Theories-of-Reaction-Rates Collision-Theory | Chemistry for EmSAT Achieve PDF Download

 Page 1


 Module 7 : Theories of Reaction Rates
Lecture 32 : Theories Of Reaction Rates - I : Collision Theory
   Objectives
  After studying this lecture you will be able to do the following.
Calculate collision frequency using kinetic theory.
  
Distinguish between collision cross section and reaction cross section.
  
Compare collision theory with Arrhenius theory and experiments.
  
Define the steric factor.
  
Rationalize the departure of collision theory from experiments for ionic reactions using the harpoon
mechanism.
   32.1Introduction
In the earlier lectures on chemical kinetics, we have studied the empirical aspects in detail. We now begin
the theoretical explanations of chemical rate processes. This is one of the most challenging areas in
chemistry. Molecular structure is fairly well understood through the advances in spectroscopy and
quantum chemistry. With the knowledge of the same molecular structures and intermolecular forces,
chemical equilibrium (thermodynamics) can be predicted quite accurately. An understanding the rates of
processes microscopically requires answers to several interesting questions. Some of the main questions
are as follows. How close should the molecules approach one another for the reaction to occur? Does the
reaction "occur" only after reaching this separation or does it start well before the distance of closest
separation is reached? Do all the reactants react identically or is there a distribution over several
possibilities? Do all "encounters" between the reactants produce the products? How closely can we
monitor the progress of the reaction theoretically (as well as experimentally)? Over the last four decades,
the time resolution of this monitoring has improved dramatically from microseconds to femtoseconds.
While most of the above questions relate to gas phase processes, solution reactions have the additional
contributions from the participation of the solvent. We shall take up these issues one by one in modest
details. We first begin with the gas phase collision theory. This is followed by the transition state theory
(lecture 33). Potential energy surfaces are introduced in lecture 34 and the trajectories over potential
energy surfaces will be considered in lecture 35.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   32.2 Bimolecular gas phase collision theory
Consider the following gas phase reaction
A + B  Products ( P ) (32.1)
The rate of the reaction, r = dP/dt, depends on the following factors: a) The concentrations or the number
densities of molecules A and B, i.e., N
A
 and N
B 
( N
A
 = numbers of molecules in unit volume of the
container, similarly N
B
 ), b) number of collisions between of A and B in unit time. This is the collision rate.
Larger the relative velocity between A an B, larger the collision rate.The relative speed between A and B is
obtained from 1/ 2  = 1/2 k
B
T or , the relative velocity = ( k
B
T /  )
1/2 
is determined by the
temperature T at equilibrium; and thirdly, c) the collision cross section, . The collision cross section is
the area within which the center of A should be around the target B so that a collision can be said to have
occurred. Let us first calculate the collision frequency between A and B.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


 Module 7 : Theories of Reaction Rates
Lecture 32 : Theories Of Reaction Rates - I : Collision Theory
   Objectives
  After studying this lecture you will be able to do the following.
Calculate collision frequency using kinetic theory.
  
Distinguish between collision cross section and reaction cross section.
  
Compare collision theory with Arrhenius theory and experiments.
  
Define the steric factor.
  
Rationalize the departure of collision theory from experiments for ionic reactions using the harpoon
mechanism.
   32.1Introduction
In the earlier lectures on chemical kinetics, we have studied the empirical aspects in detail. We now begin
the theoretical explanations of chemical rate processes. This is one of the most challenging areas in
chemistry. Molecular structure is fairly well understood through the advances in spectroscopy and
quantum chemistry. With the knowledge of the same molecular structures and intermolecular forces,
chemical equilibrium (thermodynamics) can be predicted quite accurately. An understanding the rates of
processes microscopically requires answers to several interesting questions. Some of the main questions
are as follows. How close should the molecules approach one another for the reaction to occur? Does the
reaction "occur" only after reaching this separation or does it start well before the distance of closest
separation is reached? Do all the reactants react identically or is there a distribution over several
possibilities? Do all "encounters" between the reactants produce the products? How closely can we
monitor the progress of the reaction theoretically (as well as experimentally)? Over the last four decades,
the time resolution of this monitoring has improved dramatically from microseconds to femtoseconds.
While most of the above questions relate to gas phase processes, solution reactions have the additional
contributions from the participation of the solvent. We shall take up these issues one by one in modest
details. We first begin with the gas phase collision theory. This is followed by the transition state theory
(lecture 33). Potential energy surfaces are introduced in lecture 34 and the trajectories over potential
energy surfaces will be considered in lecture 35.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   32.2 Bimolecular gas phase collision theory
Consider the following gas phase reaction
A + B  Products ( P ) (32.1)
The rate of the reaction, r = dP/dt, depends on the following factors: a) The concentrations or the number
densities of molecules A and B, i.e., N
A
 and N
B 
( N
A
 = numbers of molecules in unit volume of the
container, similarly N
B
 ), b) number of collisions between of A and B in unit time. This is the collision rate.
Larger the relative velocity between A an B, larger the collision rate.The relative speed between A and B is
obtained from 1/ 2  = 1/2 k
B
T or , the relative velocity = ( k
B
T /  )
1/2 
is determined by the
temperature T at equilibrium; and thirdly, c) the collision cross section, . The collision cross section is
the area within which the center of A should be around the target B so that a collision can be said to have
occurred. Let us first calculate the collision frequency between A and B.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 32.1 Collision cross section and the collision cylinder.
 
