Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
Subject Chemistry
Paper No and Title 6 and PHYSICAL CHEMISTRY-II (Statistical
Thermodynamics, Chemical Dynamics, Electrochemistry
and Macromolecules)
Module No and Title 20 and Theories of unimolecular reactions- Lindemann
Mechanism
Module Tag CHE_P6_M20
Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
TABLE OF CONTENTS
1. Learning outcomes
2. Introduction
3. Lindemannβs theory
3.1 Theory
3.2 Steady State hypothesis
3.3 Mechanism
3.4 Effect of inert-gas on the rate of the reaction
4. Limitations of Lindemannβs theory
5. Hinshelwood Modification
6. Summary
Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
1. Learning Outcomes
After studying this module, you shall be able to
Learn about Lindemann Mechanism for unimolecular gaseous reactions.
Derive the rate of a unimolecular reaction on the basis of Lindemann Hypothesis.
Learn about Steady State Hypothesis
Learn about limitations of Lindemann approach
Learn about Hinshelwood modification
2. Introduction
In our previous modules, we discussed in detail three approaches relating rate constant to
temperature namely,
a. Arrhenius Equation
b. Collision Theory
c. Transition State Theory
In all the above mentioned methods, we considered bimolecular gaseous reactions.
However, unimolecular gaseous reactions are very common (where molecularity = 1). A
number of gaseous reactions follow first order kinetics. Generally, decomposition reactions
and rearrangement reactions come under this category.
ππ2πΆπ2 β ππ2 + πΆπ2 (Decomposition reaction)
ππ¦πππ β πΆ3π»6 β πΆπ»3 β πΆπ» = πΆπ»2 (Rearrangement reaction)
πΆπ2βπβ 2πΆπ.
In general, for a unimolecular reaction of type
π΄ β π,
Rate will be expressed as,
π ππ‘π = π[π΄] π€βπππ π ππ π‘βπ πππ‘π ππππ π‘πππ‘
Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
In order to interpret the rate constant and rate of the reaction, one needs to know the mechanism
of the reaction.
For bimolecular reactions, as we have studied that the products are formed only when the
reactant molecules come close and collide with each other at one and same time. And only
those collisions which occur between molecules having energy equal or exceeding some
critical energy (known as activation energy) will lead to product formation and such collisions
are known as effective collisions. This implies that these collisions are bimolecular events.
But, in the case of unimolecular reactions, the question arises βHow do molecules in
unimolecular reactions attain this critical energy which leads to product formation?β..OR one
can even frame it this way βif we consider collisions, then the question becomes how the
bimolecular event leads to first order rate law?β.
The first successful explanation for unimolecular reactions was provided by Frederick
Lindemann in 1921 and therefore known as Lindemann theory. This theory is discussed in
detail in our next section.
3. Lindemannβs Theory of Unimolecular Reactions
This is the simplest theory of unimolecular reaction rates and was the first to successfully
explain the observed first order kinetics of many unimolecular reactions.
3.1 Theory
Lindemann hypothesized a situation where
Figure 1: Gaseous molecules interacting with each other in a container
First order gas phase reactions are commonly referred to as Unimolecular Reactions as they involve an elementary unimolecular step in which the reactant molecule changes into the product.
Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
βA reactant molecule A becomes energetically excited by collision with another molecule of
reactant Aβ.
π΄ + π΄ β π΄β + π΄ (πΈππππππ§ππ‘πππ) β¦(1)
This step is called energization step in which molecule A collide with another molecule A to
form energized atom A* which has acquired sufficient vibrational energy to enable it to
isomerize or decompose.
Where A represents inactive molecule while π΄β represents activated molecule. The rate for
reaction (1) becomes,
π ππ‘π =π[π΄β]
ππ‘= π1[π΄]
2 β¦(2)
Now, the energized molecule π΄β can lead to two situations:
Situation I: The energized molecule π΄β might lose its energy by colliding with another
molecule of A.
