Simple Collision Theory of Reaction Rates

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  

The notion of a collision implies at least two collision partners, but collision-based theories are applicable for theories of unimolecular reactions as well. 

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Similar problems arise when considering the simple collision theory of reaction rates. Surprisingly, no single attempt seems to have been made untill recently to define and compute a relevant classical or quantum correction corresponding to the "probability" factor in equation (3A), on the basis of an exact collision theory. 

Elementary reactions on solid surfaces are central to heterogeneous catalysis (Chapter 8) and gas-solid reactions (Chapter 9). This class of elementary reactions is the most complex and least understood of all those considered here. The simple quantitative theories of reaction rates on surfaces, which begin with the work of Langmuir in the 1920s, use the concept of sites, which are atomic groupings on the surface involved in bonding to other atoms or molecules. These theories treat the sites as if they are stationary gas-phase species which participate in reactive collisions in a similar manner to gas-phase reactants. 

The collision theory of reaction rates dealt with earlier gives a useful, even if a crude, picture of reaction rates and permits us to calculate the rates of reactions between simple molecules when the activation energies are known. However, this theory leaves much to be desired. It does not furnish a method of calculating activation energies theoretically. It provides no information on the details of reactive collisions. It also does not account for the role that the internal energy might play in the reaction. 

This probably represents the first stop on the way to a collision theory of reaction rates. If, indeed, we have the passage of a single molecule through a dilute, fixed matrix of other molecules (sometimes called a dusty gas ) with a known speed ca, and with every collision resulting in reaction, then the reaction rate would in fact be given by equation (2-9). Yet, we know that we do not have a single molecule, we do not have a fixed speed, and we do not have a dusty gas, so this simple theory needs some cosmetics at this point. 

According to the simple views of the collision theory of reaction rates, molecules are considered as hard spheres. 

A simple way of analyzing the rate constants of chemical reactions is the collision theory of reaction kinetics. The rate constant for a bimolecular reaction is considered to be composed of the product of three terms the frequency of collisions, Z a steric factor, p, to allow for the fraction of the molecules that are in the correct orientation and an activation energy term to allow for the fraction of the molecules that are sufficiently thermally activated to react. That is,... 

Simple collision theory assumes reaction occurs when molecules, with energy greater than a critical minimum, collide. Calculation of two quantities, the total rate of collision of reactant molecules and the fraction of molecules which have at least the critical energy, gives an equation to compare with the experimental Arrhenius... 

In addition to illustrating many features of monomolecular reaction networks, Wei and Prater illustrated how these results, especially the straight line reaction paths, could be helpful in planning experiments for and the determination of rate constants, and this will be discussed later. Also, these same methods have been used in the stochastic theory of reaction rates, which consider the question of how simple macroscopic kinetic relations (e.g., the mass action law) can result from the millions of underlying molecular collisions—see Widom for comprehensive reviews [14]. 

In the case of bunolecular gas-phase reactions, encounters are simply collisions between two molecules in the framework of the general collision theory of gas-phase reactions (section A3,4,5,2 ). For a random thennal distribution of positions and momenta in an ideal gas reaction, the probabilistic reasoning has an exact foundation. Flowever, as noted in the case of unimolecular reactions, in principle one must allow for deviations from this ideal behaviour and, thus, from the simple rate law, although in practice such deviations are rarely taken into account theoretically or established empirically. 

In some reactions involving gases, the rate of reaction estimated by the simple collision theory in terms of the usually infened species is much lower than observed. Examples of these reactions are the oxidation of H2 and of hydrocarbons, and the formation of HC1 and of HBr. These are examples of chain reactions in which very reactive species (chain carriers) are initially produced, either thermally (i.e., by collision) or photochemically (by absorption of incident radiation), and regenerated by subsequent steps, so that reaction can occur in chain-fashion relatively rapidly. In extreme cases these become explosions, but not all chain reactions are so rapid as to be termed explosions. The chain... 

You can use simple collision theory to begin to understand why factors such as concentration affect reaction rate. If a collision is necessary for a reaction to occur, then it makes sense that the rate of the reaction will increase if there are more collisions per unit time. More reactant particles in a given volume (that is, greater concentration) will increase the number of collisions between the particles per second. Figure 6.7 illustrates this idea. 

You can also use simple collision theory to explain why increasing the surface area of a solid-phase reactant speeds up a reaction. With greater surface area, more collisions can occur. This explains why campfires are started with paper and small twigs, rather than logs. Figure 6.8 shows an example of the effect of surface area on collision rate. 

