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.
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.
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.