Theory collision

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 

Molecular beams, chemiluminescence and laser-induced fluorescence experiments show the theory in its simple form to be fundamentally flawed, with internal states of reactants and products and the redistribution of energy on reaction being of fundamental importance. 

Molecular beam experiments study reactive and non-reactive collisions between molecules by observing the deflection of each molecule from its original path as a result of the collision. By studying the amount of scattering into various angles, much useful kinetic information can be found. 

Chemiluminesence is emission of radiation from the products of a chemical reaction, and is normally associated with exothermic reactions where the products are in excited vibrational, rotational and sometimes electronic states. The particular excited states which are produced immediately after reaction has occurred can be studied and kinetic information obtained. Such studies also throw doubts on the validity of assuming that these states play no part. 

Laser-induced fluorescence has been discussed in Chapter 2. Lasers excite molecules to excited levels from which they can lose energy by emission of radiation. This technique has been of major importance in the molecular beam experiments which force modification of the original collision theory outlined below. 

The origins of collision theory to predict the specific reaction rate lie in the kinetic theory of gases. The rate of reaction is calculated from the product of the frequency of collisions and the fraction of collisions that have enough energy to react. For the reaction involving species A and B to form species C and D, 

Both A and E may also be temperatme dependent. We can estimate the activation energy from either potential energy smfaces or various empirical relationships and the frequency factor from either collision theory, transition state theory or from computational chemistry software (see Appendix J). 

In the collision between molecttles A and B it is classically assumed that these molecttles behave as rigid spheres with radii cta and CTb, respectively. Consequently, whenever molecttle A touches molecttle B, a collision is assttmed to have taken place (Figttre G-1). With this basis the collision cross-section, S, is 

By defining a relative velocity of A with respect to B, we analyze the B molecules as if they were stationary. 

The distance molecule A travels with respect to molecttle B is 

Normally, the rate of a reaction is expressed in terms of a rate constant multiplied by a function of concentrations of reactants. As a result, it is the rate constant that contains information related to the collision frequency, which determines the rate of a reaction in the gas phase. When the rate constant is given by the Arrhenius equation. 

Ea is related to the energy barrier over which the reactants must pass as products form. For molecules that undergo collision, the exponential is related to the number of molecular collisions that have the required energy to induce reaction. The pre-exponential factor, A, is related to the frequency of collisions. Therefore, we can describe the reaction rate as 

Rate = (Collision frequency) X (Fraction of collisions with at least the 

We can define the cross-sectional area of the cylinder, ir(rA + the collisioml cross section, CTab- In 1 second, a molecule of A travels a distance of i AB (where r AB is the average molecular velocity of A relative to B) and it will collide with all molecules of B that have centers that lie within the cylinder. Therefore, the number of collisions per second will be given by 

FIGURE 4.1 Model lased for calculating collision frequency. 

If we assume that molecules can be considered as billiard balls (hard spheres) without internal degrees of freedom, then the probability of reaction between, say, A and B depends on how often a molecule of A meets a molecule of B, and also if during this collision sufficient energy is available to cross the energy barrier that separates the reactants, A and B, from the product, AB. Hence, we need to calculate the collision frequency for molecules A and B. 

We will call Pa and Pb the number of molecules A and B per unit volume, respectively. Because the probability of collision is obviously related to the size of the molecules, we also define the diameters of the effective spheres that represent A and B, namely and dg. 

According to Trautz and Lewis, who gave the first treatment of reaction rates in terms of the kinetic theory of collisions in 1916-1918, the rate of collisions (not yet reaction) between the spheres A and B is jtd u 

The effect of double counting is most easily seen in the following calculation. Suppose that the density of molecules is Pa = Pb = 10 and that A and B are identical. Consequently, the number of collisions between A and B is 

If we now take B equal to A, we have Pa = 20. The total number of collisions in the volume does not change and becomes  

A basic goal of the theory of chemical kinetics is to predict the magnitude of the reaction rate coefficient and its temperature dependence. We focus first on bimolecular reactions. The most elementary approach to bimolecular reactions is based on the collision of hard, structureless spheres. This approach is called collision theory. 