Simple collision theory recognizes that a collision between reactants is necessary for a reaction to proceed. Does every collision result in a reaction Consider a 1 mL sample of gas at room temperature and atmospheric pressure. In the sample, about 10 collisions per second take place between gas molecules. If each collision resulted in a reaction, all gas phase reactions would be complete in about a nanosecond (10 s)—a truly explosive rate As you know from section 6.2, however, gas phase reactions can occur quite slowly. This suggests that not every collision between reactants results in a reaction. 

The simple collision theory for bimolecular gas phase reactions is usually introduced to students in the early stages of their courses in chemical kinetics. They learn that the discrepancy between the rate constants calculated by use of this model and the experimentally determined values may be interpreted in terms of a steric factor, which is defined to be the ratio of the experimental to the calculated rate constants Despite its inherent limitations, the collision theory introduces the idea that molecular orientation (molecular shape) may play a role in chemical reactivity. We now have experimental evidence that molecular orientation plays a crucial role in many collision processes ranging from photoionization to thermal energy chemical reactions. Usually, processes involve a statistical distribution of orientations, and information about orientation requirements must be inferred from indirect experiments. Over the last 25 years, two methods have been developed for orienting molecules prior to collision (1) orientation by state selection in inhomogeneous electric fields, which will be discussed in this chapter, and (2) bmte force orientation of polar molecules in extremely strong electric fields. Several chemical reactions have been studied with one of the reagents oriented prior to collision. ... 

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. 

The boundary condition that Cp is zero cannot be strictly valid, as shown by Noyes [18] using as an example the recombination of iodine atoms in carbon tetrachloride solution, a quintessential diffusion-controlled reaction. The rate coefficient for collisions calculated from the simple kinetic theory of gases, 9.15 x 10-11 cm3 molecule-1 sec-1, is equated to k, which gives a value of 1.35 for the term (1 + 4irpDk l). More elaborate treatments of the rate of a diffusion-controlled reaction, as reviewed by Noyes [18], use more realistic boundary conditions, but this simple treatment gives answers certainly correct to within an order of magnitude. 

The only exception to this general conclusion is in the case of reactions which, according to all the available evidence, are elementary. This is discussed in more detail in Section 7. However, for now it can be noted, as demonstrated by the results in Table 4.1, that a simple collision theory can predict the form of the experimental rate equation for an elementary reaction involving two reactant species. For reactions which are not elementary, such as those in Table 4.2, no such theoretical approach is available. Indeed, if it were, then a large area of experimental chemical kinetics would never have come into existence. 

The ratio of collision numbers, (H2/O2) = 2.3 at 373 K. Note that the ratio obtained from Table 2.2b at 298°K is just about the same—even a little smaller. The weak dependence of collision number per unit volume on temperature is due to a compensation between collision frequency (increasing) and number density (decreasing). This should tell us that dramatic increases in reaction rate with temperature as observed in experiment surely cannot be explained solely on the basis of simple collision theory. 

The interpretation of the shock tube results is that during the induction period the branching chain reaction predominates until significant amounts of H2 and O2 have reacted and back reactions have become important. After the induction period, the concentration of free radical propagators go through a maximum and then slowly approach equilibrium values until the higher-order termination steps limit chain propagation, and the back reactions become important. All of the individual rate constants in the mechanism could be evaluated and are consistent with simple collision theory. 

The simplest version of the theory of chemical reactions rates is the kinetic collision theory of gas reactions /1/ which has been developed several decades ago by LEWIS (1918), HERZPBLD (1919), POLA-NYI (1920), HINSHELWOOD (1937) a.o./2/. For a simple bimolecular reaction of the type... 

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. 

We will consider first the accurate collisional equations (51. HI) and (67 HI) for the rate constant. The factorization of these formally similar expressions allows us a separate evaluation of the three essential factors, including the activation energy, the partition functionf , and the corrections to the simple collision theory and the activated complex theory, respectively. A complete evaluation of the rate constant in this manner is, in principle, possible, if the potential energy surface is known from accurate or approximate calculations. If there is a col, both formulations are equivalent.Both involve the reaction dynamics through the transition probabilities in the corresponding correction factors and, which can be com-... 

