Rate constant collision theory expression

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. 

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. 

Using the collision theory expression for the rate constant of a bimolecular reaction and the thermodynamic form of the TST result (in molecular units), show that, approximately ... 

In the framework of the assiamptions discussed in the previous section equation (51.Ill) is an accurate collision theory expression for the rate constant which is valid, in principle, for any potential energy surface. It represents a very useful formulation from both conceptual and practical point of view /20/. 

Expression (67.Ill) can be considered as a "statistical formulation of the rate constant in that it represents a formal generalization of activated complex theory which is the usual form of the statistical theory of reaction rates. Actually, this expression is an exact collision theory rate equation, since it was derived from the basic equations (32.Ill) and (41. HI) without any approximations. Indeed, the notion of the activated complex has been introduced here only in a quite formal way, using equations (60.Ill) and (61.Ill) as a definition, which has permitted a change of variables only in order to make a pure mathematical transformation. Therefore, in all cases in which the activated complex could be defined as a virtual transition state in terms of a potential energy surface, the formula (67.HI) may be used as a rate equation equivalent to the collision theory expression (51.III). 

III). In conditions of vibration-rotational adiabaticity, will represent an apparent tunneling correction which is necessary only to obtain the correct values of the rate constant when using the collision theory expression (51.Ill), instead of Eyring s equation (67.Ill) (corrected by the real tunneling factor ). 

The collision theory expression (51 HI) represents the most general formulation of the rate constant based on the unique assumption that the reactants are in thermal equilibrium. It does not involve any hypothesis concerning the intermediate stages of reaction and applies to any reaction regardless of the shape of the potential energy surface. 

In order to derive an exact equation for the rate constant of bimolecular reactions on the basis of the collision theory expression (1.IV), we must calculate the partition function for motion along the reaction coordinate in the reactants region of configuration space. This, however, proves to be not a trivial problem. 

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. 

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... 

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... 

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 ... 

The pre-exponential factor of a bimolecular reaction is related to the reaction cross-section (see Problem 2.3). A relation that is fairly easy to interpret can be obtained within the framework of transition-state theory. Combining Eqs (6.9) and (6.54), we can write the expression for the rate constant in a form that gives the relation to the (hard-sphere) collision frequency ... 

Within transition-state theory, Eq. (8.2) is an exact expression for the rate constant. We observe that the pre-exponential factor deviates from the simple interpretation, as being related to the collision frequency Zab via Z, due to the presence of internal degrees of freedom. Typically, the calculated value of Z is of the order of 1011 dm3 mol-1 s 1 10 16 m3 molecule-1 s 1 (see Example 4.1). The magnitude of the partition functions in Eq. (8.2) is typically small compared to this number. Thus, if we neglect the internal degrees of freedom of the reactants and the activated complex, except for rotational degrees of freedom of the activated complex (AB), and assume that the associated partition function can be approximated by QAB, we will get a pre-exponential factor given by Z. 

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 ... 

We see in Table XII. 1 that we cannot separately identify the terms in the rate-constant expression for the thermodynamics equation or the collision theories without special assumptions. A complete identification of all the terms, frequencies, energies of activation and entropies of activation from experimental data is possible only for the Arrhenius equation and the transition-state theory. 

A general expression taking into account the rotational energy was derived from RRKM theory.29 If the intermediate C is sufficiently short-lived (or the total pressure is sufficiently low) that it is not stabilized by collisions, the rate constant k for the formation of the product(s) can be written as... 

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... 

In Eq. (2-32) it is supposed that the rate is proportional to the concentration of the activated complex. Similarly, in the collision theory, Eq. (2-25), it is tacitly assumed that the concentration determines the collision frequency and the rate. However, if the results of thermodynamics were followed, the rate might be assumed proportional to activity. If the activity replaced concentration in Eq. (2-32), the activity coefficient of the activated complex would not be needed in Eq. (2-34). The final expression for the rate constant would then be... 

Transition state theory has also been applied in quite another way to reactions in solution. Reaction (7.4.2) can be described as a reaction precursor equilibrium which is characterized by the diffusion of A and B to a position close enough so that reaction can take place. This pre-equilibrium has an equilibrium constant which can also be thought of as a collision frequency. The expression for the rate constant is then... 

Collision theory for a bimolecular reaction in the gas phase treats the individual reactant species as hard spheres and introduces a threshold energy for the reaction. The expression derived for the temperature dependence of the bimolecular second-order rate constant is of the same form as that for the Arrhenius equation. The theoretical A-factor is related to the rate at which reactant species collide and is calculated to be of the order of 10 dm mol s , although experimental values can be smaller than this by several orders of magnitude. 

The collision theory predicts the value of the rate constant satisfactorily for reactions that involve relatively simple molecules if the activation energy is known. Difficulties are encountered with reactions between complicated molecules. The rates tend to be smaller than the collision theory predicts, in many cases by a factor of 10 or more. To account for this, an additional factor P, called the probability factor or the steric factor, is inserted in the expression for k ... 

Of Interest Is the absolute-rate-constant study carried out by Zellner and Stelnert (163) for the reaction of OH radicals with CH4 over the temperature range 298 to 892°K, where, as may be expected, a priori, from either collision or transition state theory, the Arrhenius plot shows strong curvature. Such effects must be expected to be a general occurrence, and this study points out (a) the need for accurate data over a wide temperature range, and (b) the need for caution In the extrapolation of Arrhenius expressions (which should be viewed as empirical experimental two-parameter fits) beyond the temperature ranges experimentally covered. 

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