting species A and B in a bimolecular reaction or between the reacting species and the surface in the case of an adsorption step and applies a reaction probability factor, Prxn, to account for the fact that not every collision leads to a chemical reaction. 

Another direction should be noted, namely the development of simple reaction models [185, 269, 372]. This approach provides qualitative and often even semi-quantitative results requiring no tedious calculation and thus permits the interpretation of some bimolecular reactions. The simple models are classified according to angular distribution of products and also by the extent to which the energy of the collision complex AB formed by collisions of reagents A and B is redistributed among various degrees of freedom before the reaction is completed. 

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