Bimolecular reactions collision theory

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,... 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

There is an inunediate coimection to the collision theory of bimolecular reactions. Introducing internal partition functions excluding the (separable) degrees of freedom for overall translation. 

We are concerned with bimolecular reactions between reactants A and B. It is evident that the two reactants must approach each other rather closely on a molecular scale before significant interaction between them can take place. The simplest situation is that of two spherical reactants having radii Ta and tb, reaction being possible only if these two particles collide, which we take to mean that the distance between their centers is equal to the sum of their radii. This is the basis of the hard-sphere collision theory of kinetics. We therefore wish to find the frequency of such bimolecular collisions. For this purpose we consider the relatively simple case of dilute gases. 

Note that A is predicted by collision theory to be proportional to For bimolecular reactions A has the units M s (liter per mole per second). 

Some of the rate constants discussed above are summarized in Table VI. The uncertainties (often very large) in these rate constants have already been indicated. Most of the rate constants have preexponential factors somewhat greater than the corresponding factors for neutral species reactions, which agrees with theory. At 2000°K. for two molecules each of mass 20 atomic units and a collision cross-section of 15 A2, simple bimolecular collision theory gives a pre-exponential factor of 3 X 10-10 cm.3 molecule-1 sec.-1... 

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. 

For complex organic molecules, geometric considerations alone lead one to the conclusion that only a small fraction of bimolecular collisions can lead to reaction. One can represent the fraction of the collisions that have the proper geometric orientation for reaction by a steric factor (Ps). Except for the very simplest reactions, this factor will be considerably less than unity. On the basis of simple collision theory, it is not possible to make numerical estimates of Ps, although it may occasionally be possible to make use of one s experience with similar reactions to determine whether Ps for a given... 

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. 

In terms of the collision theory a bimolecular reaction rate is written as... 

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

Equation (4.21) gives the rate constant for a bimolecular reaction in terms of collision theory. 

Too little attention is generally paid to the concentrations of the reactants in preparative organic work. With the exception of rare cases (e.g. in intramolecular rearrangements) we are concerned with reactions of orders higher than the first, and in these several kinds of molecules—usually two—are involved. Since, according to the kinetic molecular theory, the velocity of bimolecular reactions is proportional to the number of collisions between the various dissolved molecules and therefore to the product of the concentrations,... 

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. ... 

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... 

By collision theory this is the maximum rate of a bimolecular reaction, because we must multiply this value by the probability that reaction will occur during the coUision (a number less than unity). Thus we predict from collision theory that the pre-exponential factor in a bimolecular reaction should be no larger than 10liters/mole sec. 

The next more sophisticated theory of bimolecular reactions is called activated complex theory, which assumes that the collision of A and B forms a complex (AB) and that the rate of the reaction depends on the rate of decomposition of this complex. We write this as... 

Because a is a parameter that cannot be calculated from first principles. Equation 1-95 cannot be used to calculate reaction rate constant k from first principles. Furthermore, the collision theory applies best to bimolecular reactions. For monomolecular reactions, the collision theory does not apply. Tr3dng to calculate reaction rates from first principles for all kinds of reactions, chemists developed the transition state theory. 

Collision theory is based on the concept that molecules behave like hard spheres during a collision of two species, a reaction may occur. To estimate a rate constant for a bimolecular reaction between reactants A and B based on this theory, one needs first to calculate the number of collisions occurring in a unit volume per second (ZA1 ) when the two species, A and B, having radii rA and ru, are present in concentrations jVa and Aru, respectively. From gas kinetic theory, this can be shown to be given by Eq. (I) ... 

The primary condition for a bimolecular reaction is the close approach of iwo interacting partners. In ground state, the molecules can approach as close as their van der Waals radii and they are said to be in collision. The frequencies of collisions between unlike molecules and like molecules are given by kinetic theory of gases. 

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... 

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