Collision theory rate expression

If the potential energy surface has a col, then a separation of the reaction coordinate is possible also in the vicinity of the saddle-point. Supposing that the system remains there sufficiently long time for a stationary state to be established, we may assume the existence of a quasi-equilibrium energy distribution in that transition state, too. Then, we can apply the above statistical treatment of reactants also to the transition state of the reacting system. However, we will not introduce at this stage the above restrictive assumptions, since our aim is to derive a collision theory rate expression of a possibly general validity. 

The derivation of the exact collision theory rate expression (51 111) presumes that the transition probabilities 1[Pg.145]

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 ... 

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). 

The equation (79.Ill)is the complete classical (semiclassical) analogue to the quantum-mechanical "statistical" formulation (67.111) of the rate constant. It represents actually an exact classical (semiclassical) expression, which is equivalent to the corresponding collision theory rate equation (70,111). 

State-specific and thermal rate coefficients The general collision theory rate-coefficient expression [Eq. (3.7)] may be extended to a reaction between... 

This is the general TST expression for the thermal rate coefficient. It contains a universal frequency factor ksT/h = 6.25 x 10 s at 300 K) which is independent of the nature of the reactants. Specific molecular properties appear in the ratio of the partition functions and in Ael. Equation (3.54) differs therefore from the collision theory rate-coefficient expression [Eq. (3.7)] in that all quantities contained in this equation are at least in principle derivable from molecular properties. 

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. 

How useful is the rate expression derived from collision theory for describing adsorption For cases in which adsorption is not activated, i.e. E = 0, the collision frequency describes, in essence, the rate of impingement of a gas on a surface. This is an upper limit for the rate of adsorption. In general, the rate of adsorption is lower, because the molecules must, for example, interact inelastically with the sur-... 

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... 

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 ... 

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... 

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. 

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. 

The expression for the rate of adsorption obtained from collision theory with o°(T) = 1 is the same as that obtained from transition state theory for a mobile activated complex with AE° (T = 0) = 0. 

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... 

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. 

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 simple collision theory expression for the activation rate coefficient is in error, often underestimating by many orders of magnitude. 

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... 

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

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