The Lindemann mechanism

We are interested here in two general types of chemical process taking place in the gas phase, an isomerisation 

If such a process is a unimolecular reaction, then the general pattern of its behaviour will be describable, to zeroth order, by the celebrated Lindemann mechanism [22JL 23.C]  

By detailed balancing we know that, at equilibrium, ki[A]=k2[A ]. Thus, the ratio 2/ 1 is equal to the ratio of the populations [A]/[A ] at equilibrium, and so ilso, because [A] [A ], we can ignore 

By invoking the usual steady state hypothesis [77.V], we assume that the rate of formation of A by process (i) can be put equal to its rate of destruction by processes (ii) and (iii) combined, i.e. 

As we will see in a moment, such a reaction will always obey a first order kinetic rate law and, therefore, it is convenient to define a first order rate constant for it as 

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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... 

Thus, the competition between deactivation of the intermediate A and product formation is given in terms of the ratio a = Id lk, . When the second-order rate constants k, k2, and ki are set for the system, the ratio a is directly proportional to the pressure [M], since a = ( 2/ 3)[M]. Thus, the effect of varying [M], the variable in the Lindemann mechanism that defines the pressure, can... 

C. A. Hollingsworth, P. G. Seybold, L. B. Kier, and C.-K. Cheng, First-order stochastic cellular automata simulations of the Lindemann mechanism. Int. J. Chem. Kinet. 2004, 36, 230-237. 

According to the Lindemann-Christiansen hypothesis, formulated independently by both scientists in 1921, all molecules acquire and lose energy by collisions with surrounding molecules. This is expressed in the simplified form of the Lindemann mechanism, in which we use an asterisk to indicate a highly energetic or activated molecule, which has sufficient energy to cross the barrier towards the product side, and M is a molecule from the surroundings M may be from the same type as A ... 

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. 

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,... 

As carried out above for the Lindemann mechanism, application of the steady-state approximation gives the apparent unimolecular rate constant in Equation (24) where [Av] represents the IR photon density. Again two limits may be considered. 

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 ... 

The Lindemann mechanism of thermal collisional activation describes these systems ... 

Pressure-dependent rate constants for the syn-anti conformational process in larger alkyl nitrites provide a further test of the ability of RRKM theory to successfully model the kinetics of the internal rotation process in these molecules. Solution of the Lindemann mechanism shows that at the pressure where the rate constant is one-half of its limiting high-pressure value, Pm, the frequency of deactivating collisions is comparable to , the average rate that critically... 

Rabinovitch and co-workers found that the Lindemann mechanism is adequate for modeling the pressure dependence of bimolecular region unimolecular rate constants for extracting collision efficiencies for the methyl isocyanide isomerization [122]. For the conformer conversion of molecule A at constant temperature, it can be written as,... 

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. 

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. 

Based on the Lindemann mechanism shown above and the steady-state approximation for the intermediate A, the rate of formation of product... 

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... 

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... 

A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt d[A]/dt as a more appropriate formulation of quasi-stationarity. Furthermore, the general stationary state solution (equation (A3.4.144)) for the Lindemann mechanism contains cases that are not usually retained in the Bodenstein quasi-steady-state solution. 

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. 

What is the steady-state approximation Use the Lindemann mechanism example to discuss its validity in terms of opposing gain and loss mechanisms for A. ... 

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