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Cleaner Combustion pp 549–576Cite as

Statistical Rate Theory in Combustion: An Operational Approach

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Abstract

Statistical rate theory is a valuable tool to rationalize the microscopic mechanisms of elementary chemical steps in the gas phase, to analyze results of kinetic experiments, and to adequately parameterize the temperature and pressure dependence of rate coefficients. We briefly describe the essential elements of statistical rate theory that are relevant for the kinetic characterization of reactions under combustion conditions, emphasizing application aspects. The calculation of rate coefficients for reactions over potential energy barriers and potential energy wells is elucidated. In the former case conventional transition state theory is used, in the latter case the temperature and pressure dependence is described by means of master equations with specific rate coefficients from RRKM theory and the simplified statistical adiabatic channel model. Examples for the different types of reaction are given, and crucial quantities are discussed. The article primarily aims at readers on an intermediate level between graduate students and junior scientists, who are interested in performing practical calculations, and who are looking for a compact presentation of the topic as a guide to the extensive literature.

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Abbreviations

<X>:

Average of quantity X

A :

Vector/matrix containing elements a(E i )/a(E i , E j )

B :

Rotational constant

B cent(q):

Rotational constant for centrifugal motion in SACM

D :

Morse parameter

E i :

ith eigenvector of the matrix J

ΔE SL :

Energy transfer parameter of the stepladder model

<ΔE>:

Average energy transferred per collision (all, up and down)

<ΔE>d :

Average energy transferred per down collision

<ΔE>u :

Average energy transferred per up collision

ΔE z :

Zero-point energy correction in SACM

E 0(i) :

Threshold energy of reaction i

f(E):

Initial distribution of the intermediate in a complex-forming bimolecular reaction

F AM :

Angular momentum coupling factor in SACM

F E :

Density of states correction factor

h :

Planck′s constant

HO:

Harmonic oscillator

J :

Total angular momentum quantum number

J :

Matrix of the master equation, for definition see Eq. 21.3

k i :

Rate coefficient of reaction i

k ∞ i :

High-pressure limiting value of the rate coefficient for reaction i

k B :

Boltzmann′s constant

L i :

Reaction path degeneracy for reaction i

n(E):

Distribution of a reacting species

n s(E):

Steady-state distribution of a reacting species

ñ s(E):

Normalized steady-state distribution of a reacting species

P :

Pressure

P(E′,E) :

Collisional transition probability (density) for a collision E → E′

PST:

Phase space theory

q :

Reaction coordinate/interfragment distance

q e :

Equilibrium distance

q*HIR :

Partition function for hindered internal rotor (local coordinate)

q HO :

Partition function for harmonic oscillator (normal coordinate)

q*HO :

Partition function for harmonic oscillator (local coordinate)

Q :

Partition function

R :

Gas constant

R 1 :

Rate of reaction of a complex-forming bimolecular reaction

RRKM:

Rice, Ramsperger, Kassel, Marcus

SACM:

Statistical adiabatic channel model

T :

Temperature

TST:

Transition state theory

V :

Volume

V(q):

Classical interfragment potential

W i :

Cumulative reaction probability/sum of states for reaction/transition state i

α :

Interpolation parameter of simplified SACM or energy transfer parameter of the exponential down model

β :

Morse parameter

γ c :

Collision efficiency in a chemically activated reaction

ε(q):

Vibrational or rotational quantum in SACM

λ i :

ith eigenvalue of the matrix J

ρ :

Density of states

σ :

Symmetry number

ω :

Collision frequency (unit: s−1)

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Acknowledgments

Financial support by the Deutsche Forschungsgemeinschaft (SFB 606 “Non-Stationary Combustion: Transport Phenomena, Chemical Reactions, Technical Systems”) and by the European Cooperation in Science and Technology (COST, Action CM0901 “Detailed Chemical Kinetic Models for Cleaner Combustion”) is gratefully acknowledged.

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Authors and Affiliations

  1. Institut für Physikalische Chemie, Karlsruher Institut für Technologie (KIT), Kaiserstr. 12, 76131, Karlsruhe, Germany

    Matthias Olzmann

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  1. Matthias Olzmann
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Correspondence to Matthias Olzmann .

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Editors and Affiliations

  1. Laboratoire Reactions et Genie des Proce, ENSIC, Nancy Cedex, France

    Frédérique Battin-Leclerc

  2. Combustion Chemistry Centre, National University of Ireland, Galway, Ireland

    John M. Simmie

  3. Department of Energy Sciences, Division, University of Lund, Lund, Sweden

    Edward Blurock

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Olzmann, M. (2013). Statistical Rate Theory in Combustion: An Operational Approach. In: Battin-Leclerc, F., Simmie, J., Blurock, E. (eds) Cleaner Combustion. Green Energy and Technology. Springer, London. https://doi.org/10.1007/978-1-4471-5307-8_21

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  • DOI: https://doi.org/10.1007/978-1-4471-5307-8_21

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