Collision Theory Reaction Rate Expression

Note that limits of the integrals in Eq. 10.61 range from 0 to oo, because any molecules with negative velocities will not pass through the plane from left to right. 

We are free to place our test plane A at any point along the x axis, and the flux through the plane Zyy will be the same. Thus by letting the test plane coincide with one of the walls, we have derived the desired result the gas-wall collision frequency is Zw of Eq. 10.61 or 10.62. 

This section builds upon the previous one to derive a very simple approximation to the reaction rate constant k. Although very crude, this approach produces the familiar Arrhenius expression for the temperature dependence of k. 

In section Section 10.1.2.3 we derived formulas for the collision frequency between two unlike molecules 1 and 2. Each molecule was characterized by a radius r,-, and any time the distance between the centers of the molecules was less than or equal to the sum of the radii, a collision was said to occur. The exact nature of a collision and what the radii (or the collision cross section) depend on were not specified. For example, whether a collision happened to be head-on or just grazing did not matter in deriving Eq. 10.52 or 10.59. All types of collisions counted. 

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. 

---

As in collision theory, the rate of the reaction depends on the rate at which reactants can climb to the top of the barrier and form the activated complex. The resulting expression for the rate constant is very similar to the one given in Eq. 15, and so this more general theory also accounts for the form of the Arrhenius equation and the observed dependence of the reaction rate on temperature. 

As the system pressure is decreased at constant temperature, the time between collisions will increase, thereby providing greater opportunity for unimolecular decomposition to occur. Consequently, one expects the reaction rate expression to shift from first-order to second-order at low pressures. Experimental observations of this transition and other evidence support Linde-mann s theory. It provides a satisfactory qualitative interpretation of unimolecular reactions, but it is not completely satisfactory from a... 

Arrhenius recognized that for molecules to react they must attain a certain critical energy, E. On the basis of collision theory, the rate of reaction is equal to the number of collisions per unit time (the frequency factor) multiplied by the fraction of collisions that results in a reaction. This relationship was first developed from the kinetic theory of gases . For a bimolecular reaction, the bimolecular rate constant, k, can be expressed as... 

The simple collision theory and the activated complex theory have appeared as two alternative treatments of chemical reaction kinetics. It is clear, however, that they represent only two different kinds of approximation to an exact collision theory based either on classical or quantum mechanics. During the past few years considerable progress has been achieved in the colllsional treatment of bimole-cular reactions /7,8/. For more complicated reactions, however, the collision theory yields untractable expressions so that the activated complex theory provides a unique general method for an estimation of the rates of these reactions. Therefore, it is very important to determine well the limits of its validity. 

According to the collision theory, the rate of a reaction can be expressed as the product of three factors ... 

It is not possible to cover all of the history or the theory of the chemical kinetics in the context of this chapter. However, the authors intention is to give the student an essential minimum in the theory of chemical kinetics to be able to follow the literature and to incorporate in the design of the chemical reaction units. This chapter is divided into two sections in the first part, the homogeneous kinetics will be covered in detail, covering the collision theory and the transition state theory for the determination of the rate constants and reaction rate expressions. Old but still valid approximations of pseudo-steady-state and pseudoequilibrium concepts will be given with examples. In the second part, the heterogeneous reaction kinetics will be discussed from a mechauis-tic point of view. 

This section and the following sections will explore the transition-state expressions for the rate constants of surface reactions. We start with the adsorption of atoms. The expression for the rate of adsorption according to hard sphere collision theory was covered. Expression (4.112) will be demonstrated as to how it can be rederived within the transition-state theory. 

The collision theory of reaction rates in its simplest form (the simple collision theory or SCT) is one of two theories discussed in this chapter. Collision theories are based on the notion that only when reactants encounter each other, or collide, do they have the chance to react. The reaction rate is therefore based on the following expressions ... 

