Low-pressure limit

The apparent activation energy is then less than the actual one for the surface reaction per se by the heat of adsorption. Most of the algebraic forms cited are complicated by having a composite denominator, itself temperature dependent, which must be allowed for in obtaining k from the experimental data. However, Eq. XVIII-47 would apply directly to the low-pressure limiting form of Eq. XVIII-38. Another limiting form of interest results if one product dominates the adsorption so that the rate law becomes... 

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). 

Note that in the low pressure limit of iinimolecular reactions (chapter A3,4). the unimolecular rate constant /fu is entirely dominated by energy transfer processes, even though the relaxation and incubation rates... 

Brown N J and Miller J A 1984 Collisional energy transfer in the low-pressure-limit unimoleoular dissooiation of FIO2 J. Chem. Rhys. 80 5568-80... 

If the pressure for the process is lowered, the reaction (R3) will shift from a first-order reaction (high-pressure limit) to a second-order reaction (low-pressure limit). If (R3) is now considered a second-order reaction and assuming that the other pressure dependent reactions do not shift regime, determine expressions for d[C2H6]/dt, d[CH3]/dt, d[C2Hs]/dt and d[H]/dt. 

Equations (85) and (86) and the data shown in Figure 22 represent experimental results for flow in smooth straight tubes. Experimental data for complex porous systems sometimes differ slightly from the results given by Figure 22 in the region between the low pressure limit and a L 1. 

In the low-pressure limit the rate will increase as the square of the pressure, but at very high pressures it will fall off in a manner proportional to the reciprocal of the pressure. Consequently, the initial rate increases at first, goes through a maximum, and then declines. 

However, at some specific pressure the high-density polymorph becomes mechanically unstable. This low-pressure limit is seldom observed, since it often corresponds to negative pressures. When the mechanical stability limit is reached the phase becomes unstable with regard to density fluctuations, and it will either crystallize to the low-pressure polymorph or transform to an amorphous phase with lower density. 

In an elegant study of shock waves in dilute N2Os/Ar mixtures, Schott and Davidson278 have measured the low-pressure limit of the rate coefficient for pure N205 decomposition, /c42, by monitoring N03 and N02 formation behind the shock front. By subsequently following the decay of N03 they were able also to compute values of k30 and k43. These values, however, should be accepted with caution since there was some difficulty in separating the individual contributions of steps (30) and (43) to the total rate of N03 destruction. However, their rate... 

In the following sections, the rate coefficients dki(E)/k2, ka, and k3 will be further explored to enable the calculation of kun and isotope effects thereon from first principles, and the high pressure and low pressure limits of kunj will be explored in more detail. 

The energy level density is not important in determining the magnitude of the isotope effect at high pressure. At the low pressure limit, again for thermal activation,... 

Many systems fall in a region of pressures (and temperatures) between the high- and low-pressure limits. This region is called the fall-off range, and... 

Figure 6.7 shows the average correlation g(PQ) as a function of oxygen partial pressure. At the low-pressure limit (determined by the pair correlation only) there is no clear-cut monotonic dependence on temperature. At the high-pressure limit (as determined by the quadruplet correlations, see Section 5.8), we see that the... 

Figure 6.7. Average correlation s(Pq ) as a function of the partial pressure of oxygen Pq (in torrs), for the same system as in Fig. 6.5. (a) Low-pressure limit (b) high-pressure limit. The temperatures are indicated next to each curve.

To illustrate these issues better, the pressure at the center of fall-off (F ) is presented in Fig. 20. As seen from this figure, the unimolecular decompositions of small molecules are at their low-pressure limits at atmospheric pressure, and at process temperatures, = feo [M]- Decompositions of larger molecules, on the other hand, are closer to their high-pressure limits. It is important to recognize that the unimolecular decompositions of hydrocarbons from CH4 to CaHg exhibit differing degrees of fall-off under process conditions, and this must be properly accounted for in the development of accurate detailed chemical kinetic models. 

That is, the model assumes that all the energized A (i.e.. A ) react, and the latter terms in the preceding expression represent the fraction [A ]/[A]. Benson (1960) has also shown that Eq = E — S — )RT. That is, the low pressure limit activation energy should be less than the high pressure limit one. 

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)... 

However, the formation of the dimer in the ter-molecular reaction is sufficiently fast under stratospheric conditions that the bimolecular reactions are not important. For example, using the recommended termolecular values (DeMore et al., 1997) for the low-pressure-limiting rate constant of /c,3()0 = 2.2 X 10-32 cm6 molecule-2 s-1 and the high-pressure-limiting rate constant of k3()0 = 3.5 X 10-12 cm3 molecule-1 s-1 with temperature-dependent coefficients n = 3.1 and m = 1.0 (see Chapter 5), the effective rate constant at 25 Torr pressure and 300 K is 1.6 X 10-14 cm3 molecule-1 s-1, equal to the sum of the bimolecular channels (Nickolaisen et al., 1994). At a more typical stratospheric temperature of 220 K and only 1 Torr pressure, the effective second-order rate constant for the termolecular reaction already exceeds that for the sum of the bimolecular channels, 2.4 X 10-15 versus 1.9 X 10-15 cm3 molecule-1 s-1. 

Beyond the binary systems. Spectroscopic signatures arising from more than just two interacting atoms or molecules were also discovered in the pioneering days of the collision-induced absorption studies. These involve a variation with pressure of the normalized profiles, a(a>)/n2, which are pressure invariant only in the low-pressure limit. For example, a splitting of induced Q branches was observed that increases with pressure the intercollisional dip. It was explained by van Kranendonk as a correlation of the dipoles induced in subsequent collisions [404]. An interference effect at very low (microwave) frequencies was similarly explained [318]. At densities near the onset of these interference effects, one may try to model these as a three-body, spectral signature , but we will refer to these processes as many-body intercollisional interference effects which they certainly are at low frequencies and also at condensed matter densities. 

Energizing collisions are those with sufficient energy that molecule C obtains enough internal energy that it goes on to react. In the limit of sufficiently low pressure, the rate of energizing collisions becomes small relative to the rate of reaction of an energized molecule. As a result, in the low-pressure limit, the rate of reaction becomes bimolecular, that is, proportional to the C-M collision rate. 

At pressures between these high- and low-pressure limits, the so-called pressure fall-off regime, the rate constant of Eq. 9.105 applies. It is convenient to introduce a dimensionless parameter pr (a reduced pressure ), defined as... 

Other blending functions F have been suggested that modify the transition from the high-pressure rate constant to the low-pressure limit. Gilbert, Luther, and Troe [142] suggested the following functional form for F ... 

Therefore the low-pressure limiting form for the bimolecular rate constant is... 

Note that Mmol.0 is a bimolecular rate constant, with units m3/(mol-s). To describe the transition between the high- and low-pressure limits, we use... 

It is easy to determine the high- and low-pressure limits of the stabilization rate constant. In the limit of high pressure,... 

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