π΄β + π΄ β π΄ + π΄ (ππ β πππππππ§ππ‘πππ) β¦(3)
This step is called de-energization step. In this step the excited molecule A* is deactivated by
colliding with another A molecule. Equations (1) and (3) respectively represent rapid
equilibrium.
The rate for reaction (3) becomes,
π ππ‘π = βπ[π΄β]
ππ‘= πβ1[π΄
β][π΄] β¦(4)
Situation II: The energized molecule π΄β might lead to product formation i.e., it might
undergo unimolecular decay.
π΄β β π ( πππππ’ππ‘ ππππππ‘πππ) β¦(5)
The rate for reaction (5) becomes,
π ππ‘π = βπ[π΄β]
ππ‘=π[π]
ππ‘= π2[π΄
β] β¦(6)
If the unimolecular step (equation 5) is slow enough to be the rate determining step, then the
Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
overall reaction will follow first order kinetics.
So, the proposed mechanism actually consists of a second order bimolecular reaction (collision
activation step followed by a rate determining unimolecular step).
3.2 Steady State Hypothesis
If we assume that the unimolecular step (equation 5) is slow enough to be the rate determining
step, then we need to know [π΄β] (which is not known) in terms of [π΄]. To do this, we apply
Steady State Approximation / Principle. This principle states that when a short lived reaction
intermediate such as π΄β exists at low concentration in a system, the rate of formation of the
intermediate can be considered to be equal to its rate of disappearance. Applying this to [π΄β]
gives,
π[π΄β]
ππ‘= β
π[π΄β]
ππ‘ β¦(7)
Which means
π1[π΄]2 = πβ1[π΄
β][π΄] + π2[π΄β] β¦(8)
Rearranging equation (8) gives,
[π΄β] =π1[π΄]
2
πβ1[π΄]+π2 β¦(9)
3.3 Lindemann Mechanism
Lindemann assumed that there is time lag between activation and reaction during which the
activated or energized molecules may either react to give product or be de-energized
(deactivated) to ordinary moelcules. Only under these conditions the behavior of unimolecular
reactions can be explained on the basis of bimolecular collisions.
The Lindemann mechanism includes three basic steps:
Step 1: π΄ + π΄ β π΄β + π΄
Step 2: π΄β + π΄ β π΄ + π΄
Step 3: π΄β β π
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
The third step is the product formation step. If the unimolecular step (equation 5) is slow
enough to be the rate determining step, then the overall reaction will follow first order kinetics.
The rate for step 3 becomes,
π ππ‘π = βπ[π΄β]
ππ‘=π[π]
ππ‘= π2[π΄
β]
Substituting [π΄β] from equation (9) in the equation, modifies the above equation to,
π ππ‘π =π[π]
ππ‘=
π2π1[π΄]2
πβ1[π΄]+π2 β¦(10)
So, equation (10) is the rate law expression for unimolecular reaction based on Lindemann
theory. The rate law as given by equation (10) is not first order. Equation (10) predicts two
limiting possibilities.
Case I: πβ1[π΄] β« π2
π ππ‘π =π[π]
ππ‘=π2π1[π΄]
2
πβ1[π΄] β¦.(11)
The above equation simplifies to,
π ππ‘π =π[π]
ππ‘= (
π2π1
πβ1) [π΄] = π[π΄] β¦(12)
From equation (12), one can notice that (π2π1
πβ1) is a constant term and so the reaction becomes
first order with respect to
This case corresponds to a situation in the reaction where the [π΄] is high enough to produce
appreciable deactivation of π΄β (as shown by situation I and situation II above) i.e.,
[π΄] β« (π2
πβ1) β¦(13)
For an ideal gas, ππ = ππ π β¦(14)
Where p is the pressure, V refers to volume, n refers to the number of moles, R is the gas
constant and T represents temperature. Since, we know that number of moles upon volume
gives concentration, equation (14) modifies to,
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
π
π=
π
π π β¦(15)
Taking into account the above equation, [π΄] becomes,
[π΄] =ππ΄
π π β¦(16)
Using equation (16) into equation (13) gives,
ππ΄
π πβ« (
π2
πβ1) β¦(17)
The condition represented by equation (17) is possible only at very high pressures. When the
pressure of the gas is very high then there will be more number of collisions between the
molecules. So the energized molecule will be de-energized more by collisions rather than
decomposing into products.
