Troe factorization

Recently, Troe developed an empirical model to resolve the discrepancy between the L-H model and the actual fall-off, in which a fall-off broadening factor F X) was introduced (Gardiner and Troe, 1984) ... 

In equation (C), A() (or A 111 as used earlier) is the low-pressure limiting rate constant and Ay is the high-pressure limiting rate constant. Fc is known as the broadening factor of the falloff curve its actual value depends on the particular reaction and can be calculated theoretically. Troe (1979) suggests that for reactions under atmospheric conditions, the value of Aft will be 0.7-0.9, independent of temperature. However, values as low as 0.4 are often observed. The NASA evaluations of stratospheric reactions (DeMore et al., 1997) take Aft = 0.6 for all reactions. The IUPAC evaluation (Atkinson et al., 1997a,b) does not restrict Fc to 0.6. However, it is important to note that the values of A0 and Ay will depend on the value of Fc used to match the experimental data. For example, for reaction (11)... 

This estimate is then improved by the inclusion of a series of correction factors, which were derived and developed by Troe [67,68]. 

The strong collision correction factor Fsc is a function of two further parameters that arise in Kassel theory (see Section 3). These are the number of effective oscillators, S, and B = Eg/kT, a measure of the relative magnitudes of the threshold energy and thermal energy. Troe used an energy criterion to obtain the number of effective oscillators. 

Troe and co-workers have also developed formulae for fwc- Again these depend on Bk and 5k as well as a third variable, the collision efficiency, /3c, which is a multiplicative factor applied to the collision frequency to account for weak collisions, i.e., the effective collison frequency is /3c< where (o is the actual collision frequency. 

Troe has also suggested a simpler form of the broadening factor product (F) which is adequate for most purposes... 

Actual experimental data on the pressure variation of the pseudo-second-order rate constant k do not conform with (3.71). The reason is that the elementary rate constants k , and should have been defined for each individual quantized vibrational level of AB, and the individual rates summed to give the total rate. Also, vibrations and rotations can interconvert in the newly formed molecule. A widely used modification of the treatment of pressure-dependent reactions is due to Troe (1983). In the Troe theory, the right-hand side of (3.71) is multiplied by a broadening factor F that is itself a function of ko/k. ... 

The results of figure 7.3 might suggest that the 2n — 1 vibrational frequencies are completely equivalent to AS. This has indeed been suggested by Troe (1988) who claimed that for the case of the cycloheptatriene isomerization to toluene, it makes little difference whether a single frequency is adjusted (as in the above calculation), or whether all frequencies are simply multiplied by a common factor. The k E) curve is the same over ten orders of magnitude in the calculated rates. However, Troe s calculation is for a case in which the molecule and TS frequencies are rather similar and in which the different TS frequencies are also similar. We show in figure 7.4 what happens for a dissociation in which the transition state is loose and in which the... 

In each case a significant difference is observed, due to the introduction of the F factor, between the values calculated using the TROE formulae and those of LINDEMANN-... 

The energy-dependence factor of the density of states Fe is a function introduced by Troe [13] and it is defined for vibrational densities of... 

Troe has described how one can estimate the value of the partition function basic expression for the density of internal states at the dissociation limit, which treats the vibrations in RadiRad ) as harmonic. Multiplicative factors are then estimated to allow, in turn for (i) the anhar-monicity of the vibrations (ii) the energy dependence of the density of vibrational states (iii) an overall rotation factor, which allows for the existence of centrifugal barriers and (iv) an internal rotation factor allowing for the barriers associated with internal rotors. 

This method of Troe appears capable of reproducing experimentally determined rate constants to within a factor of about 2 or 3. Table 1.1 lists a few of the values of k ssiX) from ref. 23. These data demonstrate that the rate constants increase markedly with the size (number of atoms) of the system and they show a marked negative dependence on the temperature, which is steeper the larger the system. 

It is obvious that anharmonic corrections are often not small for molecules with a single potential minimum and attempts have been made to represent them by analytic expressions. Following work by Harhoff, Troe suggested the correction factor ... 

Troe gives an expression for kQ in terms of factors such as the harmonic density of states, the Lennard-Jones collision frequency, the vibrational partition function, and the critical energy, Eq, as well as terms to account for anharmonicity corrections, the energy dependence of the density of states, and rotational effects. We use equation (1) of reference 76 to estimate kQ. Expressions to evaluate each of the factors in that equation are summarized in reference 75. For the purpose of this discussion, 6c =1 unimolecular dis-... 

By quantum statistical mechanics one evaluates expressions for / which are more appropriate than Eqs. (3.4) and (3.5). Then, to a good approximation, the contributions to / can be factored such that Eq. (3.2.2) becomes (Troe, 1977b)... 

The factor n in Eq. (3.22) was inadvertently omitted from the corresponding equations in two earlier publications, but not in the computations reported therein. (Troe 1977b, 1979)... 

A major problem is estimation of the weak collision factor (Troe, 1977b, 1979 Quack and Troe, 1977a Tardy and Rabinovitch, 1977 Endo, Glanzer, and Troe, 1979). This quantity can be related to the average energy transferred per collision by (Troe, 1977a)... 

We have used the data on obtained by Glanzer and Troe (1975) and recalculated their k (assuming AH 29g(H02) = 2.5 kcal/mol). These data together with Howard s data above 1000 K are shown in Fig. 13. The uncertainty in the data of Glanzer and Troe is about 30% based on our estimate of the combined uncertainty in and the heat of formation of HO2. In contrast the data obtained by Howard are direct measurement of ki with less scatter and uncertainty. In this evaluation, we recommend the expression obtained by Howard (1980) with uncertainty factors of / = 0.7 and F = 1.2 for the temperature range 1000 to 2000 K. 

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