Imagine that the molecules are like billiard balls, in that there is no interaction between them until they come into contact, and they are impenetrable, so that their centers cannot 

Consider a collision between two hard spheres A and B with radii rA and r% and velocities va and vb. If we imagine B to be fixed, then the relative velocity of A is v — vA — vg. If the centers of A and B approach each other at a separation less than or equal to the sum of their radii, then collision occurs. We can therefore define a total cross section S as the effective target area presented to A by B, that is, S = nd2, where d = rA + rb An A molecule passing with velocity v through nB stationary B molecules per unit volume will collide with a B molecule whose center lies within an area nd1 around the path of A. In unit time, an A molecule sweeps out a collision volume nd2 v and undergoes collision with nd2 v ng B molecules. If there are nA molecules of A per unit volume, the total number of collisions per unit volume per unit time is 

The molecules of each species possess a Maxwell distribution of speeds. For molecules of species A, for example, their mean speed from the Maxwell distribution is vA = (%kBT lnmA)Xl 2, and the relative velocity of the A, B collision partners is 

If reaction occurred with every collision, then the rate of reaction between A and B would be just 

The Arrhenius relation means that the rate constant or the diffusivity increases with temperature. Typically, at low temperatures (0-60°C), a 10-degree increase in temperature results in a doubling of reaction rates. In this section, two theories are introduced to account for the Arrhenius relation and reaction rate laws. Collision theory is a classical theory, whereas transition state theory is related to quantum chemistry and is often referred to as one of the most significant advances in chemistry. 

The collision frequency /xb can be calculated from the kinetic theory of gases. The result is (cf., Atkins, 1982, p. 872) 

collision theory gives the correct reaction rate law (Equation 1-94) and a reaction rate constant of the form 

Although this form differs from the Arrhenius equation in that the pre-exponential term depends slightly on T, because the exponential dependence usually dominates, the weak dependence of the pre-exponential term on T may be regarded as negligible and the whole term A T regarded as a constant A. Hence, it is possible to roughly derive the Arrhenius relation from the collision theory. 

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. 

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. 

Suppose particle A moves through space with average speed v A will collide with a B particle if their center-to-center distance is less than or equal to ta -t- rg. Thus, particle A sweeps out an area irlrA + rB) v in which it can collide with B, and the corresponding volume swept out per second is irfrA -t- rg fv. If the concentration of B is B molecules cm , the number of collisions of B particles by this single A particle, per second, is 7r(rA -t- rgfngv. However, the volume also 

Equation (5-6) gives the number of bimoleeular collisions per unit time and volume, but not all of these eollisions lead to reaetion, and so we write rate = eollision frequeney X fraetion of eollisions having energy equal to or greater than that required for reaetion, or 

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). 

Let us estimate a typical value for A. Choosing ta = rt = 5 A, p. = 2 x 10 g,T = 300 K, we find A 4 x 10 M s . This is for the gas phase. In solution the situation is somewhat different because of the solvent cage effect described in Section 4.1. During each bimolecular encounter within a solvent cage, several collisions may occur. This results in a predicted A value for liquid solutions somewhat larger than that for gases.  

All matter is made up of particles, and in the study of reaction kinetics we are interested in the behaviour of atoms, molecules and ions and how they react together. It seems very obvious to say this, but two chemicals can only react if their particles come into contact with each other. 

The particles in gases and liquids are in constant motion and consequently they will collide with each other. Millions of such collisions occur every second, but not every one produces a new substance if they did reactions would be over in less than a millionth of a second. We have much everyday evidence to show that this is not so for instance, if iron rusted on its first contact with the air then bridges and towers could never be built, so evidently not all collisions are effective. 

The molecules move about randomly and at different speeds. Some may collide. 

Some molecules only catch glancing blows low energy collisions with too little energy to break bonds. So no reaction  

In order for a reaction to occur particles must coUide with each other -the more particles that are in a certain amount of space, that is a fixed volume of gas or solution, and the faster they are moving, then the greater the likelihood of collisions and so a reaction. 

Man fools himself. He prays for a long life, and he fears an old age. 

The chemical collision theory has been able to explain many of the observations related to chemical kinetics. The assumption made for this theory is that particles must physically collide for a chemical reaction to occur. In addition, these collisions must be effective without sufficient force, the electrons surrounding the nuclei of the atoms involved would just repel each other and the atoms would not combine. Thus, the rate of any step in a reaction is directly proportional to (1) the number of collisions per unit time and (2) the fraction of these collisions that are effective. 

