A simple collision model

The essential theoretical picture in chemical kinetics is that for a step in a reaction mechanism to occur, two things must happen  

The essential properties of solutions or gases tell us that the constituent particles are always in constant, random motion. We can envisage, therefore, that collisions occur continuously and this suggests that the more frequently they do so between reactant species, then the faster the consequent reaction. It is useful to look at this idea in a little more detail. 

As an example we can consider an elementary reaction between two different species A and B (which could be molecules, fragments of molecules , atoms or ions) in the gas phase 

The number of collisions between species A and species B that occur in a fixed volume in unit time (say, 1 s) is a measure of the collision rate between A and B. 

This rate will depend upon the concentration of both species. For example, doubling the concentration of B means that the number of targets for individual A species in a given volume is increased by a factor of two hence, the rate at which A species collide with B species is doubled. A similar argument holds for increasing the concentration of A. Thus, overall, the collision rate between the A and B species is directly proportional to their concentrations multiplied together, so that 

---

Such a reaction may be considered to approximate a simple collision model. The rate of the reaction is faster than cyanide exchange for either reactant so we consider the process to consist of electron transfer from one stable complex to another with no breaking of Fe—CN or Mo—CN bonds. 

Such a reaction may be considered to approximate a simple collision model. The rate of reaction is faster than cyanide... 

Before closing this section, we should remark that although this analysis of velocity relaxation effects has focused on a simple collision model, we expect that the detailed structure of the rate kernel for short times will depend on the precise form of the chemical interactions in the system under consideration. It is clear, however, that a number of fundamental questions need to be answered before more specific calculations can be undertaken form the kinetic theory point of view. 

The theoretical rate constant for Reaction 4.1, although called a constant , does depend on temperature. Increasing the temperature increases, in most circumstances, the magnitude of A theory So carrying out a reaction at a higher temperature, but with the same initial concentrations of A and B, will be expected to result in an increase in the rate of reaction. This behaviour can be understood in a qualitative way in terms of a simple collision model. 

The use of a simple collision model to predict the behaviour of elementary reactions involving two reactant species is instructive but nonetheless limited in scope. To extend such a model to chemical reactions in general would be difficult because the vast majority of these are composite. To make progress in understanding the rates of chemical reactions it is necessary to adopt an experimental approach. 

The key idea underlying a simple collision model for an elementary reaction is that the reactant species must collide before any chemical transformation can take place. 

For an elementary reaction involving two reactant species A and B, a simple collision model predicts a theoretical rate equation of the form... 

The derivation of the form of the theoretical rate equation using a simple collision model is more complex in this type of case but it turns out that the equation is of the expected form... 

A process is said to be spontaneous if it occurs without outside intervention. Spontaneous processes may be fast or slow. As we will see in this chapter, thermodynamics can tell us the direction in which a process will occur but can say nothing about the speed of the process. As we saw in Chapter 12, the rate of a reaction depends on many factors, such as activation energy, temperature, concentration, and catalysts, and we were able to explain these effects using a simple collision model. In describing a chemical reaction, the discipline of chemical kinetics focuses on the pathway between reactants and products thermodynamics considers only the initial and final states... 

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 absolute values of the sticking coefficients cannot, however, be reconciled with the measured small activation barriers in the framework of a simple collision model. They rather demonstrate the formal existence of a very small preexponential factor p whose origin must be sought in the dynamics of the process, a subject for continuing investigations. 

Incorporation of this attractive potential into a simple collision model, which we shall not present here, allows the prediction of the reaction cross section and, ultimately, the reaction rate coefficient. The Langevin rate coefficient, as it is often known [21], is given by... 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

It is useful to examine the consequences of a closed ion source on kinetics measurements. We approach this with a simple mathematical model from which it is possible to make quantitative estimates of the distortion of concentration-time curves due to the ion source residence time. The ion source pressure is normally low enough that flow through it is in the Knudsen regime where all collisions are with the walls, backmixing is complete, and the source can be treated as a continuous stirred tank reactor (CSTR). The isothermal mole balance with a first-order reaction occurring in the source can be written as... 