All the molecules that come inside the range of the collision cylinder will collide with the central molecule
on the extreme left. All those outside this cylinder do not collide with molecule A.
 
To calculate the collision frequency, consider the motion of the molecule labeled A. In time  t, the molecule
moves a distance  t where  is the average speed of the molecule. Any other molecule which is within the
volume of the cylinder   t d 
2 
spanned by molecule A, will collide with molecule A. Here d 
2
 is the area of
the cylinder spanned by A. This collision cylinder is shown in Fig 32.1. All the molecules that collide with A can be
easily identified in the figure. During collisions, the directions of molecules change and the cylinder is not a rigid
space but the effectively available space for the molecules. Since all molecules are moving, we need to use the
relative speeds between molecules. From kinetic theory of gases, the average relative speed between molecules A
and B is
 
 
rel
 = (8 k
B
T /  )
1/ 2
, 
-1
 = m 
A
-1
 + m 
B
-1
 
(32.2)
Where  is the reduced mass between A and B. For identical molecules,  
rel 
=  , where the latter is
calculated with m in the denominator of Eq. (32.2) in place of the reduced mass. The factor 2
1/2
 is because       
= m / 2. The area of the cylinder is called the collision cross section. Molecules outside this area do not collide with
A. If N
A
 is the number of molecules of A in volume V, the collision frequency is given by
 
z =    N
A
 / V
(32.3)
This z is the number of collisions encounter by a single molecule. Since there are N
A
 molecules in volume V, the
number of collisions between molecules of A in unit time and unit volume, Z
A
A
  
Z
AA
 = (1/2) z N
A
 / V
(32.4)
Substituting the values of all the factors, we have
Z
AA
 =  (4 k
B
T /  m )
1/ 2
 ( N
A
 / V )
 2
If n
A
 is the number of moles of A in volume V, then, [A] = molar concentration of A = n
A
 / V . Since N
A
 = n
A
L
where L = Avogadro number, we have for Z 
AA
 
Z
AA
 =  (4 k
B
T /  m )
1/2
 L
2
 [A]
 2
(32.5)
 
 
 
If the collision is between molecules of type A with molecules of type B, the factor of 1/2 in eq. (32.4)
drops out. If two molecules of A collide it is counted as one collision. When molecules of A collide with
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


 Module 7 : Theories of Reaction Rates
Lecture 32 : Theories Of Reaction Rates - I : Collision Theory
   Objectives
  After studying this lecture you will be able to do the following.
Calculate collision frequency using kinetic theory.
  
Distinguish between collision cross section and reaction cross section.
  
Compare collision theory with Arrhenius theory and experiments.
  
Define the steric factor.
  
