{{Short description|Microcanonic transition state theory of unimolecular reactions}} The '''Rice–Ramsperger–Kassel–Marcus''' ('''RRKM''') '''theory''' is a theory of chemical reactivity.<ref>{{GoldBookRef|title=Rice–Ramsperger–Kassel–Marcus (RRKM) theory|file=R05391}}</ref><ref name=giacomo>{{Cite journal | doi = 10.1021/ed5001312| title = A Short Account of RRKM Theory of Unimolecular Reactions and of Marcus Theory of Electron Transfer in a Historical Perspective| journal = Journal of Chemical Education| volume = 92| issue = 3| pages = 476–481| year = 2015| last1 = Di Giacomo | first1 = F. |bibcode = 2015JChEd..92..476D }}</ref><ref name=lindemann>{{Cite journal | doi = 10.1039/TF9221700598| title = Discussion on ?the radiation theory of chemical action?| journal = Transactions of the Faraday Society| volume = 17| pages = 598–606| year = 1922| last1 = Lindemann | first1 = F. A.| last2 = Arrhenius | first2 = S. | last3 = Langmuir | first3 = I. | last4 = Dhar | first4 = N. R.| last5 = Perrin | first5 = J.| last6 = Mcc. Lewis | first6 = W. C.}}</ref> It was developed by Rice and Ramsperger in 1927<ref name=Rice1927>{{citation | last1 = Rice | first1 = Oscar Knefler | year = 1927 | title = Theories of unimolecular gas reactions at low pressures | journal = Journal of the American Chemical Society | volume = 49 | issue = 7 | pages = 1617–1629 | doi = 10.1021/ja01406a001 | last2 = Ramsperger | first2 = Herman Carl | bibcode = 1927JAChS..49.1617R }}</ref> and Kassel in 1928<ref name=Kassel1928>{{citation | last = Kassel | first = Louis Stevenson | year = 1928 | title = Studies in Homogeneous Gas Reactions I | journal = The Journal of Physical Chemistry | volume = 32 | issue = 2 | pages = 225–242 | doi = 10.1021/j150284a007}}</ref> (RRK theory<ref>{{GoldBookRef|title=Rice–Ramsperger–Kassel (RRK) theory|file=R05390}}</ref>) and generalized (into the RRKM theory) in 1952 by Marcus<ref name=Marcus1952>{{citation | last = Marcus | first = Rudolph A. | year = 1952 | title = Unimolecular Dissociations and Free Radical Recombination Reactions | journal = J. Chem. Phys. | volume = 20 | issue = 3 | pages = 359–364 | doi = 10.1063/1.1700424|bibcode = 1952JChPh..20..359M | url = http://authors.library.caltech.edu/11405/1/MARjcp52b.pdf }}</ref> 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.
== Assumption == 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 {{Mvar|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
== Derivation == Assume that {{Math|''A''<sup>*</sup>}} is an excited molecule:
: <math chem>A^{*} \xrightarrow{k(E)} A^{\ddagger} \rightarrow P</math>
where {{Mvar|P}} stands for product, and {{Math|''A''<sup>‡</sup>}} for the critical atomic configuration with the maximum energy {{Math|''E''<sub>0</sub>}} along the reaction coordinate.
The unimolecular rate constant <math>k_\mathrm{uni}</math> is obtained as follows:<ref name="Steinfeld"/>
:<math>k_\mathrm{uni} = \frac{1}{h Q_{r} Q_{v}} \int\limits_{E_{0}}^{\infty} \mathrm dE \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}},</math>
where <math>k(E,J)</math> is the microcanonical transition state theory rate constant, <math>G^{\ddagger}</math> is the sum of states for the active degrees of freedom in the transition state, <math>J</math> is the quantum number of angular momentum, <math>\omega</math> is the collision frequency between <math>A^*</math> molecule and bath molecules, <math>Q_r</math> and <math>Q_v</math> are the molecular vibrational and external rotational partition functions.
==See also== *Transition state theory
==References== {{Reflist|2|refs= <ref name="Steinfeld">{{Cite book|author1=J. I. Steinfeld|author2=J. S. Francisco|author3=W. L. Hase|title=Chemical Kinetics and Dynamics|edition=2|publisher= Prentice Hall|year=1998| isbn=978-0-13737123-5}}</ref> }}
==External links== * [http://phd.marginean.net/rrkm.html An RRKM online calculator] {{Webarchive|url=https://web.archive.org/web/20170708182132/http://phd.marginean.net/rrkm.html |date=2017-07-08 }}
{{Reaction mechanisms}}
Category:Chemical physics Category:Quantum chemistry Category:Molecular physics Category:Chemical kinetics