paper no and title 6 and physical chemistry-ii

CHEMISTRY

Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)

Module No. 17: Rate constant and Collision theory

Subject Chemistry

Paper No and Title 6 and PHYSICAL CHEMISTRY-II (Statistical

Thermodynamics, Chemical Dynamics, Electrochemistry

and Macromolecules)

Module No and Title 17 and Rate constant and Collision theory

Module Tag CHE_P6_M17

CHEMISTRY



TABLE OF CONTENTS

1. Learning outcomes

2. Collision theory of bimolecular gaseous reactions

2.1 Principle of collision theory

2.2 Derivation of rate constant

2.3 Energy of activation

2.4 Effect of orientation of molecules on the rate of reaction

3. Comparison of Arrhenius equation and Collision theory

4. Summary

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1. Learning Outcomes

After studying this module you shall be able to:

Derive rate constant on the basis of collision theory.

Know about principles of collision theory.

Know about the effect of orientation of molecules on the rate constant.

Compare rate constant derived on the basis of Arrhenius Equation and collision

theory.

2. Collision Theory of bimolecular gaseous reactions

In the previous module, we discussed the relationship between rate constant and temperature

as proposed by Arrhenius. In this module, we will take up in detail Collision theory of

bimolecular reactions. Collision theory is part of Chemical dynamics and gives a quantitative

account of reaction rates for the reactions between reacting species in gas phase.

The Collision Theory explains how chemical reactions occur and why reaction rates differ for

different reactions. This theory is based on kinetic theory of gases and assumes that

a. Molecules are hard spheres and are impenetrable (can at most touch each other).

b. Reaction may occur only when molecules approach and collide with each other.

c. Reactions occur only if molecules are energetic.

d. Collisions should transfer certain minimum energy.

2.1 Principles of Collision Theory

Collision Theory predicts the rate of the reaction based on two postulates:-

Kinetic Theory of gases:

All matter is made up of tiny particles i.e. atoms or molecules which are in constant motion.

Temperature is a measure of average kinetic energy of the species participating in the reaction.

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The product formation takes place only when the reactant molecules come close and

collide with each other.

Only those collisions lead to the formation of products which satisfy the criteria of

energy of activation and specific orientation of molecules.

Thus every collision does not lead to the product formation. Only those collisions which

occur between molecules having minimum threshold energy E0 will lead to product formation

and such collisions are known as effective collisions. If every collision leads to the formation

of product, then the rate of the reaction will entirely be determined by the collision rate, i.e.,

frequency with which reactants collide. Thus, it gives the maximum rate that can be observed

experimentally for the given reaction.

Now, let us consider the reaction in which the molecules are considered to be rigid, hard

spheres with no forces of attraction and repulsion:

A + B products

Rate of the reaction can be given by the equation:

0E RT

ABRate Z e

....(1)

where ZAB refers to the number of collisions that occur per unit volume per unit time. It is

also referred to as Collision density.

The number of collisions per unit volume per unit time in such reactions are given by the

expression,

If we have two reactants, they can only react if they come into contact with each other i.e. if

they collide.

Then, they MAY react?

Collision alone is not enough.

a. They must collide the right way.

b. They must have enough energy for bonds to break.

Even if the collision happens the right way, the reaction will not happen unless the particles

collide with a certain minimum energy – activation energy. If the particles collide with less

energy than activation energy, they will just bounce and no reaction will occur.

Activation energy acts like a barrier that has to be crossed for a reaction to happen.

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𝑍𝐴𝐵 = 𝜎 (8𝑘𝑇

𝜋𝜇)

12⁄

𝑛𝐴𝑛𝐵 ...(2)

where σ is the collision cross section and it represents the closeness of approach for

molecular collisions.

𝜎 = 𝜋𝑑𝐴𝐵2 …(3)

Where 𝑑𝐴𝐵 =𝑑𝐴+𝑑𝐵

2 ,

𝑑𝐴 𝑎𝑛𝑑 𝑑𝐵 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝐴 𝑎𝑛𝑑 𝐵

(8𝑘𝑇

𝜋𝜇)

12⁄

is the average velocity of molecules and is

The collision cross section for two molecules can be regarded as the area within

which the molecule A hits molecule B for collision to occur.