Rationalize the departure of collision theory from experiments for ionic reactions using the harpoon
mechanism.
   32.1Introduction
In the earlier lectures on chemical kinetics, we have studied the empirical aspects in detail. We now begin
the theoretical explanations of chemical rate processes. This is one of the most challenging areas in
chemistry. Molecular structure is fairly well understood through the advances in spectroscopy and
quantum chemistry. With the knowledge of the same molecular structures and intermolecular forces,
chemical equilibrium (thermodynamics) can be predicted quite accurately. An understanding the rates of
processes microscopically requires answers to several interesting questions. Some of the main questions
are as follows. How close should the molecules approach one another for the reaction to occur? Does the
reaction "occur" only after reaching this separation or does it start well before the distance of closest
separation is reached? Do all the reactants react identically or is there a distribution over several
possibilities? Do all "encounters" between the reactants produce the products? How closely can we
monitor the progress of the reaction theoretically (as well as experimentally)? Over the last four decades,
the time resolution of this monitoring has improved dramatically from microseconds to femtoseconds.
While most of the above questions relate to gas phase processes, solution reactions have the additional
contributions from the participation of the solvent. We shall take up these issues one by one in modest
details. We first begin with the gas phase collision theory. This is followed by the transition state theory
(lecture 33). Potential energy surfaces are introduced in lecture 34 and the trajectories over potential
energy surfaces will be considered in lecture 35.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   32.2 Bimolecular gas phase collision theory
Consider the following gas phase reaction
A + B  Products ( P ) (32.1)
The rate of the reaction, r = dP/dt, depends on the following factors: a) The concentrations or the number
densities of molecules A and B, i.e., N
A
 and N
B 
( N
A
 = numbers of molecules in unit volume of the
container, similarly N
B
 ), b) number of collisions between of A and B in unit time. This is the collision rate.
Larger the relative velocity between A an B, larger the collision rate.The relative speed between A and B is
obtained from 1/ 2  = 1/2 k
B
T or , the relative velocity = ( k
B
T /  )
1/2 
is determined by the
temperature T at equilibrium; and thirdly, c) the collision cross section, . The collision cross section is
the area within which the center of A should be around the target B so that a collision can be said to have
occurred. Let us first calculate the collision frequency between A and B.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 32.1 Collision cross section and the collision cylinder.
 
All the molecules that come inside the range of the collision cylinder will collide with the central molecule
on the extreme left. All those outside this cylinder do not collide with molecule A.
 
To calculate the collision frequency, consider the motion of the molecule labeled A. In time  t, the molecule
moves a distance  t where  is the average speed of the molecule. Any other molecule which is within the
volume of the cylinder   t d 
2 
spanned by molecule A, will collide with molecule A. Here d 
2
 is the area of
the cylinder spanned by A. This collision cylinder is shown in Fig 32.1. All the molecules that collide with A can be
easily identified in the figure. During collisions, the directions of molecules change and the cylinder is not a rigid
space but the effectively available space for the molecules. Since all molecules are moving, we need to use the
relative speeds between molecules. From kinetic theory of gases, the average relative speed between molecules A
and B is
 
 
rel
 = (8 k
B
T /  )
1/ 2
, 
-1
 = m 
A
-1
 + m 
B
-1
 
(32.2)
Where  is the reduced mass between A and B. For identical molecules,  
rel 
=  , where the latter is
calculated with m in the denominator of Eq. (32.2) in place of the reduced mass. The factor 2
1/2
 is because       
= m / 2. The area of the cylinder is called the collision cross section. Molecules outside this area do not collide with
A. If N
A
 is the number of molecules of A in volume V, the collision frequency is given by
 
z =    N
A
 / V
(32.3)
This z is the number of collisions encounter by a single molecule. Since there are N
A
 molecules in volume V, the
number of collisions between molecules of A in unit time and unit volume, Z
A
A
  
Z
AA
 = (1/2) z N
A
 / V
(32.4)
Substituting the values of all the factors, we have
Z
AA
 =  (4 k
B
T /  m )
1/ 2
 ( N
A
 / V )
 2
If n
A
 is the number of moles of A in volume V, then, [A] = molar concentration of A = n
A
 / V . Since N
A
 = n
A
L
where L = Avogadro number, we have for Z 
AA
 
Z
AA
 =  (4 k
B
T /  m )
1/2
 L
2
 [A]
 2
(32.5)
 
 
 
If the collision is between molecules of type A with molecules of type B, the factor of 1/2 in eq. (32.4)
drops out. If two molecules of A collide it is counted as one collision. When molecules of A collide with
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
molecules of B, there are distinct AB collisions, AA and BB collisions. The total number of collisions
between A and B molecules in unit time and unit volume is given by
 
Z 
AB
 = (8 k
B
T / )
1/ 2
 L
2
 [A] [B] 
(32.6)
Substitution of the values of all the factors in eq (32.5) and (32.6) gives values of Z 
AA
 or Z 
AB
 in the
range of 10 
34
 collisions in a volume of 1 m
3
 in a second.
 
