The low pressure fall-off

Marcus approximates by evaluating the ( , ,) in the denominator with the mean centrifugal correction, as detailed above, using + = — 0 + AC j). The numerator may then be summed over j as previously, giving 

This form is convenient for computation, requiring first the evaluation of J2(e) and W(E + ), and hence ft(E) over the energy range. W(t+) is, of course, the total states of the complex for a thermal energy of +, including vibration and internal rotation. 

If necessary, the approximation to the centrifugal effect can be avoided and the individual fe(e, e j) evaluated over the range of j, with a final summation. 

In studies of thennal unimolecular reactions, it is feuni/ generally of interest, and so we have 

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To evaluate the state numbers and densities, a structure and set of frequencies have to be chosen for the transition complex. Provided the choice is made to match the experimentally derived Q lb Qrot /Qvib Qrot (obtained from the k , measured as a function of T), the computed k ( ) turn out to be insensitive to the details of the model that is selected. It means that, to a first approximation, the lifetimes of the excited molecules and the form of the low pressure fall-off are functions only of the entropies of the parent and its transition complex and that there are no adjustable parameters. This is advantageous to those whose aim is to calculate lifetimes, but evidently comparison of theory with experiment will not, in general, yield detailed information concerning the structures of the transition state. We return to these aspects later and presently consider the problem of evaluating the state densities, supposing that the structures and frequencies are known. 

The relative efficiency per collision for deactivation of excited molecules in thermal reactions increases with the number of atoms in the collider, but reaches a constant limit when this number exceeds about 12. This has been demonstrated for many thermal reactions by studying the low pressure fall-off. It may be noted from eqn. (10) that plots of k /k against pressure for different inert gases should comprise a set of curves dispersed along the pressure axis according to the various efficiencies of deactivation per unit pressure. The relative efficiency per collision can be derived by calculating the collision frequency, Z, with a hard sphere or Lennard—Jones model. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

Using mass-spectrometric techniques Vreeland and Swinehart studied the decomposition down to pressures below 10 torr. They found the normal pressure fall-off down to about 10 torr, but at lower pressures the rate coefficients were abnormally high and were essentially independent of pressure. By studying this phenomenon in packed vessels they concluded that the reaction was not becoming a surface process at these pressures, and that it occurs by two competing mechanisms. It is possible that at low pressures a chain reaction becomes important, the surface bringing about initiation and termination so that there is no overall surface effect further experimental work is needed to test this possibility. 

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. 

When k2c 0.2 torr. However, at the low pressures employed by Rolfes et al.363 one would expect361 a pressure dependence, viz. I0 = k2a [O] [SO] [M], which was not observed by those workers. The fall-off to a third-order dependence was perhaps masked by emission from SO generated at the reaction cell surface365. 

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. 

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. 

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) ... 

D.S. Ross et al, Study of the Basic Kinetics of Decomposition. . , AFRPL-70-29, SRI, Menlo Park, Contract F04611-69-C-0096 (1970) [From their work the authors conclude that there is no way to distinguish between the very low pressure pyrolysis reactions UDMH - NH3+CH2 N-CH2 (1) and UDMH ->(CH3)2N. +.NH2 (2). The reported pyrolysis fall-off rate constants kx are listed as log k(1 = 13.0 —... 

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... 

At low relative pressures p/p0 or thin adsorbate films, adsorption is expected to be dominated by the van der Waals attraction of the adsorbed molecules by the solid that falls off with the third power of the distance to the surface (FHH-regime, Eq. 3a). At higher relative pressures p/p0 or thick adsorbate films, the adsorbed amount N is expected to be determined by the surface tension y of the adsorbate vapor interface (CC-regime, Eq. 3b), because the corresponding surface potential falls off less rapidly with the first power of the distance to the surface, only. The cross-over length zcrit. between both regimes depends on the number density np of probe molecules in the liquid, the surface tension y, the van der Waals interaction parameter a as well as on the surface fractal dimension ds [100, 101] ... 

In the case of dimethyl malcate the rate was observed to fall off with pressures near 100 mm Hg, and a side reaction in which CO2 was evolved also took place. The region of pressure fall-off together with the low frocpiency factor is in dirc( t contradiction to the Slater theory or any reasonable variant. It is the feeling of the author that these are chain sensitized reactions. 

Porter et al flashed I2-NO mixtures and calculated a third-order rate coefficient for recombination of iodine atoms by nitric oxide of A = 3 x 10 P.mole. sec At very high NO pressures, the recombination rate falls off concurrently, a new transient intermediate appears in the absorption spectrum which is considered to be NOI. Porter et al found nitric oxide to be 20 times more effective than I2 in the recombination of iodine atoms. This is about a factor of ten less than observed by Engleman and Davidson but no account of possible NOI absorption was included in Engleman and Davidson s work. This could account for part of the discrepancy but not all of it. Because of the fall-off of the apparent third-order rate coefficient at high NO pressures, it is apparent that this mechanism is valid only at very low pressures. For high pressures bimolecular reactions of NOI must become rate controlling. 

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. 

The low pressure limit is identical to that of the Hinshelwood theory because the same expression for the density of states has been used. However, the fall-off is calculated from Eq. (24), which can be written in the form... 

The most obvious difficulty is that the numbers of oscillators in the classical RRK theory required to fit both the low pressure limit and the fall-off are smaller than the actual number of oscillators in the molecule, typically by a factor of about 2, although there can be wide variations [14]. Thus 14 oscillators have been used in the calculation presented in Fig. 5, but cyclobutane actually contains 30 oscillators. The problem is that the use of classical statistical mechanics overestimates the populations of excited states relative to the ground state. For example, in classical theory the mean energy of an oscillator is kT,... 

The parametric representation of unimolecular rate coefficients, both at the low pressure limit and in the fall-off region has been the result of a monumental work by Troe and his coworkers [67-71]. This work is tremendously useful for applications of unimolecular reactions to complex systems, for example to combustion or to atmospheric chemistry, as it permits accurate representation of the fall-off without having to to a time-consuming and difficult Master Equation calculation. 

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