Case II: πβ1[π΄] βͺ π2
π ππ‘π =π[π]
ππ‘=π2π1[π΄]
2
π2 β¦(18)
The above equation simplifies to,
π ππ‘π =π[π]
ππ‘= π1[π΄]
2 β¦(19)
From equation (19), we find that the reaction becomes second order with respect to A.
This case corresponds to a situation where the energization step becomes so slow as to be rate
controlling. So, the rate determining step is the bimolecular reaction.
[π΄] βͺ (π2
πβ1) β¦(20)
Using ideal gas equation condition as stated above in equation (14), (15) and (16) gives,
ππ΄
π πβͺ (
π2
πβ1) β¦(21)
The condition represented by equation (21) is possible only at very low pressures. When the
pressure of gas is so low, then the molecule will be very far apart and hence instead of the
getting de-energized by collision with another molecule, the energized molecule *A will
decomposed into products.
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
So, in all, we find that βat low pressures the rate is of second order whereas at high pressure
conditions, the rate becomes of first orderβ.
Considering the rate equation (10), we have
π ππ‘π =π[π]
ππ‘=
π2π1[π΄]2
πβ1[π΄]+π2
The above equation can be re-written as,
π[π]
ππ‘= πβ²[π΄] =
π2π1[π΄]
πβ1[π΄]+π2 β¦(22)
where πβ² is the effective rate constant.
πβ² =π2π1[π΄]
πβ1[π΄]+π2 β¦(23)
The above equation is rearranged to the form,
1
πβ²=πβ1[π΄]+π2
π2π1[π΄] β¦(24)
Or, the above can be written as,
1
πβ²=
πβ1
π2π1+
1
π1[π΄] β¦(25)
So, a test of this theory is to plot 1
πβ² against
1
[π΄] and to expect a straight line plot with:
πππππ =1
π1 and πΌππ‘ππππππ‘ =
πβ1
π2π1
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
Figure 2: Plot of 1
πβ² against
1
[π΄].
However, one can see that the straight line behavior is observed at low concentrations but
deviations are observed at higher concentrations.
3.4 Effect of inert-gas on the rate of the reaction
If we add inert gas (M) in the container containing gas molecules A, then the gas molecules
can get energized even with inert gas molecules collision.
Taking into account the Lindemannβs approach, the reaction can then be written as
Step 1: π΄ +ππ1β π΄β +π
Step 2: π΄β +ππβ1β π΄ +π
Step 3: π΄βπ2β π
Applying steady state hypothesis gives,
π1[π΄][π] = πβ1[π΄β]π + π2[π΄
β] β¦(26)
Rearranging equation (26) gives,
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
[π΄β] =π1[π΄][π]
πβ1π+π2 β¦(27)
And hence rate of the reaction becomes,
π ππ‘π =π[π]
ππ‘=π2π1[π΄][π]
πβ1[π]+π2 β¦(28)
Similar to Lindemann approach, the above equation gives rise to two limiting conditions.
Case I: πβ1[π] β« π2
π ππ‘π =π[π]
ππ‘=π2π1[π΄][π]
πβ1[π] β¦.(29)
The above equation simplifies to,
π ππ‘π =π[π]
ππ‘= (
π2π1
πβ1) [π΄] = π[π΄] β¦(30)
Again, over here we find that higher concentration of M i.e. at high pressure of inert gas, the
reaction follows first order kinetics.