If we assume that each collision leads to the formation of the products then the rate of the reaction r = -
d [A] / dt will equal Z
AB
/ L. The division by L is to get the correct molar unit of (moles / volume) / s .
Since all collisions are not effective but only those collisions with energy greater than some E'
a
 are
effective, the rate may be expressed as
 
r = - d [ A ] / dt = -( Z 
A
 
B
 / L ) e 
- E'
a
/ RT
(32.7)
E'
a
 is slightly different from the activation energy E
a
 (see below)
Expanding eq. (32.7) we get,
  
- d [A] / dt = -  (8 k
B
T /   )
1/ 2
 L [A] [B] e 
- E '
a
/ RT
(32.8)
= - k
c
 [ A] [ B] where
  
k
c
 =  (8 k
B
T /   )
1/ 2
 L e 
- E '
a
/ RT
(32.9)
Where k
c
 is the rate constant in the collision theory. Notice that unlike the Arrhenius theory, the prefactor
depends on  . If we insist on rewriting d [A] /dt = - k [A] [B] where k = A e
 - E
a
/ RT
 with A
independent of T, then the activation energy E
a
 of the Arrhenius equation is given by E
a
 = RT
2
 d ln k / dT.
Applying this formula to eq. (32.9) we get
  
E
a
 = RT
2
 d ln k
c
 / dt = E'
a
 + 1/2 RT
(32.10)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
It is for this reason that E'
a
 in (32.8) was distinguished from E
a
 of (32.10). Since (1/2) RT is much smaller
than E'
a
, the difference between E
a
 and E'
a
 is generally not of great significance.
 
Let us compare some experimental results with the results of collision theory. These are shown in Table
32.1
Reactions Arrhenius Parameters from
experiments
A / (M 
– 1
 s 
– 1
)         E
a
 / ( k J / mol)
Collision theory
A / (M 
– 1
 s 
– 1
)
      A 
Arrhenius
 /
      A
Collision Theory
     
2NOCl  2 NO + Cl
9.4 * 10 
9 102.0
5.9 * 10 
10 0.16
2ClO  Cl
2
 +O
2 6.3 * 10 
7 0.0
2.5 * 10 
10
2.5 * 10 
-3
H
2
 + C
2
 H
4
  C
2
 H
6 1.24 * 10 
6 180.0
7.3 * 10 
11
1.7 * 10 
- 6
K
2
 + Br
2
 KBr + Br
1.0 * 10 
12 0
2.1 * 10 
11 4.8
 
Table 32.1 Arrhenius parameters (from Eq. 32.10) and collision theory results
for a few gas phase reactions.
 
The comparisons are not encouraging. The above collision theory can not predict E
a
. In the first three
reactions, the experimental collision frequency is much lower than the collision theory results while in the
last reaction, it is higher. Rather than discarding the theory, we seek to improve it and learn a lot in the
process (in a manner similar to getting improved equations of state, starting with the ideal gas equation of
sate). An explanation of the above difference in the values of A is that the actual "reactive" cross section 
 is different from the collision cross section . Consider the collisions between H
2
 and C
2
 H
4
 shown in
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


 Module 7 : Theories of Reaction Rates
Lecture 32 : Theories Of Reaction Rates - I : Collision Theory
   Objectives
  After studying this lecture you will be able to do the following.
Calculate collision frequency using kinetic theory.
  
Distinguish between collision cross section and reaction cross section.
  
Compare collision theory with Arrhenius theory and experiments.
  
Define the steric factor.
  