Elementary reactions are initiated by molecular collisions in the gas phase. Many aspects of these collisions determine the magnitude of the rate constant, including the energy distributions of the collision partners, bond strengths, and internal barriers to reaction. Section 10.1 discusses the distribution of energies in collisions, and derives the molecular collision frequency. Both factors lead to a simple collision-theory expression for the reaction rate constant k, which is derived in Section 10.2. Transition-state theory is derived in Section 10.3. The Lindemann theory of the pressure-dependence observed in unimolecular reactions was introduced in Chapter 9. Section 10.4 extends the treatment of unimolecular reactions to more modem theories that accurately characterize their pressure and temperature dependencies. Analogous pressure effects are seen in a class of bimolecular reactions called chemical activation reactions, which are discussed in Section 10.5. 

We are now able to obtain the collision theory approximation to the bimolecular rate constant k (T). Recall that the mass-action kinetics expression for the reaction rate q is... 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

Derive an expression for the activation energy for the collision theory rate constant (i.e., Acoil of Eq. 10.76). Derive a similar expression for the activation energy for the unimolec-ular excitation reaction predicted by Hinshelwood theory (i.e., ke(e ) of Eq. 10.132). The activation energy is predicted to be larger for which theory ... 

Application of the collision theory of reaction rates to surface processes is not straightforward. The meaningful definition of a surface collision is difficult and the necessary assumptions, inherent in any quantitative treatment based on this approach, make the results of dubious validity and restricted usefulness. The movement of surface entities within the temperature range of interest could necessitate activation, but (in different systems) may alternatively be a rapid and facile process, and the expression defining the... 

There are a number of other variables in the collision theory expression for the rate constant and rate of reaction. These are considered explicitly in the following worked problems. 

Figure 5.39 shows the schematic diagram of the transition state for an exothermic reaction. The transition-state theory assumes that the rate of formation of a transition-state intermediate is very fast and the decomposition of the unstable intermediate is slow and is the rate-determining step. On the other hand, the collision theory states that the rate of the reaction is controlled by collisions among the reactants. The rate of formation of the intermediate is very slow and is followed by the rapid decomposition of the intermediates into products. Based on these two theories, the following expression can be derived to account for the temperature dependence of the rate constant ... 

While reactor temperature often plays a dominant role on reaction rate, it is not the only contribution to the rate expression. We also have the influence of the concentrations of the reacting species as symbolized by / (.. in Eq. (4.5). This expression can range from simple to very complex. In the simplest form the rate of reaction is proportional to the reactant concentrations raised to their stoichiometric coefficients. This is true for an elementary step where it is assumed that the molecules have to collide to react and the frequency of collisions depends upon the number of molecules in a unit volume. In reality, matters are far more complicated. Several elementary steps with unstable intermediates are usually involved, even for the simplest overall reactions. When the intermediates are free radicals, there can be a hundred or more elementary steps. From an engineering viewpoint it is impractical to deal with scores of elementary steps and intermediates and we usually seek an overall rate expression in terms of the stable, measurable (in principle) components in the reactor. In theory we can derive... 

As for bimolecular reactions, collision theory can also be used to describe the kinetics of interfacial reactions between a solid surface and solutes in the liquid phase. Astumian and Schelly have described the theory for the kinetics of interfacial reactions in detaiL The complete rate expression, derived by Astumian and Schelly, for solutes reacting with suspended solid spherical particles is given by Eq. (1)... 

The profile of the potential energy surface obtained by Brudnik et al. 25 at the G2 level is shown in Fig. 17. When the loosely bound intermediates are not stabilized by collisions, they can be omitted in the reaction mechanism. The kinetics of the reaction can, in a first approximation, be described by the rate constant obtained from classical transition state theory. The rate constant calculations of Brudnik et al 25 show that this approach is realistic at temperatures below 1000 K. The temperature dependence of the rate constants calculated for CF3O + H20 can be expressed as... 

Finally, substitution of this value of frequency factor A in Eq. (2-8) gives the collision-theory expression for the specific reaction rate,... 

---