The number of collisions for gases at standard temperature and pressure (STP 0°C and 1 atm) has been calculated to be more than 10 ° s. If all these collisions were effective, then reaction rates would be extremely fast. However, this is not true because only a small fraction of collisions are effective. The extra amount of energy required in a collision to overcome interatomic repulsive forces and produce a chemical reaction is known as the energy of activation. Its magnitude depends on properties of the reactants. 

We now identify and investigate the factors that control the value of the rate constant. In Section 6.6, we considered the Arrhenius equation 

Reactions in the gas phase introduce a number of concepts reiating to the rates of 

With the reaction profile in mind, it is quite easy to estabhsh that collision theory accounts for Arrhenius behavior. Thus, the collision frequency, the rate of collisions between species A and B, is proportional to both their concentrations if the concentration of B is doubled, then the rate at which A molecules collide with B molecules is doubled, and if the concentration of A is doubled, then the rate at which B molecules collide with A molecules is also doubled. It follows that the collision frequency of A and B molecules is directly proportional to the concentrations of A and B, and we can write 

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. 

This fraction increjises with increasing temperature at T = 0,/= e = 0 and no collisions are successful at T= °°,f= e = 1 and every collision has enough energy to result in reaction. 

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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. 

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]. 

Simple collision theories neglect the internal quantum state dependence of a. The rate constant as a function of temperature T results as a thennal average over the Maxwell-Boltzmaim velocity distribution p Ef. 

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. 

Adhi Karl S K and Kowolski K L 1991 Dynamical Collision Theory and its Applications (New York Academic)... 

Bernstein R B (ed) 1979 Atom-Molecule Collision Theory. A Guide for the Experimentalist New York Plenum)... 

Mies F H 1969 Resonant scattering theory of association reactions and unimolecular decomposition. Comparison of the collision theory and the absolute rate theory J. Cham. Phys. 51 798-807... 

Goldberger M L and Watson K M 1964 Collision Theory (New York Wiley)... 

Joachain C J 1975 Quantum Collision Theory (Amsterdam North-Holland) p 383... 

Bransden B H 1983 Atomio Collision Theory 2nd edn (Menlo Park, CA Benjamin-Cummings)... 

Baer M (ed) 1985 Theory of Chemical Reaction Dynamics (Boca Raton, FL CRC Press) vols 1-4 Bernstein R B (ed) 1979 Atom-Molecule Collision Theory A Guide for the Experimentalist (New York Plenum)... 

In chemical kinetics, it is often important to know the proportion of particles with a velocity that exceeds a selected velocity v. According to collision theories of chemical kinetics, particles with a speed in excess of v are energetic enough to react and those with a speed less than v are not. The probability of finding a particle with a speed from 0 to v is the integral of the distribution function over that interval... 

An estimate of the enthalpy change which conesponds to the activation energy of the collision theory analysis of 167kJmoP may be made by assuming that the formation of tire dimer from two molecules of the monomer is energetically equivalent to tire dipole-dipole and dispersion interactions of two HI molecules. These exothermic sources of interaction are counterbalanced... 

Child, M.S., 1974, Molecular Collision Theory (Academic Press, London). 

The collision theory considers the rate to be governed by the number of energetic collisions between the reactants. The transition state theory considers the reaction rate to be governed by the rate of the decomposition of intermediate. Tlie formation rate of tlie intermediate is assumed to be rapid because it is present in equilibrium concentrations. 

Tests of the collision theory consist of comparisons between calculated and experimental values of the preexponential factor, the comparison often being made in terms of a ratio P defined by... 

Simple collision theory does not provide a detailed interpretation of the energy barrier or a method for the calculation of activation energy. It also fails to lead to interpretations in terms of molecular structure. The notable feature of collision theoiy is that, with very simple means, it provides one basis for defining typical or normal kinetic behavior, thereby directing attention to unusual behavior. 

If the transition state theory is applied to the reaction of two hard spheres, the result is identical with that of simple collision theory. - pp Because transition state theory is an equilibrium theory, it can be inferred that collision theory is also an equilibrium theory. 

Collision theory leads to this equation for the rate constant k = A exp (-EIRT) = A T exp (,—EIRT). Show how the energy E is related to the Arrhenius activation energy E (presuming the Arrhenius preexponential factor is temperature independent). 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

Table 11.3 compares observed rate constants for several reactions with those predicted by collision theory, arbitrarily taking p = 1. As you might expect, the calculated k s are too high, suggesting that the steric factor is indeed less than 1. 

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