Rationalize the departure of collision theory from experiments for ionic reactions using the harpoon
mechanism.
   32.1Introduction
In the earlier lectures on chemical kinetics, we have studied the empirical aspects in detail. We now begin
the theoretical explanations of chemical rate processes. This is one of the most challenging areas in
chemistry. Molecular structure is fairly well understood through the advances in spectroscopy and
quantum chemistry. With the knowledge of the same molecular structures and intermolecular forces,
chemical equilibrium (thermodynamics) can be predicted quite accurately. An understanding the rates of
processes microscopically requires answers to several interesting questions. Some of the main questions
are as follows. How close should the molecules approach one another for the reaction to occur? Does the
reaction "occur" only after reaching this separation or does it start well before the distance of closest
separation is reached? Do all the reactants react identically or is there a distribution over several
possibilities? Do all "encounters" between the reactants produce the products? How closely can we
monitor the progress of the reaction theoretically (as well as experimentally)? Over the last four decades,
the time resolution of this monitoring has improved dramatically from microseconds to femtoseconds.
While most of the above questions relate to gas phase processes, solution reactions have the additional
contributions from the participation of the solvent. We shall take up these issues one by one in modest
details. We first begin with the gas phase collision theory. This is followed by the transition state theory
(lecture 33). Potential energy surfaces are introduced in lecture 34 and the trajectories over potential
energy surfaces will be considered in lecture 35.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   32.2 Bimolecular gas phase collision theory
Consider the following gas phase reaction
A + B  Products ( P ) (32.1)
The rate of the reaction, r = dP/dt, depends on the following factors: a) The concentrations or the number
densities of molecules A and B, i.e., N
A
 and N
B 
( N
A
 = numbers of molecules in unit volume of the
container, similarly N
B
 ), b) number of collisions between of A and B in unit time. This is the collision rate.
Larger the relative velocity between A an B, larger the collision rate.The relative speed between A and B is
obtained from 1/ 2  = 1/2 k
B
T or , the relative velocity = ( k
B
T /  )
1/2 
is determined by the
temperature T at equilibrium; and thirdly, c) the collision cross section, . The collision cross section is
the area within which the center of A should be around the target B so that a collision can be said to have
occurred. Let us first calculate the collision frequency between A and B.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 32.1 Collision cross section and the collision cylinder.
 
All the molecules that come inside the range of the collision cylinder will collide with the central molecule
on the extreme left. All those outside this cylinder do not collide with molecule A.
 
To calculate the collision frequency, consider the motion of the molecule labeled A. In time  t, the molecule
moves a distance  t where  is the average speed of the molecule. Any other molecule which is within the
volume of the cylinder   t d 
2 
spanned by molecule A, will collide with molecule A. Here d 
2
 is the area of
the cylinder spanned by A. This collision cylinder is shown in Fig 32.1. All the molecules that collide with A can be
easily identified in the figure. During collisions, the directions of molecules change and the cylinder is not a rigid
space but the effectively available space for the molecules. Since all molecules are moving, we need to use the
relative speeds between molecules. From kinetic theory of gases, the average relative speed between molecules A
and B is
 
 
rel
 = (8 k
B
T /  )
1/ 2
, 
-1
 = m 
A
-1
 + m 
B
-1
 
(32.2)
Where  is the reduced mass between A and B. For identical molecules,  
rel 
=  , where the latter is
calculated with m in the denominator of Eq. (32.2) in place of the reduced mass. The factor 2
1/2
 is because       
= m / 2. The area of the cylinder is called the collision cross section. Molecules outside this area do not collide with
A. If N
A
 is the number of molecules of A in volume V, the collision frequency is given by
 
z =    N
A
 / V
(32.3)
This z is the number of collisions encounter by a single molecule. Since there are N
A
 molecules in volume V, the
number of collisions between molecules of A in unit time and unit volume, Z
A
A
  
Z
AA
 = (1/2) z N
A
 / V
(32.4)
Substituting the values of all the factors, we have
Z
AA
 =  (4 k
B
T /  m )
1/ 2
 ( N
A
 / V )
 2
If n
A
 is the number of moles of A in volume V, then, [A] = molar concentration of A = n
A
 / V . Since N
A
 = n
A
L
where L = Avogadro number, we have for Z 
AA
 
Z
AA
 =  (4 k
B
T /  m )
1/2
 L
2
 [A]
 2
(32.5)
 
 
 
If the collision is between molecules of type A with molecules of type B, the factor of 1/2 in eq. (32.4)
drops out. If two molecules of A collide it is counted as one collision. When molecules of A collide with
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
molecules of B, there are distinct AB collisions, AA and BB collisions. The total number of collisions
between A and B molecules in unit time and unit volume is given by
 
Z 
AB
 = (8 k
B
T / )
1/ 2
 L
2
 [A] [B] 
(32.6)
Substitution of the values of all the factors in eq (32.5) and (32.6) gives values of Z 
AA
 or Z 
AB
 in the
range of 10 
34
 collisions in a volume of 1 m
3
 in a second.
 