𝑑𝐴 𝑎𝑛𝑑 𝑑𝐵 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝐴 𝑎𝑛𝑑 𝐵.

The radius of the contact area of molecules A and B respectively is given by the

expression:

𝑑𝐴𝐵 =𝑑𝐴 + 𝑑𝐵

2 ,

And the area of cross section is given by the expression,

𝜎 = 𝜋𝑑𝐴𝐵2

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μ is reduced mass = A B

A B

m m

(m m ), ...(4)

k is Boltzmann constant, mA is mass of single molecule A, mB is mass of single molecule B,

𝑛𝐴 and 𝑛𝐵 are the respective number densities of molecules of A and B. Number density is

given by relation:

𝑁𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠

𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒

2.2 Derivation of rate constant

According to classical collision theory (hard sphere theory), rate of the reaction as given by

equation (1) depends on collision density and exponential factor. Or one can simply say that

the rate of a bimolecular gaseous reaction depends on collision frequency and its probability

of success.

A + B products

𝑅𝑎𝑡𝑒 = −𝑑𝑛𝐴

𝑑𝑡= 𝑘𝑛𝐴𝑛𝐵 ...(5)

[𝐴] =𝑛𝐴

𝑁𝐴, [𝐵] =

𝑛𝐵

𝑁𝐴 , 𝑤ℎ𝑒𝑟𝑒 𝑁𝐴 𝑖𝑠 𝑡ℎ𝑒 𝐴𝑣𝑜𝑔𝑎𝑑𝑟𝑜 𝑛𝑢𝑚𝑏𝑒𝑟

−𝑑𝑛𝐴

𝑑𝑡= 𝑁𝐴

2𝑘[𝐴][𝐵]

−𝑁𝐴

𝑑[𝐴]

𝑑𝑡= 𝑁𝐴

2𝑘[𝐴][𝐵]

−𝑑[𝐴]

𝑑𝑡= 𝑁𝐴𝑘[𝐴][𝐵] ...(6)

Substituting the expression of collision density from equation (2) into equation (1) along with

𝑛𝐴 and 𝑛𝐵 gives the rate as,

𝑅𝑎𝑡𝑒 = 𝑁𝐴2𝜎 (

8𝑘𝑇

𝜋𝜇)

12⁄

[𝐴][𝐵]𝑒−𝐸0

𝑅𝑇⁄ ...(7)

On comparing equation (6) with equation (7), we get the expression for rate constant k as,

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𝑘 = 𝑁𝐴𝜎 (8𝑘𝑇

𝜋𝜇)

12⁄

𝑒−𝐸0

𝑅𝑇⁄ …(8)

2.3 Energy of activation

For the product formation to take place the molecules should possess sufficient energy

required for the molecular rearrangement. This energy is not the total kinetic energy of two

molecules but it is the kinetic energy corresponding to the component of the relative velocity

of the two molecules along the line of their centres at the time of collision. This is the energy

of the two molecules with which they must pressed together for the reaction to occur. This

energy should be equal to or greater than some minimum energy 𝐸0. The difference between

this minimum energy and the average energy of reacting molecules is known as energy of

activation Ea. The fraction of collisions in which the molecules have energy greater than the

minimum energy 𝐸0 is represented by Boltzmann factor 𝑒−𝐸0

𝑅𝑇⁄ .

2.4 Effect of orientation of molecules on the rate of the reaction

The probability that a collision will occur successfully is incorporated by writing collision cross

section as a function of kinetic energy of approach of two colliding entities and setting the kinetic

energy zero below a certain threshold value.

Factors affecting Collision frequency: Temperature: Faster moving particles move farther each second and collide more frequently. With increase in temperature, collision frequency increases and so the rate increases. Concentration: Particles that are packed more tightly together collide more frequently. Gases are more concentrated at high pressure than at low pressure. With increase in pressure, the concentration increases which increases the collision frequency and so the rate increases. Surface area: The greater the number of particles that are exposed for possible collisions. Decrease in particle size, increases the surface area which increases the collision frequency and so the rate increases.

