Collision theory of gas-phase reactions

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. 

According to the collision theory of gas-phase reactions, a reaction takes place only if the reactant molecules collide with a kinetic energy of at least the activation energy, and they do so in the correct orientation. 

The simplest model that accounts for the Arrhenius expression is the collision theory of gas-phase reaction rates, in which it is supposed that reaction occurs when two reactant molecules collide with at least a minimum kinetic energy (which is identified with the activation energy. Figure 2). A more sophisticated theory is the activated complex theory (also known as the transition state theory), in which it is supposed that the reactants encounter each other, form a loosened cluster of atoms, then decompose into products. 

In the collision theory of gas-phase reactions, an activation energy a is assumed for a collision to be effective, and only molecules with a kinetic energy exceeding a react. The respective portion of molecules for an arbitrarily chosen value of a of 30kj moT and a temperature of 1000°C is indicated in Figure 3.1.16b (hatched... 

For reactions taking place in the gas phase, both the Arrhenius equation and the dependence of the rate law on concentration can be accounted for using the kinetic theory of gases, giving what we call the collision theory of gas-phase chemical kinetics. In this theory, the rate of a reaction is directly proportional to the number of molecular collisions per second (i.e., to the frequency of molecular collisions) ... 

Fi 7.12 In the collision theory of gas-phase chemical reactions, reaction occurs when two molecules collide, but only if the collision is sufficiently vigorous, (a) An insufficiently vigorous collision the reactant molecules collide but bounce apart unchanged, (b) A sufficiently vigorous collision results in a reaction. 

Analysis of this system is rather straightforw ard since it is mathematically equivalent to a catalyst pellet in which reaction and diffusion occur simultaneously. The rate of reaction in this case is simply the growth rate of the film. The collision frequency of gas-phase molecules of A with the surface is given by gas kinetic theory as (V, 4)C.4, where = mean velocity of the gas phase A molecules (VSAT [7r(.Wi)]) and is the... 

M. A. Eliason and J. O. Hirschfelder, General collision theory treatment for the rate of gas phase reactions, J. Chem. Phys. 30 1426 (1959). 

In many cases, there must be energy transfer between the reacting molecules. For reactions that take place in the gas phase, molecular collisions constitute the vehicle for energy transfer, and our description of gas phase reactions begins with a kinetic theory approach to collisions of gaseous molecules. In simplest terms, the two requirements that must be met for a reaction to occur are (1) a collision must occur and (2) the molecules must possess sufficient energy to cause a reaction to occur. It will be shown that this treatment is not sufficient to explain reactions in the gas phase, but it is the starting point for the theory. 

The kinetic theory of gases can be used to understand a number of phenomena relevant to chanistry. In Chapter 14, we will use the kinetic theory to determine the collision frequency of gas-phase reacting molecules, which will be used to calculate the rate of gas-phase reactions. Problans 5.57 and 5.58 illustrate the use of kinetic theory in gas diffusion the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic motion) and effusion (escape of a gas through a small hole in a container). 

The temperature dependence of gas-phase reaction rates can be understood through collision theory. 

Now that we have a model, we must check its consistency with various experiments. Sometimes such inconsistencies result in the complete rejection of a model. More often, they indicate that we need to refine the model. In the present case, the results of careful experiments show that the collision model of reactions is not complete, because the experimental rate constant is normally smaller than predicted by collision theory. We can improve the model by realizing that the relative direction in which the molecules are moving when they collide also might matter. That is, they need to be oriented a certain way relative to each other. For example, the results of experiments of the kind described in Box 13.2 have shown that, in the gas-phase reaction of chlorine atoms with HI molecules, HI + Cl — HC1 I, the Cl atom reacts with the HI molecule only if it approaches from a favorable direction (Fig. 13.28). A dependence on direction is called the steric requirement of the reaction. It is normally taken into account by introducing an empirical factor, P, called the steric factor, and changing Eq. 17 to... 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

Termolecular Reactions. If one attempts to extend the collision theory from the treatment of bimolecular gas phase reactions to termolecular processes, the problem of how to define a termolecular collision immediately arises. If such a collision is defined as the simultaneous contact of the spherical surfaces of all three molecules, one must recognize that two hard spheres will be in contact for only a very short time and that the probability that a third molecule would strike the other two during this period is vanishingly small. 

An increase in the concentration of a reactant (or reactants) in solution, or an increase in the pressure on a gas-phase reaction, increases the rate of reaction. In terms of the collision theory ... 

The science of reaction kinetics between molecular species in a homogeneous gas phase was one of the earliest fields to be developed, and a quantitative calculation of the rates of chemical reactions was considerably advanced by the development of the collision theory of gases. According to this approach the rate at which the classic reaction... 

Simple Collision Theory (SCT) of Bimolecular Gas-Phase Reactions... 

Reactions in solution proceed in a similar manner, by elementary steps, to those in the gas phase. Many of the concepts, such as reaction coordinates and energy barriers, are the same. The two theories for elementary reactions have also been extended to liquid-phase reactions. The TST naturally extends to the liquid phase, since the transition state is treated as a thermodynamic entity. Features not present in gas-phase reactions, such as solvent effects and activity coefficients of ionic species in polar media, are treated as for stable species. Molecules in a liquid are in an almost constant state of collision so that the collision-based rate theories require modification to be used quantitatively. The energy distributions in the jostling motion in a liquid are similar to those in gas-phase collisions, but any reaction trajectory is modified by interaction with neighboring molecules. Furthermore, the frequency with which reaction partners approach each other is governed by diffusion rather than by random collisions, and, once together, multiple encounters between a reactant pair occur in this molecular traffic jam. This can modify the rate constants for individual reaction steps significantly. Thus, several aspects of reaction in a condensed phase differ from those in the gas phase ... 

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. 

Experimental evidence for the notion of an activation energy barrier comes from a comparison of collision rates and reaction rates. Collision rates in gases can be calculated from kinetic-molecular theory (Section 9.6). For a gas at room temperature (298 K) and 1 atm pressure, each molecule undergoes approximately 109 collisions per second, or 1 collision every 10 9 s. Thus, if every collision resulted in reaction, every gas-phase reaction would be complete in about 10-9 s. By contrast, observed reactions often have half-lives of minutes or hours, so it s clear that only a tiny fraction of the collisions lead to reaction. 

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