Lindemann theory

Steinfeld et al. [53] and Robinson et al. [54] explained concepts of the Lindemann theory as follows  

If the steady-state hypothesis is applied to the concentration of A, the overall rate of reaction 

The overall concept can be expressed by the equations below, where M can represent a generic bath gas molecule, an added inert gas molecule it may also represent a second molecule of reactant or product. In this Lindemann theory ki, along with k.j and 2 are taken to be energy-independent and are calculated from the simple collision theory equation. 

Application of the steady-state hypothesis to the concentration of A, allows the unimolecular rate constant and the high-pressure and low-pressure limit rate and rate constant to be determined as follows  

First order at high-pressure limit rate, [ ] = K= KK 

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Activation of Reacting Molecules by Collisions The Lindemann Theory... 

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. 

We continue our study of chemical kinetics with a presentation of reaction mechanisms. As time permits, we complete this section of the course with a presentation of one or more of the topics Lindemann theory, free radical chain mechanism, enzyme kinetics, or surface chemistry. The study of chemical kinetics is unlike both thermodynamics and quantum mechanics in that the overarching goal is not to produce a formal mathematical structure. Instead, techniques are developed to help design, analyze, and interpret experiments and then to connect experimental results to the proposed mechanism. We devote the balance of the semester to a traditional treatment of classical thermodynamics. In Appendix 2 the reader will find a general outline of the course in place of further detailed descriptions. 

The last term F in this equation is equal to 1 for the Lindemann theory (it has been added for generality). 

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. 

The treatment given in this section is analogous to the Lindemann theory of unimolecu-lar reactions. It provides a general explanation of pressure effects in bimolecular chemical activation reactions. A more sound theoretical treatment of chemical activation kinetics is given in Section 10.5. 

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. 

To estimate the rate constant pressure fall-off using Lindemann theory, typically one would estimate the excitation rate constant as... 

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 functional form of Eq. 10.102 shows the dependence of kUni on total concentration predicted by the Lindemann theory. It can be used to find the total pressure, or equivalently the total concentration of molecules [M], at which kuni drops to half its high-pressure limit (i.e., kuni,oo/2). We define this concentration as [M], and from Eqs. 10.102 and the definition of kuni,oo in Eq. 10.103 one obtains... 

Thus the Lindemann theory and Eq. 10.109 can be used to predict the fall-off concentration [M]1/2. In many tests against experiment, the predicted fall-off concentration is as much as 10 orders of magnitude greater than measured. (The experimental fall-off concentration is... 

Thus the Lindemann theory predicts that a plot of l/kUni versus 1 /[M] should yield a straight line. Experimental data consistently shows downward curvature at high pressure (small values of 1/ [M]) in plots of this type. The predicted v-intercept (1 /kam) is too large (i.e., the theory underpredicts the extrapolated infinite-pressure rate constant kUni,oo)- This is the second general breakdown of the Lindemann theory that motivated further theoretical analysis. 

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. 

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. 

Compute the fall-off curve (log kuni versus log [M]) using the Lindemann theory. Choose the range of pressures (total concentrations [M]) to illustrate the very low pressure and very high pressure extremes of the kinetics expected by the theory. 

The facts that have just been described lend considerable support to the Lindemann theory. If this theory is to be applicable, the rate of activation and deactivation at higher pressures ought to be great compared with the rate of chemical change, in order that there may be little disturbance of the statistical equilibrium and hence an absolute rate of reaction directly proportional to the total concentration. At first some difficulty was felt about this point, but the solution appears to have been found, and indeed the solution itself constitutes a rather strong piece of evidence in favour of the theory. 

Although the Lindemann theory predicts that the rate constant should be independent of bath gas pressure when [M] > k2, this is not the case at lower pressures. The expression for the unimolecular rate coefficient is... 

There are three main respects in which the Lindemann theory needs to be improved. 

Although the Lindemann theory is often satisfactory, it is incomplete since it does not fully recognise the relation between translational and internal energies. In many reactions the rate of activation by collision is not itself explicable unless it is assumed that activation can also occur by the transfer of vibrational energy from one molecule to another. This possibility was recognised by Hinshelwood and by Lewis and may be equivalent, in effect, to multiplying the frequency factor by 10" or more. 

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)... 

Demonstrate the application of the Lindemann theory to an isomerization reaction A <-> B under conditions that might be encountered in practice. 

In terms of the Lindemann theory, what must be the effeetive eollision number (A ) if the ealeulation is to agree with the experimental results shown in Figure 2.10 Use ( / oo) = 0.525 at logF(em) = 0 as a basis. The temperature is 492°C. 

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