In the Classroom edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
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Pressure Dependence of Gas-Phase Reaction Rates Stéphanie de Persis, Alain Dollet, and Francis Teyssandier* Institut de Science et de Génie des Matériaux et Procédés, CNRS UPR8521, TECNOSUD, Rambla de la Thermodynamique, F-66100, Perpignan, France; *
[email protected] The Arrhenius equation, which is universally used to describe the temperature dependence of chemical reaction rates, is familiar to any student who has learned the basic concepts of chemical kinetics. In contrast, students generally ignore the pressure dependence of most gas-phase reaction rates; dependence that sometimes is very large. Although the basic principles governing the influence of pressure on the rate of thermally or chemically activated elementary gasphase reactions could be easily introduced in the chemistry curriculum, the pressure dependence of gas-phase reaction rates has never been addressed in this Journal. This short article shows that only simple concepts, mainly taken from activated-complex or transition-state theory, are required to explain and analytically describe the influence of pressure on gas-phase reaction kinetics. Reaction Categories The simplest kind of elementary gas-phase reaction is a unimolecular decomposition reaction. Distinction is usually made between dissociation, elimination, and isomerization reactions: the difference lies in the form of the potential surface. Three types of bimolecular reactions can be distinguished: metathesis reactions, in which an atom or a group of atoms is transferred from one of the reactants to the other (for example: CH4 + •H •CH3 + H2); exchange or disC 6H 6 placement reactions (for example: •H + CH3C6H5 + •CH3); and association reactions, either recombination or
Origin of the Pressure-Dependent Behavior of Reaction Rates It is well established that the pressure dependence of gasphase reaction rates mainly arises from collisional energy transfer. The phenomenon can be understood and described from very simple considerations and concepts that are presented below.
Unimolecular Reactions According to the mechanism proposed by Lindemann– Christiansen (1), unimolecular decomposition reactions (AB → products) involve the following steps: • Collisional energy transfer between reactant (AB) and bath-gas (M) molecules •
collisional activation
AB + M → AB* + M
•
collisional deactivation
AB* + M → AB + M
• Intramolecular Rearrangement •
molecular fragmentation
AB* → products
B
AB
products high-pressure limit
kc
falloff regime
log(bimolecular rate constant)
log(unimolecular rate constant)
A
addition reaction. Termolecular reactions correspond to the simultaneous collision of three species. As the probability of such an event remains generally low, these reactions will not be considered in this article. Among the reactions listed above, only unimolecular and association bimolecular reaction rates are pressure dependent.
(stabilization channel) A + B AB
(dissociation channel) A + B
C + D
high-pressure limit
falloff regime
low-pressure limit
low-pressure limit Pc
log(pressure)
log(pressure) Figure 1. Typical pressure dependence of unimolecular (A) and bimolecular (B) reaction rate coefficients.
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The typical timescales involved either in the collision processes or in the reaction process are significantly different (10᎑13 s and 10᎑9 s, respectively). A consequence of the separation of the collision and reaction timescales is that the rate coefficient for the overall reaction (kuni ) has a dependence on bath-gas pressure. This dependence is governed by the competition between collisional energy transfer and intramolecular rearrangements processes and results in three domains of pressure (see Figure 1A). High-Pressure Limit For sufficiently high pressures (the exact limit depends on the nature of the reactant and bath-gas molecules, and also on temperature) the number of collisions between reactant and bath-gas molecules is so large that energy and angular momentum populations maintain equilibrium Boltzmann distribution. This high-pressure rate coefficient is pressure-independent and information about the collisional activation and deactivation is not required in this domain. Low-Pressure Limit When decreasing pressure, the reactant bath-gas collisions are no longer sufficiently frequent to maintain the Boltzmann distribution of energy and angular momentum. Populations from levels above the critical energy E0 (minimum energy required for an isolated molecule to undergo reaction, i.e., enthalpy difference between transition state and reactant, at 0 K) are depleted by reactions and kuni falls below the high-pressure limit. Below a certain value of pressure, collisional activation and deactivation processes become rate limiting and the rate coefficient is proportional to the bath-gas pressure. Intermediate Domain of Pressure or Falloff Region The pressure domains that correspond to collisional energy transfer (low-pressure domain) and intramolecular rearrangement (high-pressure domain) overlap in the medium range domain of pressure. The pressure dependence of the reaction rate constant is no longer linear in this intermediate domain, which is referred to as the “falloff region”. Isomerization Reactions (AB → BA) Isomerization reactions do not behave exactly as decomposition reactions. While the fragmentation of the energized complex into products is an irreversible elementary process in the case of a decomposition reaction (see above), the energized product BA* of an isomerization reaction can either be stabilized by collisional deactivation or revert to the energized reactant AB* (and even decompose, in the case of multichannel reactions) (2). Thus, the mechanism proposed by Lindemann–Christiansen must be modified as follows for isomerization reactions: • Collisional energy transfer •
collisional activation
AB + M → AB* + M
•
collisional deactivation
AB*+ M → AB + M BA* + M → BA + M
• Intramolecular Rearrangement • •
isomerization (forward) (reverse)
AB* → BA* BA* → AB*
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Bimolecular Reactions Among bimolecular reactions, only those association reactions that form vibrationally excited species leading to stable species by collision deactivation are pressure dependent. An association reaction between two species A and B involves the formation of an energized collision complex AB* that can be stabilized by collision with some chemically inert species (bath gas). This complex may have a chemical activation energy distribution different from the thermal energy distribution because of the energy released by making the new bond. If this energy distribution extends above the barrier for new dissociations, these channels will be in competition with the collisional stabilization channel. Such a reaction is referred to as “chemically activated reaction”. The general pressure dependence behavior of bimolecular reactions is presented in Figure 1B for both the dissociation and stabilization channels. Derivation of Analytical Expressions for the Rate Constants Simple analytical expressions of pressure-dependent reaction rate coefficients are easily derived from the Lindeman– Hinshelwwod theory (1). Sophisticated calculations based on Rice–Ramsperger and Kassel (2–6) theories are also commonly performed to predict the pressure dependence of reaction rate coefficients k with a good accuracy, and several parametrization methods (7–9) based on the Lindemann– Hinshelwood formalism can be used to derive analytical expressions of k from these calculations. Determining the quantity of energy transferred per collision between reactant and bath-gas molecules is essential to perform any calculation of a reaction rate coefficient that is pressure dependent. This is a difficult task that has been the subject of intense experimental and theoretical investigation (1). No accurate model is available to predict the effect of collisional energy transfer and only empirical models can be used for that purpose. For a more detailed review of the derivation of analytical expressions for k(T, P ), parameterization methods, and collisional energy transfer models, the reader is invited to refer to our full article in the Supplemental Material.W WSupplemental
Material A detailed version of this article is available in this issue of JCE Online. Literature Cited 1. Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell Scientific Publications: Oxford, UK, 1990. 2. Dean, A. M. J. Phys. Chem. 1985, 89, 4600–4608. 3. Rice, O. K.; Ramsperger, H. C. J. Am. Chem. Soc. 1927, 49, 1617–1629. 4. Kassel, L. S. J. Phys. Chem. 1928, 32, 225–242. 5. Kassel, L. S. J. Phys. Chem. 1928, 32, 1065–1079. 6. Marcus, R. A. J. Chem. Phys. 1952, 20, 359–364. 7. Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 161–169. 8. Larson, C. W.; Patrick, R.; Golden, D. M. Combust. Flame 1984, 58, 229–237. 9. Oref, I. J. Phys. Chem. 1989, 93, 3465–3469.
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