Collision theory - simple

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. 

The SCT model considers reaction between chemical species A and B, each considered to be structureless, spherical masses that interact according to the hard sphere potential V(r) — 0, r dAB V(r) = CO, r — dAB, and all collisions result in reaction. The last may be restated as a reaction probability the probability of chemical reaction, F(r), is 1 when r — 7ab and 0 otherwise. The collision diameter, 7ab — ( 5 a + 5 b)/ 2, where 7a and 7b are the molecular diameters of A and B, respectively, defines the interaction distance for these billiard ball-like collisions. The collision rate, Zab, is 

In these equations, — mAmB/(wA + WB) is the reduced mass of the collision pair, m the species molecular mass, Icb the Boltzmann constant, T the absolute temperature, and a and b the molecular densities. The 

For an elementary reaction of species A with species B, the rate of disappearance of A is given by the kinetic rate expression 

If reaction occurs on every hard sphere collision, the rate of disappearance of A is equal to the colhsion rate, Zab- Comparison of equations (56) and (58) shows that the SCT expression for the rate coefficient, kscT fot reaction between A and B is 

The simplest model for the rates of chemical reactions assumes that every time there is a bimolecular collision, there is a reaction. From calculations of the frequency of collisions in this system, we can then determine the rate constant. 

To calculate the number of collisions per unit time, we need a model for the behaviour of molecules in these systems. The simplest approach involves a system of two gases, A and B, whose molecules behave as hard spheres, which are characterised by impenetrable radii and r. A collision between A and B occurs when their centres approach within a 

If we assume that the molecules of B are fixed and that those of A move with an average velocity each molecule A sweeps a volume nd per unit time which contains stationary molecules of B. The area is known as the collision cross section. If 

If the total number of molecules of A per unit volume is NJV, then the total number of collisions of A with B per unit volume is given by 

The value of cos 6 can vary between -1 and 1. As all values of 6 between 0 and 360° are equally probable, the positive and negative values of cos 6 will cancel out, and the mean value will be zero. Thus we obtain 

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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. 

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. 

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... 

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 ... 

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

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... 

SCT simple collision theory ST stirred tank (CSTR)... 

The value of first order rate constant (k1), i.e. k ko/ki (say k,J at high concentration is determined from experiments and has also been calculated from simple collision theory (i.e. rate constant = Ze E ,c IRT). In all cases it has been observed that rate constant dropped at much higher concentration than is actually observed. Since there can be no doubt about kwhich is an experimental quantity, the error must be in the estimation of rate constant. 

Thus, El is higher than the experimental activation energy and can give much higher rates of activation and, therefore, much higher value of k lk[ than simple collision theory. 

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. ... 

On simple collision theory, this ratio should be invarient and close to 2. This has been shown to be the case for a number of pairs of radicals.58 In the original paper,48 this ratio varied from 0.44 to 1.32, whereas in the more recent study48 an average value of 1.8 was obtained with only a 10% variation over wide concentration ranges. The hypothetical reaction of radical addition to the C=0 double bond has recently been shown to occur in the photolysis of hexafluoroacetone and will be discussed below. It is sufficient at this stage to point out that such a reaction could lead to radical interchange. 

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. 

This result is identical to the hard-sphere rate constant Eq. 10.82 derived from the simple collision theory introduced in Section 10.2. 

The first of the shortcomings of the Lindemann theory—underestimating the excitation rate constant ke—was addressed by Hinshelwood [176]. His treatment showed that ke can be much larger than predicted by simple collision theory when the energy transfer into the internal (i.e., vibrational) degrees of freedom is taken into account. As we will see, some of the assumptions introduced in Hinshelwood s model are still overly simplistic. However, these assumptions allowed further analytical treatment of the problem in an era long before detailed numerical solution was possible. 

This expression can be compared to the simple collision theory formula of Eq. 10.83 ... 

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 simple collision theory, for a bimolecular reaction this critical energy, eo, is the kinetic energy of relative translational motion along the line of centres of the colliding molecules, loosely described as the violence of the collision. 

As the complexity of the reactants increases the p factor decreases, and the discrepancy with collision theory increases. This reflects the inadequacy of simple collision theory, especially the neglect of the internal structures of the reactants, and intermolecular interactions. 

In simple collision theory, if the impact parameter 6 < rA + rB, then collision will occur if 6 > rA + rB collision cannot occur. In molecular beam studies a collision is still defined in terms of a distance apart of the trajectories, but this is no longer rA + rB, but is a distance 6,nax. If the impact parameter b < 6max, collision occurs, and the numerical value of 6max is found from scattering experiments and is closely related to the minimum angle of scattering able to be detected. 

This simple collision theory thus predicts preexponential factors of about 10 cc/mole-sec, since we expect P < 1. Values of P < 1 are interpreted kinetically as due to improperly oriented collisions ( steric hindrance) or thermodynamically as a negative entropy of activation, i.e., a loss of freedom of A and B in forming the collision complex. As we shall see, these results are in good qualitative agreement with observations and Zab does indeed seem to be an upper limit for bimolecular frequency factors. ... 

In comparing both these theories with the simple collision theory we see that what has happened is that the rather vague steric factor P of the latter has been replaced by the—in principle more meaningful—ratios of partition functions of the species A, B, and AB". In the case that A and B are atoms then P = 1, and all theories have the same factors. The same is true if we assume that the groups A and B interact so weakly in forming the transition state that their rotational and vibrational modes are unaltered. 

Lindemann assumed that the rate coefficient for this process could be calculated from simple collision theory. 

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