Case II: πβ1[π] βͺ π2
π ππ‘π =π[π]
ππ‘=π2π1[π΄][π]
π2 β¦(31)
The above equation simplifies to,
π ππ‘π =π[π]
ππ‘= π1[π΄][π] β¦(32)
From equation (32), we find that the reaction becomes second order at low pressures.
3.5 Hinshelwood modification
Hinshelwood modified Lindemann Mechanism by stating that βevery energized molecule will
not enter into product formation but will go into activated molecule. And these activated
molecules will lead to product formationβ
π΄ + π΄π1β π΄β + π΄ (πΈππππππ§ππ‘πππ)
π΄β + π΄πβ1β π΄ + π΄ (π·π β πππππππ§ππ‘πππ)
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
π΄βπ2β π΄# (π΄ππ‘ππ£ππ‘πππ)
π΄#π3β π (πππππ’ππ‘ ππππππ‘πππ)
4. Limitations of Lindemann
Lindemann approach breaks down for two reasons:
The bimolecular step takes no account of the energy dependence of activation, the internal
degrees of freedom of the molecule are completely ignored and the theory underestimates
the rate of activation.
The unimolecular step fails to take into account that a unimolecular reaction specifically
involves one particular form of molecular motion (for example rotation around a double
bond).
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PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
5. Summary
1. Lindemann approach is the simplest theory of unimolecular reaction rates and
Lindemann was the first to successfully explain the observed first order kinetics of many
unimolecular reactions.
2. Lindemann assumed that there is time lag between activation and reaction during which
the activated or energized molecules may either react to give product or be de-energized
(deactivated) to ordinary moelcules. Only under these conditions the behavior of
unimolecular reactions can be explained on the basis of bimolecular collisions.
3. Lindemann hypothesized a situation where βA reactant molecule A becomes
energetically excited by collision with another molecule of reactant Aβ.
π΄ + π΄ β π΄β + π΄ (πΈππππππ§ππ‘πππ)
Where A represents inactive molecule while π΄β represents activated molecule. The rate for
reaction (1) becomes,
π ππ‘π =π[π΄β]
ππ‘= π1[π΄]
2
The energized molecule π΄β can lead to two situations:
Situation I: The energized molecule π΄β might lose its energy by colliding with another
molecule of A.
π΄β + π΄ β π΄ + π΄ (ππ β πππππππ§ππ‘πππ)
The rate for reaction becomes,
π ππ‘π = βπ[π΄β]
ππ‘= πβ1[π΄
β][π΄]
Situation II: The energized molecule π΄β might lead to product formation i.e., it might
undergo unimolecular decay.
π΄β β π (πππππ’ππ‘ ππππππ‘πππ)
The rate for reaction becomes,
π ππ‘π = βπ[π΄β]
ππ‘=π[π]
ππ‘= π2[π΄
β]
If the unimolecular step is slow enough to be the rate determining step, then the
Chemistry
PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
MODULE No. 20: Theories of unimolecular reactions- Lindemann Mechanism
overall reaction will follow first order kinetics.
4. Using Steady State Approximation / Principle which states that when a short lived
reaction intermediate such as [π΄β] exists at low concentration in a system, the rate of
formation of the intermediate can be considered to be equal to its rate of disappearance.
Applying this to [π΄β] we got,
[π΄β] =π1[π΄]
2
πβ1[π΄] + π2
5. Rate of the reaction
π ππ‘π =π[π]
ππ‘=
π2π1[π΄]2
πβ1[π΄]+π2
This equation gives the rate law expression for unimolecular reaction based on Lindemann
theory. This equation predicts two limiting possibilities:
At low pressure 21
k]A[k
thus, 21
]A[kdt
]P[d which is second order. Thus according
to this rate determining step is first step.
Whereas at high pressure 21
k]A[k
thus ]A[k
kk
dt
]P[d
1
21
which is first order reaction
and hence rate determining step is the second step.