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Only those collisions taking place between molecules in proper orientation and possessing

certain minimum amount of energy can lead to the formation of products. Thus it is

important to study the effect of orientation of molecules on the rate of the reaction.

To study the effect of orientation of molecules we will consider reaction involving a collision

between two molecules - ethene, CH2=CH2, and hydrogen chloride, HCl.

CH2=CH2 + HCl → CH3CH2Cl (Chloroethane)

As a result of the collision, the double bond between the two carbons of ethene is converted

into a single bond. A hydrogen atom gets attached to one of the carbons and a chlorine atom

to the other.

It is important to realize that this reaction will only occur if the hydrogen end of the H-Cl

bond approaches the carbon-carbon double bond. Any other collision between the two

molecules will not work.

Figure: Orientation of two molecules. Only first case results in successful collision and

not the ways

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Thus, we conclude that the rate of formation of products is not only dependent on

collision energy but also on the relative orientation of the molecules at the time of

collision.

This criteria of specific orientation of molecules at the time of collision is taken into account

by multiplying p factor to equation (8). The constant p is called the steric factor and is

usually less than 1.

𝑘 = 𝑝𝑁𝐴𝜎 (8𝑘𝑇

𝜋𝜇)

12⁄

𝑒−𝐸0

𝑅𝑇⁄ ...(9)

3. Comparison of Arrhenius equation and Collision theory

We have discussed in previous module Arrhenius equation in detail.

𝑘 = 𝐴𝑒−𝐸𝑎

𝑅𝑇⁄ ...(10)

And according to collision theory, the rate constant is given by expression:


𝜋𝜇)

12⁄

𝑒−𝐸0

𝑅𝑇⁄ ...(11)

The above equation can be written in the following form:

𝑘 = 𝐵(𝑇)1

2⁄ 𝑒−𝐸0

𝑅𝑇⁄ ...(12)

Where 𝐵 = 𝑝𝑁𝐴𝜎 (8𝑘

𝜋𝜇)

12⁄

...(13)

So, we can say that B is constant and is independent of temperature.

Now, let us derive the expression 𝜕𝑙𝑛𝑘

𝜕𝑇 from both the equations for finding the relation

between activation energy Ea and minimum energy 𝐸0 .

Let us first derive the expression from Arrhenius Equation:

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𝑘 = 𝐴𝑒−𝐸𝑎

𝑅𝑇⁄

Applying logarithm to both the sides gives the expression,

ln 𝑘 = ln 𝐴 −𝐸𝑎

𝑅𝑇⁄

∂ln 𝑘

𝜕𝑇=

𝐸𝑎𝑅𝑇2⁄ ...(14)

Let us know derive the expression from Collision Theory:

𝑘 = 𝐵(𝑇)1

2⁄ 𝑒−𝐸0

𝑅𝑇⁄

Applying logarithm to both the sides gives the expression,

ln 𝑘 = ln 𝐵 +1

2ln 𝑇 −

𝐸0𝑅𝑇⁄

∂ln 𝑘

𝜕𝑇=

1

2𝑇+

𝐸0𝑅𝑇2⁄ ...(15)

On comparing equation (14) and (15) we find,

𝐸𝑎

𝑅𝑇2⁄ =1

2𝑇+

𝐸0𝑅𝑇2⁄

or

𝐸𝑎 = 𝐸0 +𝑅𝑇

2 ....(16)

where 𝐸0 is the threshold energy or barrier energy leading to the formation of products, and

Ea is activation energy which is required to activate the reactants. Thus, from the above

relation, we find that the activation energy is dependent on temperature.

Substituting above relation (16) in equation (12) gives the expression,

𝑘 = 𝐵(𝑇)1

2⁄ 𝑒−(𝐸𝑎−

𝑅𝑇2

)𝑅𝑇

⁄

𝑘 = 𝐵(𝑇)1

2⁄ 𝑒1

2⁄ 𝑒−𝐸𝑎

𝑅𝑇⁄ …(17)

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On comparing the above relation with Arrhenius equation, we find that the pre-exponential

factor can be expressed as,

𝐴 = 𝐵(𝑇)1

2⁄ 𝑒1

2⁄

Substituting the value of B in the above expression gives,

𝐴 = 𝑝𝑁𝐴𝜎 (8𝑘

𝜋𝜇)

12⁄

(𝑇)1

2⁄ 𝑒1

2⁄ …(18)

Question: Does this theory apply to the enzymes in the human body?

