Unimolecular reactions Lindemann theory

We have calculated the addition channel rate constant using the RRKM approach to unimolecular reaction rate theory, as formulated by Troe ( ) to match RRKM results with a simpler computational approach. The pressure dependence of the addition reaction (1) can be simply decribed by a Lindemann-Hinshelwood mechanism, written most conveniently in the direction of decomposition of the stable adduct ... 

Rice and Ramsperger and independently Kassel proposed the theories to explain unimolecular reaction, in which both (k2) and (kfk[) have been treated as dependent on the energy of an individual energized molecule E. These theories jointly are referred as RRK theory. According to the theory the expression for the first order rate constant given by Lindemann theory i.e. 

The theory of Lindemann explains most of the trends observed in the kinetics of uni-molecular reactions. It has been very useful in understanding the qualitative behavior of this class of reactions. It provides the starting point for all modem theories of unimolecu-lar reactions. The theoretical basis for unimolecular reaction rates is treated in much more detail in Chapter 10. 

The Lindemann treatment for association reactions is analogous to the theory just given for unimolecular reactions. For convenience, rewrite reactions 9.100 and 9.101 in the reverse directions... 

The theoretical analysis of chemical activation reactions is similar to the Lindemann theory of unimolecular and association reactions. There are a number of competing reaction pathways. Depending on total pressure, concentrations of the participating species, and temperature, the outcome of the competition can change. 

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. 

The Lindemann theory thus has the correct behavior at the high- and low-pressure limits. However, quantitative comparisons between this theory and experiment revealed a number of problems. The remainder of this section discusses more detailed theoretical treatments of unimolecular reaction kinetics. 

The Hinshelwood model thus corrects one of the major deficiencies in the Lindemann theory of unimolecular reactions. The greater excitation rate constant of Eq. 10.132 brings the predicted fall-off concentration [M]j/2 of Eq. 10.109 into much better accord with experiment. However, because of the many simplifying assumptions invoked in the Hinshelwood model, there are still a number of shortcomings. 

Perrin s argument that the very nature of a unimolecular reaction demands independence of collisions, and therefore dependence on radiation, is adequately met both by the theory of Lindemann and by that of Christiansen and Kramers. Both these theories have the essential element in common that the distribution of energy among the molecules is not appreciably disturbed by the chemical transformation of the activated molecules thus the rate of reaction is proportional simply to the number of activated molecules and therefore to the total number of molecules, sinc in statistical equilibrium the activated molecules are a constant fraction of the whole. Thus the radiation theory is not necessary to explain the existence of reactions which are unimolecular over a wide range of pressures. 

Thermal unimolecular reactions usually exhibit first-order kinetics at high pressures. As pointed out originally by Lindemann [1], such behaviour is found because collisionally energised molecules require a finite time for decomposition at high pressures, collisional excitation and de-excitation are sufficiently rapid to maintain an equilibrium distribution of excited molecules. Rice and Ramsperger [2] and, independently, Kassel [3] (RRK), realised that a detailed theory must take account of the variation of decomposition rate of an excited molecule with its degree of internal excitation. Kassel s theory is still widely used and is valid for the chosen model of a set of coupled, classical, harmonic oscillators. 

It is easy to understand a bimolecular reaction on the basis of collision theory. Thus, when two molecules A and B collide their relative kinetic energy exceeds the threshold energy, the collision may result in the breaking of bonds and the formation of new bonds. But how can one account for a unimolecular reaction If we assume that in such a reaction (A — P) the molecule A acquires the necessary activation energy by colliding with another molecule, then the reaction should obey second-order kinetics and not the first-order kinetics which is actually observed in several unimolecular gaseous reactions. A satisfactory theory of these reactions was proposed by F.A. Lindemann in 1922. According to him, a unimolecular reaction... 

Note The Lindemann mechanism was also suggested independently by Christiansen. Hence, it is also sometimes referred to as the Lindemann-Christiansen mechanism. The theory of unimolecular reactions was further developed by Hinshelwood and refined by Rice, Rampsberger, Kassel and Marcus. 

Conditions necessary for neglecting dc i/dt in the manner employed above may be investigated through formal approximations in reaction-rate theory. This will be considered further, with application to the Lindemann mechanism, in Section B.2.5. The mechanism itself generally contains fundamental inaccuracies and is best viewed as a simplified approximation to more-complex mechanisms. In particular, molecules capable of experiencing unimolecular decomposition or isomerization may exist in many different vibrationally excited states, and the rate constant for the reaction may differ in each state. Approximate means for summing over states to obtain average rate constants have been developed an introduction to these considerations maybe found in [3]. 

Some of the continuing approaches to reaction-rate theory that differ from either the simple collisional theory or the transition-state theory discussed here are cited on pages 98-112 of [4]. Examples of differing approaches may be found in particular in theories for rates of three-body radical-recombination processes [61]. Advances in methods for calculating rate constants relevant to the Lindemann view of unimolecular processes also are providing new information relevant unimolecular and bimolecular rates. Future work may be expected to produce further results of use in combustion problems. 

Although the theory does need to be improved in a number of details before it can provide a quantitative description of experiment, the observation of fall-off from first order at high pressures to second order at low pressures is correctly explained by the Lindemann-Christiansen mechanism, and modem theories of unimolecular reactions are based on this mechanism. 

Figure 2.13 is a sketch of the pressure dependence of a unimolecular reaction showing the two limiting conditions. The region joining the two extremes is known as the fall off region. Theories of unimolecular reactions have advanced considerably since Lindemann s initial proposal but they are still based on the same physical ideas so clearly highlighted in the Lindemann mechanism. 

M is Br2 or any other gas that is present. By the principle of microscopic reversibility , the reverse processes are also pressure-dependent. A related pressure effect occurs in unimolecular decompositions which are in their pressure-dependent regions (including unimolecular initiation processes in free radical reactions). According to the simple Lindemann theory the mechanism for the unimolecular decomposition of a species A is given by the following scheme (for more detailed theories see ref. 47b, p.283)... 

The formulation of reaction rate theory used in the previous sections applies to bimolecular and higher-order reactions in general, but to unimolecular reactions only at high pressures. We shall, therefore, reconsider the problem of isotope effects in unimolecular gas reactions. We start with the recent elaboration of the Lindemann hypothesis given by Marcus.42... 

The first successful theory of unimolecular reactions (2.46) was proposed by Lindemann in 1922. He introduced the idea that apparently unimolecular reactions were really the result of the following processes ... 

With decrease in pressure, the chance that an activated molecule will lose its energy decreases more rapidly than the chance of the energy becoming so distributed as to allow a reaction to occur. Consequently, at low pressures the rates of unimolecular reactions cannot remain independent of the pressure. The activated molecule tends to become a Van t Hoff intermediate, and the rate to depend on the collision frequency. This corollary of Lindemann s theory has been... 

Collision theory does not deal directly with unimolecular reactions but touches on the subject through the Lindemann mechanism. Once the molecule has been provided with sufficient energy by collision, the problem is to calculate the rate constant for the unimolecular decomposition,... 

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