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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
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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.
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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 β 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.
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
ππ΄π΅ = π (8ππ
ππ)
12β
ππ΄ππ΅ ...(2)
where Ο is the collision cross section and it represents the closeness of approach for
molecular collisions.
π = πππ΄π΅2 β¦(3)
Where ππ΄π΅ =ππ΄+ππ΅
2 ,
ππ΄ πππ ππ΅ πππ π‘βπ πππ ππππ‘ππ£π ππππππ‘πππ ππ πππππ‘πππ‘ ππππππ’πππ π΄ πππ π΅
(8ππ
ππ)
12β
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.
ππ΄ πππ ππ΅ πππ π‘βπ πππ ππππ‘ππ£π ππππππ‘πππ ππ πππππ‘πππ‘ ππππππ’πππ π΄ πππ π΅.
The radius of the contact area of molecules A and B respectively is given by the
expression:
ππ΄π΅ =ππ΄ + ππ΅
2 ,
And the area of cross section is given by the expression,
π = πππ΄π΅2
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
ΞΌ 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,
ππ΄ and ππ΅ are the respective number densities of molecules of A and B. Number density is
given by relation:
ππ’ππππ ππππ ππ‘π¦ =ππ’ππππ ππ ππππππ’πππ
π’πππ‘ π£πππ’ππ
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
π ππ‘π = βπππ΄
ππ‘= πππ΄ππ΅ ...(5)
[π΄] =ππ΄
ππ΄, [π΅] =
ππ΅
ππ΄ , π€βπππ ππ΄ ππ π‘βπ π΄π£ππππππ ππ’ππππ
βπππ΄
ππ‘= ππ΄
2π[π΄][π΅]
βππ΄
π[π΄]
ππ‘= ππ΄
2π[π΄][π΅]
βπ[π΄]
ππ‘= ππ΄π[π΄][π΅] ...(6)
Substituting the expression of collision density from equation (2) into equation (1) along with
ππ΄ and ππ΅ gives the rate as,
π ππ‘π = ππ΄2π (
8ππ
ππ)
12β
[π΄][π΅]πβπΈ0
π πβ ...(7)
On comparing equation (6) with equation (7), we get the expression for rate constant k as,
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
π = ππ΄π (8ππ
ππ)
12β
πβπΈ0
π πβ β¦(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 πΈ0. 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 πΈ0 is represented by Boltzmann factor πβπΈ0
π πβ .
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.
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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 β 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
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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.
π = πππ΄π (8ππ
ππ)
12β
πβπΈ0
π πβ ...(9)
3. Comparison of Arrhenius equation and Collision theory
We have discussed in previous module Arrhenius equation in detail.
π = π΄πβπΈπ
π πβ ...(10)
And according to collision theory, the rate constant is given by expression:
π = πππ΄π (8ππ
ππ)
12β
πβπΈ0
π πβ ...(11)
The above equation can be written in the following form:
π = π΅(π)1
2β πβπΈ0
π πβ ...(12)
Where π΅ = πππ΄π (8π
ππ)
12β
...(13)
So, we can say that B is constant and is independent of temperature.
Now, let us derive the expression ππππ
ππ from both the equations for finding the relation
between activation energy Ea and minimum energy πΈ0 .
Let us first derive the expression from Arrhenius Equation:
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
π = π΄πβπΈπ
π πβ
Applying logarithm to both the sides gives the expression,
ln π = ln π΄ βπΈπ
π πβ
βln π
ππ=
πΈππ π2β ...(14)
Let us know derive the expression from Collision Theory:
π = π΅(π)1
2β πβπΈ0
π πβ
Applying logarithm to both the sides gives the expression,
ln π = ln π΅ +1
2ln π β
πΈ0π πβ
βln π
ππ=
1
2π+
πΈ0π π2β ...(15)
On comparing equation (14) and (15) we find,
πΈπ
π π2β =1
2π+
πΈ0π π2β
or
πΈπ = πΈ0 +π π
2 ....(16)
where πΈ0 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,
π = π΅(π)1
2β πβ(πΈπβ
π π2
)π π
β
π = π΅(π)1
2β π1
2β πβπΈπ
π πβ β¦(17)
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
On comparing the above relation with Arrhenius equation, we find that the pre-exponential
factor can be expressed as,
π΄ = π΅(π)1
2β π1
2β
Substituting the value of B in the above expression gives,
π΄ = πππ΄π (8π
ππ)
12β
(π)1
2β π1
2β β¦(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.
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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 π½πΎβ1πππβ1)(700 πΎ)
(3.14)(1.28 Γ 10β3 πππππβ1)]
12β
= 340.4 ππ β1
πβ = ππ΄
ππ=
(6.023Γ1023πππβ1)
(8.314 π½πΎβ1πππβ1)(700 πΎ)/ (101.325Γ103ππ)= 1.05 Γ 1025πβ3
Now substituting this in the expression of collision frequency
2 * *1Z N N u
2
= (1
1.414) (3.14)(3.5 Γ 10β9π)2(340.4 ππ β1)(1.05 Γ 1025πβ3)2 = 1.02 Γ 1036πβ3π β1
The exponential factor is given by:
πβπΈπ
π πβ = exp [β183.9 Γ 103π½πππβ1/{(8.314 π½πΎβ1πππβ1)(700 πΎ)}] exp(β31.6) = 1.89 Γ 10β14
Hence, β1
2
πππ΄
ππ‘= ππ
βπΈππ πβ = (1.02 Γ 1036πβ3π β1)(1.89 Γ 10β14) = 1.93 Γ 1022πβ3π β1
β1
2
π[π΄]
ππ‘=
1.93 Γ 1022πβ3π β1
6.023 Γ 1023πππβ1= 0.032 πππ πβ3π β1
Now the concentration of reactant molecules is given by:
[π΄] =πβ
ππ΄=
1.05Γ1025πβ3
6.023Γ1023πππβ1 = 17.43 πππ πβ3
Therefore, the rate constant k will be:
π =β(
1
2)π[π΄]/ππ‘
[π΄]2 = 0.032 πππ πβ3π β1
(17.43 πππ πβ3)2 = 1.053 Γ 10β4πππβ1π3π β1
CHEMISTRY
Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)
Module No. 17: Rate constant and Collision theory
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
π = πππ΄π (8ππ
ππ)
12β
πβπΈ0
π πβ
The rate of a bimolecular gaseous reaction depends on collision frequency and its
probability of success.
Comparison of Arrhenius Equation and Collision Theory:
πΈπ = πΈ0 +π π
2
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