Modified Collision Theory

Transition state theory, especially with its recent developments, has proved a very powerful tool, vastly superior to collision theory. It has only recently been challenged by modem advances in molecular beams and molecular dynamics which look at the microscopic details of a collision, and which can be regarded as a modified collision theory. These developments along with computer techniques, and modem experimental advances in spectroscopy and lasers along with fast reaction techniques, are now revolutionizing the science of reaction rates. 

Modified collision theory along with molecular beam studies has greatly expanded the knowledge of reaction at the microscopic level for many simple elementary reactions. 

The fact that the same expression modifies both collision theory and transition state theory should not cause surprise. This is because the modification is taking account of the same physical phenomenon in both theories, i.e. the interaction between the ions. 

Table 7.1 lists values of log10 A for some ionic reactions, and shows qualitative agreement of experiment with this electrostatically modified collision theory. At first sight this indicates the correctness of the electrostatic treatment. 

Have you ever seen a demolition derby in which the competing vehicles are constantly colliding Each collision may result in the demolition of one or more vehicles as shown in Figure 17-2a. The reactants in a chemical reaction must also come together in order to form products, as shown in Figure 17-2b. The collision theory states that atoms, ions, and molecules must collide in order to react. The collision theory, summarized in Table 17-2, explains why reactions occur and how the rates of chemical reactions can be modified. 

Equation (252) can be solved by a modified Enskog or similar procedure. In particular. Present has taken a classical with a reaction cross-section compatible with classical collision theory (e.g.. Reference 39). For areaction of type A -f A- -B - - C he finds that for E IkT > 5, the effect causes a difference in rates <8%. When EjkT < 5 his first-order approximation becomes progressively worse, and the method is no longer appropriate. 

The data obtained on reaction rates may be interpreted through either the collision theory or the theoiy of absolute reaction rates. The former places emphasis on the energy of activation as the rate-determining factor. This may be related to the temperature (T) and the rate constant (k) by a modified form of the Arrehenius equation ... 

The al ini. o potential energy surface of SHAVITT, STEVENS, MINN, and KARPLUS (SSMK), as modified by TRUHLAR and KUPPERMANN / 4/ has been used by these authors for exact quantal calculations of the transition probabilities for the colinear H2 + H and D2 + D reactions. On the basis of these data, CHRISTOV and PARLAPANSKI /132/ directly computed the values of the factor le. in the collision theory expression (23.IV) in the temperature range 300-1000 K. The corresponding values of the factor in the statistical expression (67,... 

The collision theory, which has been modified by BENSON, allows the pre-exponential factor A for the combination of two free radicals, to be calculated. 

The collision theory is a useful one not only in the sense that it has provided insight into the nature of chemical reactions, but also because it is a theory that can be readily tested. The mark of a scientific theory is that it can be tested and falsified. So far, collision theory has been supported by experimental evidence, but if new data were produced that could not be explained using the collision theory then it would need to be modified or dismissed in favour of a new theory that did explain all the evidence. Currently collision theory is the best explanation of the experimental data produced so far (at this working level). It should be noted here that we have not begun to distinguish between elementary and complex, multi-step reactions. That discussion is developed in Chapter 16 with the introduction of the idea of the rate-determining step in a sequence of stages. This is an example of how the theory is modified to explain more complex situations. Note that unimolecular reactions are an apparent exception which require special treatment. 

Simple collision theory can, in fact, be modified and extended to reactions in solution. In solutions, which contain solvated molecules and ions rather than simple molecules or atoms, interactions are known as encounters rather than collisions. It would be expected that encounter rates should be smaller than collision frequencies because the solvent molecules reduce the collision rate between reactants. However, encounters may be more likely than collisions where molecules are trapped in a temporary cage of solvent molecules (Figure 6.20). 

One of the central issues in studying reaction dynamics isf how you describe the flux over the reaction barrier, whether from simple collision theory, or transition state theory, to modified generalized Langevin theory, Kramer s theory...or even to quantum mechanical tunnelling - This is a selective not exhaustive list of possibilities (7,8). When the timescale of the photo chemical event (be it photodissociation, isomerization, charge transfer)... 

In many cases, the observed rate does not agree with the value calculated on the basis of Eq.(9.17). In order to account for deviation from the collision theory, Eq.(9.15) is modified to... 

Meanwhile, the pre-exponential factor A in the Arrhenius Eq. (2.39) is the temperature independent factor related to reaction frequency. Comparing the Eq. (2.33) for the collision theory and Eq. (2.38) with the transition state theory, the pre-exponential factors in these theories contain temperature dependences of T and T respectively. Experimentally, for most of reactions for which the activation energy is not close to zero, the temperature dependence of the reaction rate constants are known to be determined almost solely by exponential factor, and the Arrhenius expression holds as a good approximation. Only for the reaction with near-zero activation energy, the temperature dependence of the pre-exponential factor appears explicitly, and the deviation from the Arrhenius expression can be validated. In this case, an approximated equation modifying the Arrhenius expression can be used. 

Collision integral for self-diffusion in kinetic theory (—) Collision integral for diffusion in kinetic theory (—) Modified particle collision density ( 3 [ ] [ ,[)... 

Hence, the problem is reduced to whether g(co) has its maximum on the wings or not. Any model able to demonstrate that such a maximum exists for some reason can explain the Poley absorption as well. An example was given recently [77] in the frame of a modified impact theory, which considers instantaneous collisions as a non-Poissonian random process [76]. Under definite conditions discussed at the end of Chapter 1 the negative loop in Kj(t) behaviour at long times is obtained, which is reflected by a maximum in its spectrum. Insofar as this maximum appears in g(co), it is exhibited in IR and FIR spectra as well. Other reasons for their appearance are not excluded. Complex formation, changing hindered rotation of diatomic species to libration, is one of the most reasonable. 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

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 ... 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

Kramers [67], Northrup and Hynes [103], and also Grote and Hynes [467] have considered the less extreme case of reaction in the liquid phase once the reactants are in collision where such energy diffusion is not rate-limiting. Let us suppose we could evaluate the (transition state) rate coefficient for the reaction in the gas phase. The conventional transition state theory needs to be modified to include the effect of the solvent motion on the motion of the reactants as they approach the top of the activation barrier. Kramers [67] used a simple model of the... 

For polyatomic molecules equation (18) is employed with the vibrational matrix elements modified as described above. For vibrational exchange, in equation (18) the single vibrational matrix element is replaced by the product of the squares of the matrix elements for each molecule. In general, the theory leads to collision probabilities which are in good agreement with experiment. 

If redissociation into reactants is faster than stabilization, equations (3.15) and (3.16) simplify into a product of k,/k, and either kr or kcoll. Under these conditions, to obtain a theory for a total association rate coefficient, one must calculate both k,/k i and kr or kco . Three levels of theory have been proposed to calculate k, /k, . In the simplest theory, one assumes (Herbst 1980 a) that k, /k 3 is given by its thermal equilibrium value. In the next most complicated theory, the thermal equilibrium value is modified to incorporate some of the details of the collision. This approach, which has been called the modified thermal or quasi-thermal treatment, is primarily associated with Bates (1979, 1983 see also Herbst 1980 b). Finally, a theory which takes conservation of angular momentum rigorously into account and is capable of treating reactants in specific quantum states has been proposed. This approach, called the phase space theory, is associated mainly with Bowers and co-workers... 

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