If we assume that each collision leads to the formation of the products then the rate of the reaction r = -
d [A] / dt will equal Z
AB
/ L. The division by L is to get the correct molar unit of (moles / volume) / s .
Since all collisions are not effective but only those collisions with energy greater than some E'
a
 are
effective, the rate may be expressed as
 
r = - d [ A ] / dt = -( Z 
A
 
B
 / L ) e 
- E'
a
/ RT
(32.7)
E'
a
 is slightly different from the activation energy E
a
 (see below)
Expanding eq. (32.7) we get,
  
- d [A] / dt = -  (8 k
B
T /   )
1/ 2
 L [A] [B] e 
- E '
a
/ RT
(32.8)
= - k
c
 [ A] [ B] where
  
k
c
 =  (8 k
B
T /   )
1/ 2
 L e 
- E '
a
/ RT
(32.9)
Where k
c
 is the rate constant in the collision theory. Notice that unlike the Arrhenius theory, the prefactor
depends on  . If we insist on rewriting d [A] /dt = - k [A] [B] where k = A e
 - E
a
/ RT
 with A
independent of T, then the activation energy E
a
 of the Arrhenius equation is given by E
a
 = RT
2
 d ln k / dT.
Applying this formula to eq. (32.9) we get
  
E
a
 = RT
2
 d ln k
c
 / dt = E'
a
 + 1/2 RT
(32.10)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
It is for this reason that E'
a
 in (32.8) was distinguished from E
a
 of (32.10). Since (1/2) RT is much smaller
than E'
a
, the difference between E
a
 and E'
a
 is generally not of great significance.
 
Let us compare some experimental results with the results of collision theory. These are shown in Table
32.1
Reactions Arrhenius Parameters from
experiments
A / (M 
– 1
 s 
– 1
)         E
a
 / ( k J / mol)
Collision theory
A / (M 
– 1
 s 
– 1
)
      A 
Arrhenius
 /
      A
Collision Theory
     
2NOCl  2 NO + Cl
9.4 * 10 
9 102.0
5.9 * 10 
10 0.16
2ClO  Cl
2
 +O
2 6.3 * 10 
7 0.0
2.5 * 10 
10
2.5 * 10 
-3
H
2
 + C
2
 H
4
  C
2
 H
6 1.24 * 10 
6 180.0
7.3 * 10 
11
1.7 * 10 
- 6
K
2
 + Br
2
 KBr + Br
1.0 * 10 
12 0
2.1 * 10 
11 4.8
 
Table 32.1 Arrhenius parameters (from Eq. 32.10) and collision theory results
for a few gas phase reactions.
 
The comparisons are not encouraging. The above collision theory can not predict E
a
. In the first three
reactions, the experimental collision frequency is much lower than the collision theory results while in the
last reaction, it is higher. Rather than discarding the theory, we seek to improve it and learn a lot in the
process (in a manner similar to getting improved equations of state, starting with the ideal gas equation of
sate). An explanation of the above difference in the values of A is that the actual "reactive" cross section 
 is different from the collision cross section . Consider the collisions between H
2
 and C
2
 H
4
 shown in
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig 32.2.
 
 
Figure 32.2 Various collisions of H
2
 with C
2
 H
4
 
 
Most collisions are in random directions and only a small fraction of collisions with the correct relative alignments
lead to the products. This requirement of "proper relative" alignment between molecules is called the steric
effect and we can define the steric factor P as
 
 = P 
(32.11)
For the four reactions in Table 32., the values of P are 0.16, 2.5 * 10 
-3
, 1.7 * 10 
- 6 
and 4.8 respectively.
Before we consider  being larger than , we will consider the dependence of  on energy or velocity of
molecules. When the steric factor is small, very few collisions occur with suitable relative orientations of
reactants.
 
The above discussion brings out the central role of reaction cross sections (  ) in chemical dynamics. In
addition to the dependence of the reaction cross section on the relative orientations of reactants,  depends on
the relative velocity (or relative kinetic energy) as well. For smaller relative velocities, the reactants often have
more time to feel the presence of one another.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


 Module 7 : Theories of Reaction Rates
Lecture 32 : Theories Of Reaction Rates - I : Collision Theory
   Objectives
  After studying this lecture you will be able to do the following.
Calculate collision frequency using kinetic theory.
  
Distinguish between collision cross section and reaction cross section.
  
