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Bimolecular Master Equations for a Single and Multiple Potential Wells with Analytic Solutions

Ghaderi, Nima (2018) Bimolecular Master Equations for a Single and Multiple Potential Wells with Analytic Solutions. Journal of Physical Chemistry A, 122 (14). pp. 3506-3534. ISSN 1089-5639. https://resolver.caltech.edu/CaltechAUTHORS:20180327-155535091

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Abstract

The analytic solutions, that is, populations, are derived for the K-adiabatic and K-active bimolecular master equations, separately, for a single and multiple potential wells and reaction channels, where K is the component of the total angular momentum J along the axis of least moment of inertia of the recombination products at a given energy E. The analytic approach provides the functional dependence of the population of molecules on its K-active or K-adiabatic dissociation, association rate constants and the intermolecular energy transfer, where the approach may complement the usual numerical approaches for reactions of interest. Our previous work, Part I, considered the solutions for a single potential well, whereby an assumption utilized there is presently obviated in the derivation of the exact solutions and farther discussed. At the high-pressure limit, the K-adiabatic and K-active bimolecular master equations may each reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high-pressure limit expressions) for bimolecular recombination rate constant, for a single potential well, and augmented by isomerization terms when multiple potential wells are present. In the low-pressure limit, the expression for population above the dissociation limit, associated with a single potential well, becomes equivalent to the usual presumed detailed balance between the association and dissociation rate constants, where the multiple well case is also considered. When the collision frequency of energy transfer, Z_(LJ), between the chemical intermediate and bath gas is sufficiently less than the dissociation rate constant k_d(E′J′K′) for postcollision (E′J′K), then the solution for population, g(EJK)+, above the critical energy further simplifies such that depending on Z_(LJ), the dissociation and association rate constant k_r(EJK), as g(EJK)+ = k_r(EJK)A·BC/[Z_(LJ)+k_d(EJK)], where A and BC are the reactants, for example, relevant for O_3 formation from O + O_2 + Ar up to ∼100 bar; otherwise, additional contributions from postcollision are present and especially relevant at high pressures. In the aforementioned regime Z_(LJ) < k_d(E′J′K) the physical connection of recombination rate constants, k_(rec) based on either utilizing population from the master equation approach or a collision based bimolecular RRKM theory is traced and elucidated analytically that the rate constants are equal. Recombination rate constants, k_(rec), based on the population, are also given and considered for an adiabatic or active K. For example, for O_3 formation in Ar bath gas, the K-adiabatic-based k_(rec) for O_3 yields 4.0 × 10^(–34) cm^6 molecule^(–2) s^(–1) at T = 300 K and 1 bar, in agreement with the experimental value, where the contribution from the population of metastable ozone is discussed and the adiabaticity of Khighlighted. A facile numerical implementation of the formalism for g(EJK) and k_(rec) for O_3 is noted. The application of the expressions to ozone recombination as a function of pressure and temperature is given elsewhere, beyond the selection considered here, for studying the physical features, including the contributions from the K and intermolecular energy transfer to the k_(rec), and the puzzles reported from experiments for this reaction.


Item Type:Article
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https://doi.org/10.1021/acs.jpca.7b09244DOIArticle
https://pubs.acs.org/doi/10.1021/acs.jpca.7b09244PublisherArticle
ORCID:
AuthorORCID
Ghaderi, Nima0000-0002-6257-1373
Additional Information:© 2018 American Chemical Society. Received: September 16, 2017; Revised: December 15, 2017; Publication Date (Web): March 27, 2018. It is a great pleasure to acknowledge Prof. R. A. Marcus for the valuable and constructive comments, and kindly noting a great teacher and a remarkable mind. The critically valuable comments of Dr. S. J. Klippenstein are appreciated very much. The very thoughtful and the insightful comments of the anonymous reviewer carrying depth and knowledge are greatly appreciated and duly noted. The various thoughtful comments and/or support of Profs. W. A. Goddard III, H. B. Gray, S. I. Chan, J. Troe, B. Widom, and T. F. Rosenbaum are kindly noted and appreciated very much. The author gratefully thanks Mrs. M. Vojdani (grandmother) for the inquiries, the stimulating discussions regarding bimolecular reactions, and the kind encouragement, as noted with much appreciation. It is a pleasure to gratefully acknowledge support from CalTech and kindly acknowledge the NSF. The author declares no competing financial interest.
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Issue or Number:14
Record Number:CaltechAUTHORS:20180327-155535091
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180327-155535091
Official Citation:Bimolecular Master Equations for a Single and Multiple Potential Wells with Analytic Solutions. Nima Ghaderi. The Journal of Physical Chemistry A 2018 122 (14), 3506-3534. DOI: 10.1021/acs.jpca.7b09244
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85469
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:28 Mar 2018 18:01
Last Modified:03 Oct 2019 19:31

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