Bimolecular collision theory

To calculate the energy of activation and the probability factor for the dimerization of butadiene in the gas phase. 

The dimerization of butadiene in the gas phase is a homogeneous second-order reaction (see problem 142). Values of the second-order rate constant at several temperatures (Kistiakowsky and Ransom, J. Chem. Phys. 1939, 7, 725) are given in table 1, where is defined by 

We diall assume that on a collision between two butadiene molecules the probability of reaction has a constant value a when the rdative translational energy along the line of centres exceeds a definite energy e and is otherwise zero. We call the energy of activation and a the probability factor . The formula for is then (Fowler and Guggenheim, Statistical thermodynamics , Cambridge University 

formula 1206.5 Frost and Pearson, mechanism , John Wiley, 1953, p. 65) 

We plot log (kJT ) against T and obtain a line of slope —elk In 10. Then, since all the quantities in (1) except a are known, we can calculate a. 

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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... 

Among four-atom reactions this system has become a prototype test case for the comparison between quantum state resolved dynamics experiments and bimolecular collision theory. In 1973 the first semi-empirical LEPS and BEBO PESs were constructed by Zellner and Smith [56]. The subsequent development of a global H-HOH(A ground state PES in 1980 [20b], based on ab initio calculations [20ak was favoured by the fact that three of the four atoms involved are H atoms. The... 

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). 

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... 

If hu0 is small compared with kT, the partition function becomes kT/hv0. The function kT/h which pre-multiplies the collision number in the transition state theory of the bimolecular collision reaction can therefore be described as resulting from vibration of frequency vq along the transition bond between the A and B atoms, and measures the time between each potential transition from reactants to product which will only occur provided that the activation energy, AE°0 is available. 

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. 

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. 

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) ... 

In particular, the question has been raised about whether Reaction 4 occurs as a simple atom transfer in a bimolecular collision between the radical R and 02, or whether it goes thru the same type of peroxy radical structure as Reaction 1. This problem has its roots deep in the history of oxidation kinetics, and early examples occur during the 1930 s. At that time several authors, inspired more by the need to explain products than by the constraints of any detailed transition state theory, were... 

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... 

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,... 

The mechanism of unimolecular decomposition can be described semi-quali-tatively by the simple theory of Kassel156 which is adequate for this brief review. Molecules exchange energy in bimolecular collisions, and if a molecule acquires... 

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