This document summarizes Rice-Ramsperger-Kassel-Marcus (RRKM) theory, which describes the rates of unimolecular reactions. RRKM theory originated from Lindemann-Christiansen theory and considers how energy is distributed among vibrational modes. During the 1950s-1952, R.A. Marcus merged transition state theory with RRK theory. RRKM theory accounts for individual vibrational frequencies and classifies energy as either active or inactive. It can explain high pre-exponential factors and has been used to study reactions like isomerization, though limitations include uncertainty in activated complex vibrational frequencies.

1. SEMINAR ON
MSc I INORGANIC CHEMISTRY
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2. Contents
 Introduction
 Rice-Ramsperger-Kassel-Marcus (RRKM) Theory
 Energy scheme for RRKM theory
 Significance
 Applications
 Limitations
 Bibliography

3. Introduction
 Rice-Ramsperger-Kassel-Marcus (RRKM) Theory traces its
origin from Lindemann-Christiansen approach to
unimolecular reactions.

4. Assuming steady state approximation,
 The difficulty with the original Lindemann-Christiansen
hypothesis was that the first order rates are maintained down
to lower concentrations than the theory appeared to permit.
 This difficulty was overcome successfully by Hinshelwood.

5.  Hinshelwood modified this treatment by considering
collision theory:

6.  Rice-Ramsperger-Kassel (RRK) considered the energy
dependence.

7.  Here M is any molecule including A, that can transfer energy
to A when collision occurs.
 is the energized molecule and is the activated molecule.
 According to RRK, a molecule is treated as a system of loosely
coupled oscillator, where a amount of energy is distributed
among normal modes of vibrations.
 During vibrations, energy get transferred from one vibrational
level to next vibrational level randomly.
 And if one of the vibrational mode get sufficient energy for
activation then it will lead to product formation.

8. Rice-Ramsperger-Kassel-Marcus (RRKM) Theory
 During 1951-52, R.A. Marcus merged Transition State Theory (TST) with
RRK theory and came up with Rice-Ramsperger-Kassel-Marcus
(RRKM) Theory.
 Marcus followed the same mechanism as that in RRK theory but considered the
energy to be distributed in active (vibrations) and in inactive (translations)
states.
 In RRKM theory, the individual vibrational frequencies of the energized species
and activated complexes are considered.

9.  Account is taken on the way various normal modes of
vibration and rotation contribute to reaction and allowance is
made for zero point energies.
 The total energy contained in the energized molecule is
classified as either active or inactive (also referred to as
adiabatic).
 The inactive energy is the energy that remains in the same
quantum state during the course of reaction and, therefore,
cannot contribute to the breaking of bonds.
 The zero point energy is inactive, as is the energy of overall
translation and rotation, since this energy is preserved as such
when the activated molecule A# is formed.

10.  Vibrational energy and the energy of internal rotations are
active.
 In RRKM theory, the distribution function ∫ (𝜀*)d 𝜀* is
calculated using quantum statistics and is given by
Where 𝜌(𝜀*) is the density of states (DOS) having energy
between 𝜀* and 𝜀*+ d 𝜀* .
 Density of states (DOS) is defined as the number of states per
unit energy range.

11.  The denominator of this equation is the partition function
relating to the active energy contributions.
 According to the RRKM theory, the rate constant k2(𝜀*) is
given by,
Where 𝓁# is the statistical factor and ΣP(𝜀#
active) is the number of
vibration-rotation quantum states for the activated molecule
corresponding to all energies upto and including 𝜀#
active .

12. The factor Fr is introduced to correct for the fact that the
rotations may not be the same in the activated molecule as in the
energized molecule.
 A noteworthy feature of the RRKM theory is that it leads to the
same expression for the limiting high pressure unimolecular
(first-order) rate constant as is given by Transition State
Theory (TST):
Where q# and qi are the partition functions for the activated and
initial states.

13.  The energy scheme for the RRKM mechanism:

14. Significance
 RRKM theory can explain the abnormally high pre-
exponential factors that are sometimes observed.
 In order to use RRKM theory for detailed calculations, we
must decide on models for energized and activated molecules.
 Vibrational frequencies for the various normal modes must be
estimated and decision made as to which energies are active
and which are inactive.
 Numerical methods are used to calculate the rate constants k1
at various concentrations.

15. Applications
 The reactions which have been successfully investigated using
RRKM theory include;
• Isomerization of cyclopropane
• Isomerization of cyclobutane
• Dissociation of cyclobutane into two ethylene molecules
 The Isomerization of cyclopropane to propylene was
the first unimolecular gaseous reaction investigated in the
1920s.

16. Limitations
 A major difficulty in applying RRKM theory is that the
vibrational frequencies of the activated complexes usually
cannot be estimated very reliably and there is evidence for
non-RRKM behavior.
 In nonthermal activated unimolecular reactions, it has been
found that the translational energy distribution of reaction
fragments is non-statistical, contrary to the predictions of the
RRKM theory.
 This implies that not all degrees of freedom participate in the
fragmentation of the complex.

17. Bibliography
 Physical Chemistry by Puri, Sharma, Pathania

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