Compare collision theory with Arrhenius theory and experiments.
  
Define the steric factor.
  
Rationalize the departure of collision theory from experiments for ionic reactions using the harpoon
mechanism.
   32.1Introduction
In the earlier lectures on chemical kinetics, we have studied the empirical aspects in detail. We now begin
the theoretical explanations of chemical rate processes. This is one of the most challenging areas in
chemistry. Molecular structure is fairly well understood through the advances in spectroscopy and
quantum chemistry. With the knowledge of the same molecular structures and intermolecular forces,
chemical equilibrium (thermodynamics) can be predicted quite accurately. An understanding the rates of
processes microscopically requires answers to several interesting questions. Some of the main questions
are as follows. How close should the molecules approach one another for the reaction to occur? Does the
reaction "occur" only after reaching this separation or does it start well before the distance of closest
separation is reached? Do all the reactants react identically or is there a distribution over several
possibilities? Do all "encounters" between the reactants produce the products? How closely can we
monitor the progress of the reaction theoretically (as well as experimentally)? Over the last four decades,
the time resolution of this monitoring has improved dramatically from microseconds to femtoseconds.
While most of the above questions relate to gas phase processes, solution reactions have the additional
contributions from the participation of the solvent. We shall take up these issues one by one in modest
details. We first begin with the gas phase collision theory. This is followed by the transition state theory
(lecture 33). Potential energy surfaces are introduced in lecture 34 and the trajectories over potential
energy surfaces will be considered in lecture 35.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   32.2 Bimolecular gas phase collision theory
Consider the following gas phase reaction
A + B  Products ( P ) (32.1)
The rate of the reaction, r = dP/dt, depends on the following factors: a) The concentrations or the number
densities of molecules A and B, i.e., N
A
 and N
B 
( N
A
 = numbers of molecules in unit volume of the
container, similarly N
B
 ), b) number of collisions between of A and B in unit time. This is the collision rate.
Larger the relative velocity between A an B, larger the collision rate.The relative speed between A and B is
obtained from 1/ 2  = 1/2 k
B
T or , the relative velocity = ( k
B
T /  )
1/2 
is determined by the
temperature T at equilibrium; and thirdly, c) the collision cross section, . The collision cross section is
the area within which the center of A should be around the target B so that a collision can be said to have
occurred. Let us first calculate the collision frequency between A and B.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 32.1 Collision cross section and the collision cylinder.
 
All the molecules that come inside the range of the collision cylinder will collide with the central molecule
on the extreme left. All those outside this cylinder do not collide with molecule A.
 
To calculate the collision frequency, consider the motion of the molecule labeled A. In time  t, the molecule
moves a distance  t where  is the average speed of the molecule. Any other molecule which is within the
volume of the cylinder   t d 
2 
spanned by molecule A, will collide with molecule A. Here d 
2
 is the area of
the cylinder spanned by A. This collision cylinder is shown in Fig 32.1. All the molecules that collide with A can be
easily identified in the figure. During collisions, the directions of molecules change and the cylinder is not a rigid
space but the effectively available space for the molecules. Since all molecules are moving, we need to use the
relative speeds between molecules. From kinetic theory of gases, the average relative speed between molecules A
and B is
 
 
rel
 = (8 k
B
T /  )
1/ 2
, 
-1
 = m 
A
-1
 + m 
B
-1
 
(32.2)
Where  is the reduced mass between A and B. For identical molecules,  
rel 
=  , where the latter is
calculated with m in the denominator of Eq. (32.2) in place of the reduced mass. The factor 2
1/2
 is because       
= m / 2. The area of the cylinder is called the collision cross section. Molecules outside this area do not collide with
A. If N
A
 is the number of molecules of A in volume V, the collision frequency is given by
 
z =    N
A
 / V
(32.3)
This z is the number of collisions encounter by a single molecule. Since there are N
A
 molecules in volume V, the
number of collisions between molecules of A in unit time and unit volume, Z
A
A
  
Z
AA
 = (1/2) z N
A
 / V
(32.4)
Substituting the values of all the factors, we have
Z
AA
 =  (4 k
B
T /  m )
1/ 2
 ( N
A
 / V )
 2
If n
A
 is the number of moles of A in volume V, then, [A] = molar concentration of A = n
A
 / V . Since N
A
 = n
A
L
where L = Avogadro number, we have for Z 
AA
 
