RRKM theory (original) (raw)

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Microcanonic transition state theory of unimolecular reactions

The Rice–Ramsperger–Kassel–Marcus (RRKM) theory is a theory of chemical reactivity.[1][2][3] It was developed by Rice and Ramsperger in 1927[4] and Kassel in 1928[5] (RRK theory[6]) and generalized (into the RRKM theory) in 1952 by Marcus[7] who took the transition state theory developed by Eyring in 1935 into account. These methods enable the computation of simple estimates of the unimolecular reaction rates from a few characteristics of the potential energy surface.

Assume that the molecule consists of harmonic oscillators, which are connected and can exchange energy with each other.

• Assume the possible excitation energy of the molecule to be E, which enables the reaction to occur.
• The rate of intra-molecular energy distribution is much faster than that of reaction itself.
• As a corollary to the above, the potential energy surface does not have any "bottlenecks" for which certain vibrational modes may be trapped for longer than the average time of the reaction

Assume that A* is an excited molecule:

A ∗ → k ( E ) A ‡ → P {\displaystyle A^{*}{\xrightarrow {k(E)}}A^{\ddagger }\rightarrow P}

where P stands for product, and _A_‡ for the critical atomic configuration with the maximum energy _E_0 along the reaction coordinate.

The unimolecular rate constant k u n i {\displaystyle k_{\mathrm {uni} }} is obtained as follows:[8]

k u n i = 1 h Q r Q v ∫ E 0 ∞ d E ∑ J = 0 ∞ ( 2 J + 1 ) G ‡ ( E , J ) exp ( − E k b T ) 1 + k ( E , J ) ω , {\displaystyle k_{\mathrm {uni} }={\frac {1}{hQ_{r}Q_{v}}}\int \limits _{E_{0}}^{\infty }\mathrm {d} E\sum _{J=0}^{\infty }{\frac {(2J+1)G^{\ddagger }(E,J)\exp \!\left({\frac {-E}{k_{b}T}}\right)}{1+{\frac {k(E,J)}{\omega }}}},}

where k ( E , J ) {\displaystyle k(E,J)} is the microcanonical transition state theory rate constant, G ‡ {\displaystyle G^{\ddagger }} is the sum of states for the active degrees of freedom in the transition state, J {\displaystyle J} is the quantum number of angular momentum, ω {\displaystyle \omega } is the collision frequency between A ∗ {\displaystyle A^{*}} molecule and bath molecules, Q r {\displaystyle Q_{r}} and Q v {\displaystyle Q_{v}} are the molecular vibrational and external rotational partition functions.

• Transition state theory

1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Rice–Ramsperger–Kassel–Marcus (RRKM) theory". doi:10.1351/goldbook.R05391
2. ^ Di Giacomo, F. (2015). "A Short Account of RRKM Theory of Unimolecular Reactions and of Marcus Theory of Electron Transfer in a Historical Perspective". Journal of Chemical Education. 92 (3): 476–481. Bibcode:2015JChEd..92..476D. doi:10.1021/ed5001312.
3. ^ Lindemann, F. A.; Arrhenius, S.; Langmuir, I.; Dhar, N. R.; Perrin, J.; Mcc. Lewis, W. C. (1922). "Discussion on ?the radiation theory of chemical action?". Transactions of the Faraday Society. 17: 598–606. doi:10.1039/TF9221700598.
4. ^ Rice, Oscar Knefler; Ramsperger, Herman Carl (1927), "Theories of unimolecular gas reactions at low pressures", Journal of the American Chemical Society, 49 (7): 1617–1629, doi:10.1021/ja01406a001
5. ^ Kassel, Louis Stevenson (1928), "Studies in Homogeneous Gas Reactions I", The Journal of Physical Chemistry, 32 (2): 225–242, doi:10.1021/j150284a007
6. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Rice–Ramsperger–Kassel (RRK) theory". doi:10.1351/goldbook.R05390
7. ^ Marcus, Rudolph A. (1952), "Unimolecular Dissociations and Free Radical Recombination Reactions" (PDF), J. Chem. Phys., 20 (3): 359–364, Bibcode:1952JChPh..20..359M, doi:10.1063/1.1700424
8. ^ J. I. Steinfeld; J. S. Francisco; W. L. Hase (1998). Chemical Kinetics and Dynamics (2 ed.). Prentice Hall. ISBN 978-0-13737123-5.

• An RRKM online calculator