Reaction mechanisms Lindemann mechanism

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

This approxunation is generally valid if For the Lindemann mechanism of unimolecular reactions... 

Lindemann Mechanism of unimolecular reactions — activation by collisions... 

If the stoichiometric equation for unimolecular reaction is A -> B + C, and if the energized molecules are denoted by A, the Lindemann mechanism consists of the following sequence of events. 

Thus, according to this (Lindemann) mechanism, a unimolecular reaction is first-order at relatively high concentration (cM) and second-order at low concentration. There is a... 

A test of the Lindemann mechanism is normally applied to observed apparent first-order kinetics for a reaction involving a single reactant, as in A - P. The test may be used in either a differential or an integral manner, most conveniently by using results obtained by varying the initial concentration, cAo (or partial pressure for a gas-phase reaction). In the differential test, from equations 6.4-20 and -20a, we obtain, for an initial concentration cAo = cM, corresponding to the initial rate rPo,... 

The isomerization of cyclopropane follows the Lindemann mechanism and is found to be unimolecular. The rate constant at high pressure is 1.5 x 10- s- and that at low pressure is 6 X 10- torr- s-K The pressure of cyclopropane at which the reaction changes its order, found out ... 

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 activation and its interplay with chemical reaction is often described by the so-called (generalized) Hinshelwood Lindemann mechanism ... 

In the Lindemann mechanism, a time lag exists between the energisation of A to A and the decomposition (or isomerisation) of A to products. During this time lag, A can be de-energised back to A. According to the steady-state approximation (s.s.a), whenever a reactive (i.e., short-lived) species is produced as an intermediate in a chemical reaction, its rate of formation is equal to its rate of decomposition. Here, the energised species A is short-lived. Its rate of formation = kJAp and its rate of decomposition k t [A][A ] + k2[A ]. Thus,... 

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. 

The simplest type of system that obeys equation (17) is the unimolecular process 95li products. Since a stable molecule should not spontaneously break up into reaction products, the mechanism by which the unimolecular process occurs must be explained. Many unimolecular reactions are believed to follow the mechanism proposed by Lindemann, namely. 

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]. 

The Lindemann mechanism for unimolecular reactions, discussed in Section B.2.2, provides a convenient vehicle for illustrating partial-equilibrium approximations and for comparing them with steady-state approximations, even though this mechanism is not a chain reaction. To use the partial-equilibrium approximation for the two-body production of SRJ, select for example, as the species whose concentration is to be determined by partial equilibrium and use... 

Radical decompositions are unimolecular reactions and show complex temperature and pressure dependence. Section 2.4.l(i) introduces the framework (the Lindemann mechanism) with which unimolecular reactions can be understood. Models of unimolecular reactions are vital to provide rate data under conditions where no experimental data exist and also to interpret and compare experimental results. We briefly examine one empirical method of modelling unimolecular reactions which is based on the Lindemann mechanism. We shall return to more detailed models which provide more physically realistic parameters (but may be unrealistically large for incorporation into combustion models) in Section 2.4.3. 

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. 

If we further take fe = 0 this becomes the Lindemann mechanism that is used to explain the observation that many gas-phase reactions of the type A product that appear unimolecular at high pressure change their character to bimolecular at low pressure. Lindemann has postulated that such unimolecular reactions proceed... 

More complicated interpretations of B -1- M -> C -h M in the limit >>1 can be made that still fall in the realm of simplified modeling, for example, the two-step Lindemann mechanism, B M <=> B + M B -> C [8]. However, such interpretations haven t been shown to lead to better agreement with observations for homogeneous energetic solids than the single step, irreversible reaction and since they represent an additional level of complexity they are not considered here. 

The Lindemann mechanism for thermally activated unimolecular reactions is a simple example of a particular class of compound reaction mechanisms They are mechanisms whose constituent reactions individually follow first-order rate laws [JT, 20, 36,48,40, 5f, 52, 53, 54, 55 and 56] ... 

As an example we take again the Lindemann mechanism of unimolecular reactions. The system of differential equations is given by equation (A3.4.127). equation (A3.4.128) and equation (A3 A. 1291. The rate coefficient matrix is... 

When it is necessary for a reaction to proceed through several successive elementary steps before the product is formed, the rate of the reaction is determined by the rates of all these steps. If one of these reactions is much slower than any of the others, then the rate will depend on the rate of this single slowest step. The slow step is the rate-determining step. The situation is analogous to water flowing through a series of pipes of different diameters. The rate of delivery of the water will depend on the rate at which it can pass through the narrowest pipe. An apt illustration of this feature of consecutive reactions is offered by the Lindemann mechanism of activation for unimolecular decompositions. 

The apparent first-order rate constant decreases at low pressures. Physically the decrease in value of the rate constant at lower pressures is a result of the decrease in number of activating collisions. If the pressure is increased by addition of an inert gas, the rate constant increases again in value, showing that the molecules can be activated by collision with a molecule of an inert gas as well as by collision with one of their own kind. Several first-order reactions have been investigated over a sufficiently wide range of pressure to confirm the general form of Eq. (32.61). The Lindemann mechanism is accepted as the mechanism of activation of the molecule. 

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,... 

The Lindemann mechanism consists of three reaction steps. Reactions (1.4) and (1.5) are bimolecular reactions so that the true unimolecular step is reaction (1.6). Because the system described by Eqs. (1.4)-(l. 6) is at some equilibrium temperature, the high-pressure unimolecular rate constant is the canonical k T). This can be derived by transition state theory in terms of partition functions. However, in order to illustrate the connection between microcanonical and canonical systems, we consider here the case of k(E) and use Eq.(1.3) to convert to k(T). 

Livermore and Phillips (1966) also studied the thermal decomposition of (32H5O in the presence of NO at 2(K) C at very low xessures in a flow system. They found reaction 2a to be pressure dependent and follow a Lindemann mechanism with a half-reaction pressure of 0.08-1.6 Torr of (C2H50)2 as a chaperone. This agrees exactly with the results of Steacie and Calder (1936), who found a similar half-reaction pressure for the reverse reaction. 

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