J Phys Chem. 1990, 94, 3328-3332
3328
NonequMrium Kinetics of Btmolecular Exchange Reactions. 3. Application to Some Combustion Reactions H.Teitelbaum Department of Chemistry, University of Ottawa, Ottawa, Canada K I N 6N5 (Received: October 6 , 1989; In Final Form: January 24, 1990)
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The elementary chemical reactions 0 H2 OH H and OH + H2 H 2 0 H are examined for self-disturbance of the vibrational level population distribution during steady-statethermal reaction. Calculations are performed using an analytical solution of the master equation that accounts for reaction from levels v = 0, 1, and 2 of H2 and for vibrational-translational (V-T) energy transfer by H1. OH, 0, and Ar. Microscopic rate constants are taken from the literature. It is found that the nonequilibrium rate coefficient, k , differs from the equilibrium rate coefficient, k,, by no more than 20% over the temperature range 300-4000 K and the composition range [O]/[H2] or [OH]/[H2] = 0.01-1 and [O]/[Ar] or [OH]/[Ar] = 0-m, but that the population of u = 2 is reduced by as much as a factor of 5 . The effect is most pronounced at intermediate temperatures -2000-3000 K, at intermediate ratios of [O]/[H2] or [OH]/[H2],and at high values of [O]/[Ar] or [OH]/[Ar]. The analytical expression for k / k , makes routine evaluation of nonequilibrium kinetics of many elementary reactions feasible now.
Introduction The rate of an elementary thermal chemical reaction A + BC - A B + C
simple mechanism, such as the following one thought to be sufficient for describing the first H 2 / 0 2 explosion limit, becomes a nightmare to analyze:
rate = k[A][BC]
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OH(u)
OH(U) H 2 ( ~ 9
HZO
reactants
can be written as
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(1)
where k is termed the thermal rate coefficient. Equation 1 is an empirical relationship, and k is an average over the reagent electronic, translational, rotational, and vibrational energies. Experimenters tend to suppose that k is a pure constant, depending only on temperature. Modelers of complex reaction mechanisms rely on the existence of a unique set of such parameters and on the correctness of the form of ( I ) . However, in actual fact, whenever a chemical reaction proceeds primarily via a specific excited level of the reagent, that level population becomes selectively depleted. Just as in the case of unimolecular reactions with weak colliders at low pressure,’ a bimolecular reaction will occur with non-Boltzmann or “nonequilibrium” distributions of reagent levels.2 Unlike the case of unimolecular reactions, though, a bimolecular rate coefficient depends on the concentration of all species in the mixture playing a role in relaxing the reagentsmaking k not ~ n i q u e . ~Therefore it is important that we systematically examine elementary reactions, in order to identify the conditions where nonequilibrium effects are importantparticularly involving the vibrational degree of freedom. A rule of thumb that indicates if a given elementary reaction will proceed under vibrationally nonequilibrium conditions requires the four following conditions to be satisfied: (a) reaction is “fast”; (b) the reagents relax “slowly”; (c) the reaction is enhanced by vibrational energy; and (d) the temperature is high enough 2 ‘ / , ( h u l k ) so that a significant fraction of vibrationally excited states are populated and can be potentially depleted. I f some or all of these requirements are met, then ( 1 ) is not valid under these nonequilibrium conditions; Le., k is not a true constant in concentration or time. Consequently, from the point of view of a modeler of complex reaction mechanisms, it becomes necessary to measure many state-selected rate constants instead of a single thermal rate coefficient. Also, it becomes necessary to solve a far more complicated set of simultaneous differential rate equations than otherwise. The final result will be a complicated set of falloff curves depending on the chemical environment (amount and kind of diluent M and noninert relaxers). A ( I ) Troe, J . J . Chem. Phys. 1977, 66,4745. ( 2 ) Bowes, C.; Mina, N.; Teitelbaum. H . J . Chem. SOC.,Farodoy Trons. 2, submitted for publication. (3) Teitelbaum. H . J . Chem. Soc.. Faraday Trans. 2 1988, 84, 1889.
0022-3654/90/2094-3328$02.50/0
+H OH(U) + 0 O H ( U )+ H +
+ 02(~’9 0 + H~(u’)
H
+
H
(2)
(3)
(4) (5)
wall
(6)
In combustion reactions temperatures up to 3000 K can easily be attained. If 3 levels for each of H2, 02,and O H play a role in the above mechanism, then there are 31 rate constants to determine. At this point one has a choice: (a) solve 12 differential equations, some of them having a large number of terms, e.g., 28 for H, or (b) for each effective “elementary” reaction write an improved rate law, allowing us to reduce the complexity to five differential equations and five rate coefficients. It is the latter strategy that we choose, keeping in mind that a full combustion mechanism of a small multimode species, such as CH,, consists of about 150 elementary reactions4 (and consequently involves myriads of rate constants). How this is done is the subject of the present study. I n the first paper of this series of studies3 we examined the reaction Br HCI H B r CI, and we found surprisingly large effects. A simple model led to an improved rate equation of the form
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(7)
where f ( T ) is a simple function of reactive and inelastic rate constants and R is the major relaxer of BC. The second paper of this series examined the reaction H + H 2 H 2 H and its isotopic analogues2 Several approximations were removed, and a real calculation was performed using accurate microscopic rate constants. The essential validity of (7) was confirmed. The present study continues with an examination of two key combustion reactions, (3) and ( S ) , with the aim of identifying the conditions where nonequilibrium effects play a role. Even if no role is played in laboratory experiments, it is important to know how to extrapolate to real combustion systems. The main quantity we calculate is the phenomenological rate coefficient, k.
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(4) Lutz, A . E.; Kee, R. J.: Miller, J. A,; Dwyer, H. A,; Oppenmheim, A. K . Twenty-Second Symposium (International) on Combustion; The Combustion Institute: Pittsburgh. 1988; p 1683.
Q 1990 American Chemical Societv
Kinetics of Bimolecular Exchange Reactions
The Journal of Physical Cheniistry, Vol. 94, No. 8, 1990 3329
Solution of the Steady-State Master Equation We treat each elementary reaction as isolated from all others in a complex mechanism, and we decompose it into its vibrationally reactive and inelastic components: A + BC(u) AB + C (8) R
+ BC(U)
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+
R
+ BC(o-I)
(9)
where R represents all relaxer species present, e.g., A, BC, inert diluent M, etc. The rate constant for process 8 is denoted k,; that for process 9 is denoted ku,wl.u can take on any value, but in practice is limited to a quantum number V beyond which no microscopic rate constants are available. Then the differential equation describing the rate of change of the population, Nu,of level u is given by
The above equations are mathematically exact solutions of the master equation and make use of experimentally precise or theoretically accurate rate constants, as far as they are available. A desirable improvement would be the including of vibrationalvibrational (V-V) energy transfer processes, (whose rate constants - i e generally not available). We suspect, though, that V-V 'messes will not alter the qualitative conclusions because, at least for harmonic or nearly harmonic oscillators, they cannot affect the total vibrational energy of BC. At best, they can lead to the establishment of a vibrational temperature which, as seen earlier2 and below, seems to be established by the processes we have already considered. Finally, it is important to realize that if the reverse reaction is fast enough, it may also set in even before a steady-state distribution is established for BC, and then we may not treat the reaction A BC AB C in isolation. We presume, though, that at least in the laboratory, rate constants are measured for processes which, for some conditions, have indeed been isolated. Nevertheless, we always check for this using the criterion developed
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k , k [ A ][BC]
