Fall off behavior

In spite of the proper qualitative features of the Lindemann-Hinshelwood model, it does not correctly predict the much broader experimental fall-off behavior this is shown in Fig. 18, in which log(fe/fc ,) is plotted as a function of log(M = P/RT/Mj = Pc/RT). As evident from this figure, the actual rate at the center of fall-off (i.e., at PJ is depressed relative to the L-H model consequently, the transition of rate from low- to high-pressure limit occurs more gradually. 

Troe s analysis summarized above requires the knowledge of both low- and high-pressure rate constants, in addition to an empirically determined to describe the actual fall-off behavior. We already discussed methods for the estimation of high-pressure rate parameters. The low-pressure rate parameters can be estimated by recognizing the fact that ko represents pure energy transfer limitations, and thus can be determined from rate of collisional energization of A and from the thermal energy distribution function K E, T) ... 

The QRRK approach illustrated above also constitutes the basis to analyze the behavior of the reverse, i.e., association, reactions that proceed through chemically activated transition states. Recently Dean (1985) reformulated the unimolecular quantum-RRK method of Kassel and devised a practical method for the proper description of the fall-off behavior of bimolecular reactions, including reactions when multiple product channels are present. The method developed was shown to describe the behavior of a large variety of bimolecular reactions with considerable success (Dean and Westmoreland, 1987 Westmoreland et ai, 1986). 

Pressure fall-off behavior in an association reaction is illustrated in Fig. 9.1, namely... 

Fig. 9.1 Rate constant for the association reaction CCI3+O2 - CCl3C>2, as a function of pressure for three different temperatures. Fits to the fall-off behavior were determined by Luther et al. [250], and experimental data are from Refs. [126, 250,345].

The QRRK rate constant in Fig. 10.7 certainly fits the experimental data well. However, this is to be expected given the origin of the parameters in the model. Specifically, the high-pressure Arrhenius parameters were obtained from fits to the experimental data. The number of oscillators was taken as an adjustable parameter, as was the collision cross section used in ks. Thus the QRRK curve in Fig. 10.7 should match the experiment in the high-pressure limit, and two parameters were varied to enable a fit to the pressure fall-off behavior. 

These models were used to evaluate the necessary sums, densities, and moments of inertia, and eqs. (22) and (27) were integrated numerically to give k, (Fig. 12) and the calculated values of Table XVI. As was mentioned earlier in connection with ethane decomposition, we believe the thermally activated ethyl radical decomposition at 600°C. to be well into the fall-off region at pressures below atmospheric. In fact, comparison of Tables XII and XVI indicates that the fall-off behavior at 600°C. is very similar in its pressure dependence for ethyl and ethane, i.e., k, is a closely similar function of energy in both cases as shown explicitly... 

The Marcus (rrkm) treatment is, at the present time, the most satisfactory formulation, although its application sometimes presents a considerable amount of difficulty. It interprets the fall-off behavior in a very satisfactory manner, with all vibrational degrees of freedom being considered as active. 

The results of the rate constant calculations by d Anna et al,156 seem to confirm this reaction mechanism. In Fig. 25 is shown the temperature dependence of the observed and calculated rate constants. The rate constant k describes the rate of formation of the post-reaction adduct under the assumption that the pre-reactive adducts are not stabilized by collisions, whereas kadd describes the kinetics of formation of the stable pre-reactive complexes at a total pressure of 1 bar. Thus the overall rate constant for the decay of reactants (denoted in the figure by a solid line) is given by the sum k + k. The values of k predicted by d Anna et al.156 distinctly underestimate the reaction rate at low temperatures, but they approach the results of measurements at temperatures above 700 K. The limiting rate constants kadd, and kadd,0 for the addition channels were analyzed in terms of statistical unimolecular rate theory. Results of the calculations show a fall-off behavior of the reaction kinetics under typical atmospheric conditions corresponding to a total pressure of 1 bar. Therefore, all kadd values were derived from the... 

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... 

