subject chemistry paper no and title 6 and physical

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



TABLE OF CONTENTS

1. Learning outcomes

2. Introduction

3. Lindemann’s 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 Lindemann’s theory

5. Hinshelwood Modification

6. Summary

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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.

𝑆𝑂2𝐶𝑙2 → 𝑆𝑂2 + 𝐶𝑙2 (Decomposition reaction)

𝑐𝑦𝑐𝑙𝑜 − 𝐶3𝐻6 → 𝐶𝐻3 − 𝐶𝐻 = 𝐶𝐻2 (Rearrangement reaction)

𝐶𝑙2ℎ𝜈→ 2𝐶𝑙.

In general, for a unimolecular reaction of type

𝐴 → 𝑃,

Rate will be expressed as,

𝑅𝑎𝑡𝑒 = 𝑘[𝐴] 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Chemistry



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 “How do molecules in

unimolecular reactions attain this critical energy which leads to product formation?”..OR one

can even frame it this way “if we consider collisions, then the question becomes how the

bimolecular event leads to first order rate law?”.

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. Lindemann’s 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



“A reactant molecule A becomes energetically excited by collision with another molecule of

reactant A”.

𝐴 + 𝐴 → 𝐴∗ + 𝐴 (𝐸𝑛𝑒𝑟𝑔𝑖𝑧𝑎𝑡𝑖𝑜𝑛) …(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 𝐴∗ represents activated molecule. The rate for

reaction (1) becomes,

𝑅𝑎𝑡𝑒 =𝑑[𝐴∗]

𝑑𝑡= 𝑘1[𝐴]

2 …(2)

Now, the energized molecule 𝐴∗ can lead to two situations:

Situation I: The energized molecule 𝐴∗ might lose its energy by colliding with another

molecule of A.

𝐴∗ + 𝐴 → 𝐴 + 𝐴 (𝑑𝑒 − 𝑒𝑛𝑒𝑟𝑔𝑖𝑧𝑎𝑡𝑖𝑜𝑛) …(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,

𝑅𝑎𝑡𝑒 = −𝑑[𝐴∗]

𝑑𝑡= 𝑘−1[𝐴

∗][𝐴] …(4)

Situation II: The energized molecule 𝐴∗ might lead to product formation i.e., it might

undergo unimolecular decay.

𝐴∗ → 𝑃 ( 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛) …(5)

The rate for reaction (5) becomes,


𝑑𝑡=𝑑[𝑃]

𝑑𝑡= 𝑘2[𝐴

∗] …(6)

If the unimolecular step (equation 5) is slow enough to be the rate determining step, then the

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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 [𝐴∗] (which is not known) in terms of [𝐴]. To do this, we apply

Steady State Approximation / Principle. This principle states that when a short lived reaction

intermediate such as 𝐴∗ 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 [𝐴∗]

gives,

𝑑[𝐴∗]

𝑑𝑡= −

𝑑[𝐴∗]

𝑑𝑡 …(7)

Which means

𝑘1[𝐴]2 = 𝑘−1[𝐴

∗][𝐴] + 𝑘2[𝐴∗] …(8)

Rearranging equation (8) gives,

[𝐴∗] =𝑘1[𝐴]

2

𝑘−1[𝐴]+𝑘2 …(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: 𝐴 + 𝐴 → 𝐴∗ + 𝐴

Step 2: 𝐴∗ + 𝐴 → 𝐴 + 𝐴

Step 3: 𝐴∗ → 𝑃

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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,


𝑑𝑡=𝑑[𝑃]


∗]

Substituting [𝐴∗] from equation (9) in the equation, modifies the above equation to,

𝑅𝑎𝑡𝑒 =𝑑[𝑃]

𝑑𝑡=

𝑘2𝑘1[𝐴]2

𝑘−1[𝐴]+𝑘2 …(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: 𝑘−1[𝐴] ≫ 𝑘2


𝑑𝑡=𝑘2𝑘1[𝐴]

2

𝑘−1[𝐴] ….(11)

The above equation simplifies to,


𝑑𝑡= (

𝑘2𝑘1

𝑘−1) [𝐴] = 𝑘[𝐴] …(12)

From equation (12), one can notice that (𝑘2𝑘1

𝑘−1) 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 [𝐴] is high enough to produce

appreciable deactivation of 𝐴∗ (as shown by situation I and situation II above) i.e.,

[𝐴] ≫ (𝑘2

𝑘−1) …(13)

For an ideal gas, 𝑝𝑉 = 𝑛𝑅𝑇 …(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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𝑛

𝑉=

𝑃

𝑅𝑇 …(15)

Taking into account the above equation, [𝐴] becomes,

[𝐴] =𝑝𝐴

𝑅𝑇 …(16)

Using equation (16) into equation (13) gives,

𝑝𝐴

𝑅𝑇≫ (

𝑘2

𝑘−1) …(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: 𝑘−1[𝐴] ≪ 𝑘2


𝑑𝑡=𝑘2𝑘1[𝐴]

2

𝑘2 …(18)




2 …(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.

