3 The rate theory of unimolecular reaction
Unimolecular reaction is elementary reaction with only
one reactant molecule. It is obvious that a single molecule at
ground state will not undergo any reaction except that it was
activated by energy of some types.
A  A*  P
For unimolecular reaction, the puzzling question is that
how do reactant molecules acquire necessary activation
energy.
At early 19th century, it was found that all unimolecluar
reaction, such as gaseous decomposition and isomerization, is
of first-order.
For bimolecular reaction, before reaction takes place,
reactants must collide with each other to acquire enough
activation energy. This may lead to a conclusion that all
gaseous reaction should be second-ordered.
In 1919, Perrin proposed that, for unimolecular reaction, the
reactant molecule was activated by absorption of infrared
radiation from the container or other molecules – radiation
activation theory.
k1
radiation
A 
 A* 
P
dP
 k1[A* ]
dt
The result obtained from the radiation activation theory is in
good accordance with the early observation of kinetic
characteristics of the unimolecular reaction, i.e., unimolecular
reaction is of first-order.
Problems proposed by Langmuir :
1) the energy of the infrared radiation of the wall of
container is very low and is not sufficient for activation.
2) some reactant molecules do not have an absorption
band in the wave-length region of infrared radiation.
3) Latterly, it was observed
that the unimolecular reaction
is of second-order at low
pressure and first-order at
high pressure.
k∞
A
In 1921, Christiansen postulated that the activation of
molecules in unimolecular reaction is also through
intermolecular collision.
For this theory, it is easy to explain the second-ordered
feature of unimolecular reaction at low pressure but it is
impossible to gave any reasonable explanation to its firstordered feature at high pressure.
In 1922 Lindemann and Christiansen postulated that the
activated molecules react long after the collision. There is a
time lag between activation and reaction. During the stay of
activated molecules, some of them may lose their energy due
to the further collision (deactivation). Only part of the
activated molecules form product.
3.1 Lindemann mechanism
1) Activation through collision:
A  M 
 A*  M
k1
2) deactivation through collision
during time lag
k1
A*  M 
A  M
3) decomposition of activated
molecule after time lag
A 
P
*
k2
It is obvious that the activation and the deactivation
processes are bimolecular reactions and the decomposition
process is truly a unimolecular reaction.
3.2 Rate equation of Lindemann mechanism
The rate of the reaction can be given as:
d [A* ]
r
 k2 [A* ]
dt
To determine the concentration of activated molecules, the
stationary-state approximation should be used.
the stationary-state approximation
If A* is very active, its concentration is very low and after a
short time, its concentration does not vary with time:
d [A* ]
0
dt
Lindemann mechanism
d [A* ]
 k1[A][M]  k1[A* ][M]  k2 [A* ]
dt
k1[A][M]
d [A* ]
*
 0 [A ] 
k1[M]  k2
dt
*
d [A ]
r
 k2 [A* ]
dt
k1k2 [A][M]
r
k1[M]  k2
This equation suggests that unimolecular reaction have no
definite reaction order.
discussion
k1k2 [A][M]
r
k1[M]  k2
Case I:
When the pressure of the system is low enough to satisfy:
k1[M]  k2
r  k1[A][M]
When the pressure and the concentration of A and M is low,
the collision frequency is low and thus the deactivation is rare.
discussion
k1k2 [A][M]
r
k1[M]  k2
Case II:
At high pressure, the collision frequency is high and thus the
deactivation is fast. When the rate of deactivation is much
higher than the rate of decomposition
k1[M]  k2
k1k2 [A][M]
r
 k '[A]
k1[M]
This is an expression for the pseudo-first-order reaction.
k1k2
r
[A]  k [A]
k1
r  k [A]
r
k1k2 [A][M]
k1[M]  k2
r  k1[A][M]
The Lindemann mechanism can satisfactorily explain the
phenomena of the unimolecular reaction.
experimental
Lindemann
The quantitative calculation of Lindermann mechanism was
poor. To modify Lindemann mechanism, Hinshelwood used
theory of energy partition to calculate k1, but the result is
still not satisfactory.
3.3 RRKM theory
In 1930s, Rice-Ramsperger-Kassel
established another mechanism for
unimolecular reaction.
In 1950, Marcus innovated the
mechanism and introduced TST into
the treatment of unimolecular reaction.
Rudolph A. Marcus
1992 Noble Prize
USA
Theories of electron transfer
In which A is energized molecules
They postulated that, before A* decomposed to product, the
energy they attained must be transferred to the chemical bond
to break and the molecules must attain the transition state (A,
energized molecule) with special configuration similar to
product.
The time needed for configuration transformation corresponds
to the time lag proposed by Lindemann.
experimental
RRKM