Answer: Enzymes are the biological catalysts. They increase the rate of the reaction by

decreasing the activation energy needed for the reaction to occur.

By decreasing the activation energy, greater number of collisions takes place and thus the

reaction occurs at faster rate.

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Question: The equation representing the decomposition of hydrogen iodide is

2HI H2 + I2. Calculate collision rate, rate of reaction and rate constant for the given

reaction at 700 K and 1 atm pressure if collision diameter is assumed to be 3.5 nm and

activation energy be 183.8 kJ mol-1.

Answer: Collision frequency per unit volume per unit time between two identical molecules

is given by:

2 * *1Z N N u

2

In this, u is average velocity. It is given by:

8RT

uM

=[8(8.314 𝐽𝐾−1𝑚𝑜𝑙−1)(700 𝐾)

(3.14)(1.28 × 10−3 𝑘𝑔𝑚𝑜𝑙−1)]

12⁄

= 340.4 𝑚𝑠−1

𝑁∗ = 𝑁𝐴

𝑉𝑚=

(6.023×1023𝑚𝑜𝑙−1)

(8.314 𝐽𝐾−1𝑚𝑜𝑙−1)(700 𝐾)/ (101.325×103𝑃𝑎)= 1.05 × 1025𝑚−3

Now substituting this in the expression of collision frequency

2 * *1Z N N u

2

= (1

1.414) (3.14)(3.5 × 10−9𝑚)2(340.4 𝑚𝑠−1)(1.05 × 1025𝑚−3)2 = 1.02 × 1036𝑚−3𝑠−1

The exponential factor is given by:

𝑒−𝐸𝑎

𝑅𝑇⁄ = exp [−183.9 × 103𝐽𝑚𝑜𝑙−1/{(8.314 𝐽𝐾−1𝑚𝑜𝑙−1)(700 𝐾)}] exp(−31.6) = 1.89 × 10−14

Hence, −1

2

𝑑𝑁𝐴

𝑑𝑡= 𝑍𝑒

−𝐸𝑎𝑅𝑇⁄ = (1.02 × 1036𝑚−3𝑠−1)(1.89 × 10−14) = 1.93 × 1022𝑚−3𝑠−1

−1

2

𝑑[𝐴]

𝑑𝑡=

1.93 × 1022𝑚−3𝑠−1

6.023 × 1023𝑚𝑜𝑙−1= 0.032 𝑚𝑜𝑙 𝑚−3𝑠−1

Now the concentration of reactant molecules is given by:

[𝐴] =𝑁∗

𝑁𝐴=

1.05×1025𝑚−3

6.023×1023𝑚𝑜𝑙−1 = 17.43 𝑚𝑜𝑙 𝑚−3

Therefore, the rate constant k will be:

𝑘 =−(

1

2)𝑑[𝐴]/𝑑𝑡

[𝐴]2 = 0.032 𝑚𝑜𝑙 𝑚−3𝑠−1

(17.43 𝑚𝑜𝑙 𝑚−3)2 = 1.053 × 10−4𝑚𝑜𝑙−1𝑚3𝑠−1

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4. Summary

The Collision Theory explains how chemical reactions occur and why reaction rates

differ for different reactions. This theory is based on kinetic theory of gases.

Collision Theory predicts the rate of the reaction based on two postulates:

The product formation takes place only when the reactant molecules come close and

collide with each other.

Only those collisions lead to the formation of products which satisfy the criteria of

energy of activation and specific orientation of molecules.

According to collision theory the rate constant is given by


𝜋𝜇)

12⁄

𝑒−𝐸0

𝑅𝑇⁄

The rate of a bimolecular gaseous reaction depends on collision frequency and its

probability of success.

Comparison of Arrhenius Equation and Collision Theory:

𝐸𝑎 = 𝐸0 +𝑅𝑇

2

paper no and title 6 and physical chemistry-ii

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