[𝐴] ≪ (𝑘2

𝑘−1) …(20)

Using ideal gas equation condition as stated above in equation (14), (15) and (16) gives,

𝑝𝐴

𝑅𝑇≪ (

𝑘2

𝑘−1) …(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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So, in all, we find that “at low pressures the rate is of second order whereas at high pressure

conditions, the rate becomes of first order”.

Considering the rate equation (10), we have


𝑑𝑡=

𝑘2𝑘1[𝐴]2

𝑘−1[𝐴]+𝑘2

The above equation can be re-written as,

𝑑[𝑃]

𝑑𝑡= 𝑘′[𝐴] =

𝑘2𝑘1[𝐴]

𝑘−1[𝐴]+𝑘2 …(22)

where 𝑘′ is the effective rate constant.

𝑘′ =𝑘2𝑘1[𝐴]

𝑘−1[𝐴]+𝑘2 …(23)

The above equation is rearranged to the form,

1

𝑘′=𝑘−1[𝐴]+𝑘2

𝑘2𝑘1[𝐴] …(24)

Or, the above can be written as,

1

𝑘′=

𝑘−1

𝑘2𝑘1+

1

𝑘1[𝐴] …(25)

So, a test of this theory is to plot 1

𝑘′ against

1

[𝐴] and to expect a straight line plot with:

𝑆𝑙𝑜𝑝𝑒 =1

𝑘1 and 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =

𝑘−1

𝑘2𝑘1

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Figure 2: Plot of 1

𝑘′ against

1

[𝐴].

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 Lindemann’s approach, the reaction can then be written as

Step 1: 𝐴 +𝑀𝑘1→ 𝐴∗ +𝑀

Step 2: 𝐴∗ +𝑀𝑘−1→ 𝐴 +𝑀

Step 3: 𝐴∗𝑘2→ 𝑃

Applying steady state hypothesis gives,

𝑘1[𝐴][𝑀] = 𝑘−1[𝐴∗]𝑀 + 𝑘2[𝐴

∗] …(26)

Rearranging equation (26) gives,

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[𝐴∗] =𝑘1[𝐴][𝑀]

𝑘−1𝑀+𝑘2 …(27)

And hence rate of the reaction becomes,


𝑑𝑡=𝑘2𝑘1[𝐴][𝑀]

𝑘−1[𝑀]+𝑘2 …(28)

Similar to Lindemann approach, the above equation gives rise to two limiting conditions.

Case I: 𝑘−1[𝑀] ≫ 𝑘2



𝑘−1[𝑀] ….(29)



𝑑𝑡= (

𝑘2𝑘1

𝑘−1) [𝐴] = 𝑘[𝐴] …(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: 𝑘−1[𝑀] ≪ 𝑘2



𝑘2 …(31)



𝑑𝑡= 𝑘1[𝐴][𝑀] …(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 “every energized molecule will

not enter into product formation but will go into activated molecule. And these activated

molecules will lead to product formation”

𝐴 + 𝐴𝑘1→ 𝐴∗ + 𝐴 (𝐸𝑛𝑒𝑟𝑔𝑖𝑧𝑎𝑡𝑖𝑜𝑛)

𝐴∗ + 𝐴𝑘−1→ 𝐴 + 𝐴 (𝐷𝑒 − 𝑒𝑛𝑒𝑟𝑔𝑖𝑧𝑎𝑡𝑖𝑜𝑛)

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𝐴∗𝑘2→ 𝐴# (𝐴𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛)

𝐴#𝑘3→ 𝑃 (𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛)

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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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 “A reactant molecule A becomes

energetically excited by collision with another molecule of reactant A”.

𝐴 + 𝐴 → 𝐴∗ + 𝐴 (𝐸𝑛𝑒𝑟𝑔𝑖𝑧𝑎𝑡𝑖𝑜𝑛)

Where A represents inactive molecule while 𝐴∗ represents activated molecule. The rate for

reaction (1) becomes,

𝑅𝑎𝑡𝑒 =𝑑[𝐴∗]


2

The energized molecule 𝐴∗ can lead to two situations:

Situation I: The energized molecule 𝐴∗ might lose its energy by colliding with another

molecule of A.

𝐴∗ + 𝐴 → 𝐴 + 𝐴 (𝑑𝑒 − 𝑒𝑛𝑒𝑟𝑔𝑖𝑧𝑎𝑡𝑖𝑜𝑛)

The rate for reaction becomes,


𝑑𝑡= 𝑘−1[𝐴

∗][𝐴]

Situation II: The energized molecule 𝐴∗ might lead to product formation i.e., it might

undergo unimolecular decay.

𝐴∗ → 𝑃 (𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛)

The rate for reaction becomes,


𝑑𝑡=𝑑[𝑃]


∗]

If the unimolecular step is slow enough to be the rate determining step, then the

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overall reaction will follow first order kinetics.

4. Using Steady State Approximation / Principle which states that when a short lived

reaction intermediate such as [𝐴∗] 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 [𝐴∗] we got,

[𝐴∗] =𝑘1[𝐴]

2

𝑘−1[𝐴] + 𝑘2

5. Rate of the reaction


𝑑𝑡=

𝑘2𝑘1[𝐴]2

𝑘−1[𝐴]+𝑘2

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.

subject chemistry paper no and title 6 and physical

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