For all alkenes for which rate constants have been studied as a function of temperature the rate constants decrease with Increasing temperature, equivalent to Arrhenius activation energies of [except for propadlene (187)] approximately -1000 cal mole l (101, 169, 184, 187, 189) the Individual values are given In Tables 3a and 4. This temperature dependence Is equivalent to < iT over the temperature ranges studied (297-426°K). For propadlene a smaller temperature dependence, equivalent to an Arrhenius activation energy of -305 cal mole, has been obtained (187). From product studies, and the observation of fall-off behavior for ethene, the reactions of the OH radical with these alkenes proceed mainly via addition to the alkene (see below). 

OH-alkene adduct back to reactants at low pressures, which gives rise to the observed rate-constant fall-off behavior for ethene. It is expected that under the almost single-collision conditions employed by Gutman and co-workers (158) the addition products and their fragmentation products would be much diminished. Thus, their conclusions concerning H-atom abstraction are not totally relevant to higher pressure systems. 

Hence, for the reaction of OH radicals with vinyl fluoride and other fluoroalkenes containing no Cl or Br atoms, the rate constant will, as observed for ethene, exhibit fall-off behavior from second- to third-order kinetics at low total pressures. 

For isomerization of small hydrocarbons (fewer than five carbons), the observed rate constants are generally independent of total pressure at pressure greater than about 1 atm. However, at lower pressures the first-order rate constants fall ofiF and are dependent on pressure. Importantly, the smaller the molecule the higher the pressure at which the fall off behavior occurs. 

The fall off behavior as a function of molecular size is explained by arguing that the energy in the activated molecule is randomly distributed among a number of vibrational modes, and for a reaction to occur, i.e. k2, the energy must be localized in the appropriate vibration. Therefore, 2 is proportional to [(E — Eact)/E] where is the number of vibrations in the molecule, and the energy factor is the fractional... 

The two isomers n-CsHy and FC3H7 are separated by a barrier of 37kcal/mol (measured with respect to n-CsH ) and they can easily interconvert at sufficiently high temperatures. Although in reality both radicals dissociate to propene + H and ethylene+ CH3 (see Fig. 8), we will ignore these channels here and focus exclusively on the isomerization part. The steady-state analysis with CARRA yields one apparent pressure-dependent rate constant, since the rate constant for the reverse reaction is determined by the equilibrium constant. The predictions with both versions (MSC and ME) for T = 1200 K and various pressures are shown in Fig. 9. The results are very similar and show the expected fall-off behavior. The MSC treatment—despite its simplicity—captures the pressure dependence well. 

Several studies cover the fall-off region of unimolecular dissociation. The pressure at which the observed rate constant reaches one-half of the limiting high-pressure value is around 4 atm (at ca. 400 K) [14]. The fall-off behavior does not allow a direct comparison to low-pressure, termolecular NF2 recombination rate constants, see a. comment [20] on earlier dissociation... 

Figure 2. Reduced fall-off curve of a dissociation or recombination reaction. The dashed curve is given by Eq. (2.11) the solid one corresponds to observed fall-off behavior.

From the limiting low- and high-pressure rate constants ko and respectively, one immediately derives the pressure at which fall-off behavior is to be expected. The center of the fall-off curve, i.e., that bath-gas concentration [M]c at which the extrapolated limiting rate coefficients intersect, is given by rearranging Eq. (2.10) to... 

The reverse of this exothermic reaction (Fig. 31) has been discussed before, including a presentation of its fall-off behavior (Fig. 30). Flame propagation is rather sensitive to the rate coefficient of this reaction (Fig. 4) because of the importance of CH3 as an intermediate in the combustion of aliphatic hydrocarbons (Warnatz, 1981) and the chain-terminating character of this step. As no information on the efficiency of third bodies other than noble gases is available, the collision numbers given in Section 1.2 are recommended for use with the low-pressure rate coefficient. 

The reverse of this reaction has been discussed above, including the fall-off behavior (Fig. 60). There exist numerous measurements at room temperature (Fig. 61), but no determination of the activation energy. The review of Kerr and Parsonage (1972) recommends an Arrhenius expression derived from an estimated preexponential factor. This leads to a large rate coefficient for high temperature, in disagreement with values given by Baldwin et al (1966) and with determinations from the reverse reaction, which are the basis of the Arrhenius expression recommended here. 

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