Fall-off Curves

Figure A3.4.9. Pressure dependence of the effective unimolecular rate constant. Schematic fall-off curve for the Lindemaim-FIinshelwood mechanism. A is the (constant) high-pressure limit of the effective rate constant...

Figure B2.5.3. The fall-off curve of reaction (B2.5.14) with M = He between 0.3 bar and 200 bar. The dashed lines represent the extrapolated low- and high-pressure limits, /r r, = (2.1 0.2) x [He] cm moU s ...

The rate constants were determined at a series of pressures in the fall-off region, and the fall-off curve was very similar to that obtained for the structural isomerization to propylene. The similarity of the two sets of data suggests that both reactions may proceed through similar reaction paths. One obvious possibility is that once again the trimethylene biradical is formed, which can undergo internal rotation followed by recyclization. An alternative transition state has been suggested which involves, as an activated complex, a much expanded cyclopropane ring in which hindered internal rotation occurs (see also Smith, 1958). 

In the preceding expression, log(FJ is related to the depression of the fall-off curve at the center relative to the L-H expression in a og k/k ) vs. log(2f/(l -I- X)) plot. The values for F<. can then be related to the properties of specific species and reaction and temperature using methods discussed in Gardiner and Troe (1984). In Fig. 19, values of F for a variety of hydrocarbon decompositions are presented. As evident from this figure, in the limit of zero or infinite temperatures and pressures, all reactions exhibit Lindemann-Hinshelwood behavior and F approaches unity. From this figure, it is clear that L-H analysis generally does an adequate job in... 

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. 

Compute the fall-off curve using Hinshelwood theory. For this calculation, take the effective number of vibrational modes s to be one-half of the actual number of vibrational modes for the the species. vacluai ... 

Compute the fall-off curve using QRRK theory. For this calculation, assume a collision diameter of 4.86 A. Assume that the average energy transfer per N2-C-C5H5 collision is -0.69 kcal/mol (needed to calculate the parameter /5 used in the model). Take the number of oscillators to be, v = actual, with the frequency calculated above. Assume the reaction barrier to be E0, given above. 

The result of a fall-off curve under maximum inhibition was compared with the theoretical curve estimated by the RRKM formulation. From the foregoing observations, the authors offered a reaction mechanism which may explain satisfactorily the process of the 1,2-dibromopropane pyrolysis. 

In this way, fall-off curves for the rate coefficient, k, are characterized by three numbers k°, k°° and Fin. k° is the rate coefficient at low concentrations. For example, this formalism has been used in a compilation of kinetic parameters of elementary processes occurring in the middle atmosphere [65]. 

In the Lindemann-Hinshelwood theory the Lindemann expression for the uni-molecular rate constant, Eq. (9), is still assumed to be correct, but an improved activation rate coefficient is obtained from the Hinshelwood formulation. The shape of the fall-off curve should therefore still be the simple form predicted by Lindemann. Reference to Fig. 2 shows that, for the cyclobutane decomposition reaction, the change in the activation rate coefficient brings the theory much closer to the experimental results, particularly at low pressure. However, the shape of the fall-off curve is still not correct the Lindemann-Hinshelwood model predicts a faU-off region that is too narrow, the true fall-off is broader. 

The RRK theory has the virtue that it is very simple to apply and it does get close to the correct shape of the fall-off curve. As an example. Fig. 5 shows the fall-off curve calculated from classical RRK theory for the dissociation of cyclobutane using 14 oscillators. It can be seen that the theory is a considerable improvement on the Lindemann-Hinshelwood model. There are, however, some remaining problems. 

U ii,/RT [16]. However, it seems that no unique value of s is capable of modelling the fall-off curve properly [17], and some attempts have been made to introduce models with two values of s [18]. 

Troe proposed a similar approach to the calculation of the fall-off curve. In this case the zero-order approximation is the Lindemann-Hinshelwood model, formulated with the correct high and low-pressure limiting rate coefficients ... 

Although these equations are empirical, they do give a good representation of experimental fall-off curves, and are frequently used to analyse and represent data. 

Forst treated the decomposition of azomethane with the quantum harmonic version of the Marcus-Rice theory of unimolecular reactions. He used different models which comprised three dimensional free rotations of the two methyls and of the central nitrogen molecule and adjusted the overall moments of inertia to give the correct total entropy. For planar or tetrahedral methyls calculations gave an increase of six in the number of active rotations in going from the molecule to the complex. With a minimum of assumption it was also possible to reproduce the pressure fall-off curve of the experimental first order rate coefficient for planar and tetrahedral complexes. A further result of the computations is the conclusion that the vibrational frequency pattern of the complex is so much less important than the number of active rotations that both tetrahedral and planar complexes lead to identical fall-off behaviour. 

Chang and Rice have also calculated the pressure fall-off curve of the first order rate coefficient assuming a three-centered D-honded model,... 

Fig. 2.27. Fall-off curves for reaction (31) as a function of helium pressure. A, 300 K, O,...

Fig. 1. The fall-off curve for a thermal dissociation/recombination reaction.

Under the assumption that M is temperature independent, the above equation describes the temperature dependence of (3C. The temperature dependence of ko for a given bath-gas is then completely determined by the value of m, which is varied to refine the fall-off curve.17 23... 

The theoretical fall-off curves by Delbos et al 82 shown in Fig. 27 reproduce the results of their measurements very well. The high-pressure value... 

Fig. 27. Fall-off curves and results of experiments179,182 for the CH2CHO + NO (+ M) reaction.

The mechanism of the reaction CH2C(CH3)0 + NO is very similar to that discussed for CH2CHO + NO. The formation of 0N-CH2C(CH3)0 is the dominant reaction pathway. The fall-off behavior of the reaction system was also analyzed within the Troe formalism.17"23 The fall-off curves of Delbos et al.m were constructed on the basis of their experimental results. The best fit leads to results corresponding to the fall-off parameters... 

The analysis of the pressure dependence of the rate constants for the CH3 + OH and OH + OH recombination reactions was based on the construction of the theoretical fall-off curve. The fall-off curves obtained theoretically by Fagerstrom et al. 92 are shown in Fig. 30. The temperature dependence of the limiting rate constants for both the CH3 + OH (+ SF6) and OH + OH (+ SF6) recombination reactions can be expressed in the form... 

Fig. 31. A comparison of the theoretical fall-off curves of Jodkowski et al.212 and results of experiments205 212 for CH3 + NO (+ M) obtained at room temperature for other bath-gases.

Low-pressure limiting rate constants for other bath-gases were also derived by Jodkowski et al.212 by fitting the theoretical fall-off curve to available... 

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