Now we turn to calculation of the susceptibility component Xst( ) in Eq. (17). To extract it from Eq. (14c), one should replace there p for and account for an inhomogeneity of the induced distribution F(y). The latter is determined by a chosen collision model. Such models are described in detail in GT, Section IV.B, and in VIG, Section VI, where they are separated into the self-consistent and non-self-consistent models. For one simple example they are considered also in Section VII.C... 

Even in the domain of inorganic redox chemistry relatively little use has been made of the full potential of the Marcus theory, i.e. calculation of A, and A0 according to (48) and (52) and subsequent use of (54) and (13) to obtain the rate constant (for examples, see Table 5). Instead the majority of published studies are confined to tests of the Marcus cross-relations, as given in (62)-(65) (see e.g. Pennington, 1978), or what amounts to the same type of test, analysis of log k vs. AG° relationships. The hesitation to try calculations of A is no doubt due to the inadequacy of the simple collision model of Fig. 4, which is difficult to apply even to species of approximately spherical shape. 

Isolated atoms show spherical symmetry, and it is natural to model atoms by spheres of some suitably defined radii. The potential energy of interaction between two atoms rises very sharply at short internuclear distances during atomic collisions, not unlike the potential energy increase in the collisions of hard, macroscopic bodies. In a somewhat crude, approximate sense, atoms behave as hard balls. This analogy can be used for a simple molecular model where atoms are represented by hard spheres. Once a choice of atomic radii is made, the approximate atomic surfaces can be defined as the surfaces of these spheres. 

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. 

Both the Arrhenius and the Eyring equation describe the temperature dependence of reaction rate. Strictly speaking, the Arrhenius equation can be applied only to gas reactions. The Eyring equation is used in the smdy of gas, condensed and mixed phase reactions - aU places where the simple collision model is not very helpful. The Arrhenius equation is founded on the empirical observation that conducting a reaction at a higher temperature increases the reaction rate. The Eyring equation is a theoretical construct, based on transition state model. 

Much remains to be done in this area. The role of the many-body polarizability has yet to be explored, a simple dynamical model has yet to be presented, and a microscopic justification of Eq. (14.2.5) has yet to be developed. Moreover the important practical question concerning the collision-induced scattering from molecular liquids has yet to be answered. Only after this effect is assessed can it be subtracted from the depolarized scattering in such a manner that the remaining spectrum gives information about molecular tumbling. 

According to the simple collision model, the fraction of collisions with a kinetic energy sufficient to overcome the energy barrier to reaction increases with increasing temperature. This behaviour largely accounts for the temperature dependence of the theoretical rate constant. 

A most interesting recent development is the work of Augustin and Rabitz, who obtained a transition between statistical and perturbation theories for any type of collision, not only complex-forming ones. More general stochastic aspects of unimolecular reactions have been discussed by Sole and Widom. An application of a phase-space model to electronic transitions in atomic collisions has been reported, as well as a simple RRKM model for electronic to vibrational energy transfer in 0( Z)) -I- Nj collisions. ... 

A simple PR model adopted in [32] simulates the repulsive interaction of the H atom on its "rotational" approach to the A atom by an angle-dependent potential oc l/cosh (t /2y) where d is the angle between the molecular axis and the collision axis. For this PR model, the following formula for the vibrational deactivation rate coefficient is valid ... 

With increasing interparticle collisions the probability of formation of floes from dispersed (nonflocculated) particles increases. Thus the horizontal axis can also be interpreted to mean a change from weakly flocculated particles on the left to increasingly flocculated particles toward the right. An outcome is that, irrespective of the degree of interparticle interaction, at low values of cp the viscosity rises slowly, but tends to increase rapidly when particle packing becomes dense.For randomly packed spheres this change occurs at about 95 = 0.60. A simple viscoplastic model is the Herschel-Bulkley equation... 

Dahneke B. Particle bounce or capture — search for an adequate theory I conservation-of-energy model for a simple collision process. Aerosol Sci Technol 1995 23 25-39. 

---