Our treatment, based on both the collision and the statistical formulations of reaction rate theory, shows that there exist two possibilities for an interpretation of the experimental facts concerning the Arrhenius parameter K for unimolecular reactions. These possibilities correspond to either an adiabatic or a non-adiabatic separation of the overall rotation from the internal molecular motions. The adiabatic separability is accepted in the usual treatment of unimolecular reactions /136/ which rests on transition state theory. To all appearances this assumption is, however, not adequate to the real situation in most unimolecular reactions.The nonadiabatic separation of the reaction coordinate from the overall rotation presents a new, perhaps more reasonable approach to this problem which avoids all unnecessary assumptions concerning the definition of the activated complex and its properties. Thus, for instance, it yields in a simple way the rate equations (7.IV), corresponding to the "normal Arrhenius parameters (6.IV), which are both direct consequences of the general rate equation (2.IV). It also predicts deviations from the normal values of the apparent frequency factor K without any additional assumptions, such that the transition state (AB)" (if there is one) differs more or less from the initial state of the activated molecule (AB). ... 

Quantum-corrections to the simple collision theory () and activated complex theory ( ac) for the isotopic reactions AH + H and BD + B (A and B hypothetical superheavy hydrogen isotopes) based on Weston potential energy surface Vtt/Vj. ratio of the rate constants ... 

The interpretation of these reactions was a considerable triumph for conventional transition-state theory. Simple collision theory proved unsatisfactory for trimolecular reactions, owing to the difficulty of defining a collision between three molecules, and usually led to very serious overestimations (by several powers of ten) of the rate constants. Similar difficulties are encountered with dynamical treatments, and these have still not been satisfactorily resolved. Conventional transition-state theory, by regarding the activated complex as being in equilibrium with the reactants, leads to a very simple formulation of the rate constant and to values in good agreement with experiment. It also very neatly explains the rather marked negative temperature dependence of the pre-exponential factors for these reactions. 

Simple collision theory (SCT) is an early theory of bimolecular reactions that was developed in the hrst decades of the twentieth century. Although SCT oversimplihes collision dynamics, and is of limited predictive power, it provides a beginning point for the collision dynamics approach to bimolecular reactions, and the beginnings of insight into factors that affect chemical reactivity. SCT also permits estimates to be made of the upper limit expected for the value of the bimolecular gas phase reaction rate coefficients from the rate of gas phase collisions. For these reasons SCT is worthy of examination. 

An example from kinetics is simple collision theory developed from the kinetic theory of gases to account for the influence of concentration and temperature on reaction rates. The theory is based on several postulates ... 

Simple collision theory can, in fact, be modified and extended to reactions in solution. In solutions, which contain solvated molecules and ions rather than simple molecules or atoms, interactions are known as encounters rather than collisions. It would be expected that encounter rates should be smaller than collision frequencies because the solvent molecules reduce the collision rate between reactants. However, encounters may be more likely than collisions where molecules are trapped in a temporary cage of solvent molecules (Figure 6.20). 

The concept of a transition state originates from transition-state theory. Before transition-state theory, chemists had explained rates of reactions in terms of collision theory, which is based on the kinetic theory of gases. It treats collisions by regarding the reacting molecules as hard spheres colliding with one another. Transition-state theory does not conflict with collision theory. It assumes that reactions involve collisions but takes into account some of the details of the collision, such as how the reacting molecules must approach one another for a reaction to be possible and what the effect of a solvent might be. None of these factors are taken into account by simple collision theory. 

The effect of a catalyst on the rate equation is to increase the value of the rate constant. Table 16.11 summarizes the changes that affect the rate of reaction and the rate constant. Rate constants are unaffected by changes in concentration and are only affected by temperature (as described by the Arrhenius equation) or the presence of a catalyst, which provides a new pathway or reaction mechanism. Rates increase with concentration and pressure (if gaseous reactants are involved), which can be accounted for by simple collision theory (Chapter 6). 

In this chapter we focus upon the dynamics of the reactive collision process in the gas phase. To this end we first show how the basic measurable parameter in collision chemistry, the reaction cross section, may be related to its thermal counterpart, the rate constant. We then discuss the experimental approach to measurement of reaction cross section as well as a naive model. The remainder of the chapter is devoted to the interpretation of a number of experiments in terms of simple dynamical models for reactive collisions. We make no attempt to develop a theory of reaction cross sections. 

Table 2 lists the available rate data for non-associative reactions between a nximber of small radicals. At 298 K, the rate constants are all within about a factor of ten of the simple collision theory estimate of the rate constant for all collisions. As was... 

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