Lindemann mechanism, pressure

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... 

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 ... 

The same result is traditionally derived from the high-pressure limit of the Hinshelwood-Lindemann mechanism see Section 7.4.1. 

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,... 

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... 

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. 

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. 

Although it is rarely possible to study any particular unimolecular reaction all the way from the first order (high pressure) limit to the second order (low pressure) limit, many genuine unimolecular reactions have now been characterised over at least part of the fall-off region [72.R]. Thus, we can easily compare the observed shape of the fall-off curve, and its position on the pressure axis with the behaviour suggested by the Lindemann mechanism. 

Figure 14.21 Plot of the pressure dependence of the rate for a unimolecular decomposition that follows the Lindemann mechanism.

In this case, the rate law is first order in A and independent of the total pressure. A plot of the pressure dependence of a unimolecular decay that obeys the Lindemann mechanism is shown in Figure 14.21. This dependence is consistent with that which is experimentally observed in many unimolecular decompositions in the gas phase. 

Depending upon the concentration of A, the predicted rate law varies between first and second order. At very low pressure ka k [A] and the rate law is second order with an apparent rate constant k. However, when the pressure is high and A [A] k the reaction is first order with an apparent rate constant k = k K, where K is the equilibrium constant for the activation process. To test the Lindemann mechanism express the data in first-order form... 

In Fig. 5.5, log(A ex/ oo) is plotted for the decomposition of cyclopropane. While it appears to be a linear function of log [A] at low pressure, suggesting that the low-pressure limit has been reached, the slope is closer to than 1, the value predicted by (5.21). Such differences are typical the actual falloff with pressure is more gradual than is consistent with the simple Lindemann mechanism. Improvement requires considering the specific... 

Therefore, the termolecularreaction rate constant has pressure dependence, and it is explained by the following scheme, called the Lindemann mechanism. According to the mechanism. 

The curve (a) in Fig. 2.9 is the schematic graph of the pressure dependence of a termolecular reaction rate constant according to the Lindemann mechanism. From the figure, it can be seen that the reaction rate constant is proportional to [M] (pressure) in the low-pressure limit, and gets nearly constant independent on the pressure in the high-pressure limit. The intermediate region between these two limits is called the fall-off region. 

The correct treatment of the mechanism (equation (A3.4.25), equation (A3.4.26) and equation (A3.4.27), which goes back to Lindemann [18] and Hinshelwood [19], also describes the pressure dependence of the effective rate constant in the low-pressure limit ([M] < [CHoNC], see section A3.4.8.2). 

Pressure effects are also seen in a class of bimolecular reactions known as chemical activation reactions, which were introduced in Section 9.5. The treatment in that chapter was analogous to the Lindemann treatment of unimolecular reactions. The formulas derived in Section 9.5 provide a qualitative explanation of chemical activation reactions, and give the proper high- and low-pressure limits. However, that simple treatment neglected many quantum mechanical effects, namely the energy dependence of various excitation/de-excitation steps. 

Lindemann s mechanism further demands that the velocity constant shall begin to fall at some pressure sufficiently low, but does not predict at what pressure the effect should begin to be observable, since this depends upon a specific factor, namely, the average life of an activated molecule. 

How thermal activation can take place following the Lindemann and the Lindemann-Hinshelwood mechanisms. An effective rate constant is found that shows the interplay between collision activation and unimolecular reaction. In the high-pressure limit, the effective rate constant approaches the microcanonical rate... 

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. 

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