Z
AA
 =  (4 k
B
T /  m )
1/2
 L
2
 [A]
 2
(32.5)
 
 
 
If the collision is between molecules of type A with molecules of type B, the factor of 1/2 in eq. (32.4)
drops out. If two molecules of A collide it is counted as one collision. When molecules of A collide with
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
molecules of B, there are distinct AB collisions, AA and BB collisions. The total number of collisions
between A and B molecules in unit time and unit volume is given by
 
Z 
AB
 = (8 k
B
T / )
1/ 2
 L
2
 [A] [B] 
(32.6)
Substitution of the values of all the factors in eq (32.5) and (32.6) gives values of Z 
AA
 or Z 
AB
 in the
range of 10 
34
 collisions in a volume of 1 m
3
 in a second.
 
If we assume that each collision leads to the formation of the products then the rate of the reaction r = -
d [A] / dt will equal Z
AB
/ L. The division by L is to get the correct molar unit of (moles / volume) / s .
Since all collisions are not effective but only those collisions with energy greater than some E'
a
 are
effective, the rate may be expressed as
 
r = - d [ A ] / dt = -( Z 
A
 
B
 / L ) e 
- E'
a
/ RT
(32.7)
E'
a
 is slightly different from the activation energy E
a
 (see below)
Expanding eq. (32.7) we get,
  
- d [A] / dt = -  (8 k
B
T /   )
1/ 2
 L [A] [B] e 
- E '
a
/ RT
(32.8)
= - k
c
 [ A] [ B] where
  
k
c
 =  (8 k
B
T /   )
1/ 2
 L e 
- E '
a
/ RT
(32.9)
Where k
c
 is the rate constant in the collision theory. Notice that unlike the Arrhenius theory, the prefactor
depends on  . If we insist on rewriting d [A] /dt = - k [A] [B] where k = A e
 - E
a
/ RT
 with A
independent of T, then the activation energy E
a
 of the Arrhenius equation is given by E
a
 = RT
2
 d ln k / dT.
Applying this formula to eq. (32.9) we get
  
E
a
 = RT
2
 d ln k
c
 / dt = E'
a
 + 1/2 RT
(32.10)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
It is for this reason that E'
a
 in (32.8) was distinguished from E
a
 of (32.10). Since (1/2) RT is much smaller
than E'
a
, the difference between E
a
 and E'
a
 is generally not of great significance.
 
Let us compare some experimental results with the results of collision theory. These are shown in Table
32.1
Reactions Arrhenius Parameters from
experiments
A / (M 
– 1
 s 
– 1
)         E
a
 / ( k J / mol)
Collision theory
A / (M 
– 1
 s 
– 1
)
      A 
Arrhenius
 /
      A
Collision Theory
     
2NOCl  2 NO + Cl
9.4 * 10 
9 102.0
5.9 * 10 
10 0.16
2ClO  Cl
2
 +O
2 6.3 * 10 
7 0.0
2.5 * 10 
10
2.5 * 10 
-3
H
2
 + C
2
 H
4
  C
2
 H
6 1.24 * 10 
6 180.0
7.3 * 10 
11
1.7 * 10 
- 6
K
2
 + Br
2
 KBr + Br
1.0 * 10 
12 0
2.1 * 10 
11 4.8
 
Table 32.1 Arrhenius parameters (from Eq. 32.10) and collision theory results
for a few gas phase reactions.
 
The comparisons are not encouraging. The above collision theory can not predict E
a
. In the first three
reactions, the experimental collision frequency is much lower than the collision theory results while in the
last reaction, it is higher. Rather than discarding the theory, we seek to improve it and learn a lot in the
process (in a manner similar to getting improved equations of state, starting with the ideal gas equation of
sate). An explanation of the above difference in the values of A is that the actual "reactive" cross section 
 is different from the collision cross section . Consider the collisions between H
2
 and C
2
 H
4
 shown in
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig 32.2.
 
 
Figure 32.2 Various collisions of H
2
 with C
2
 H
4
 
 
Most collisions are in random directions and only a small fraction of collisions with the correct relative alignments
lead to the products. This requirement of "proper relative" alignment between molecules is called the steric
effect and we can define the steric factor P as
 
 = P 
(32.11)
For the four reactions in Table 32., the values of P are 0.16, 2.5 * 10 
-3
, 1.7 * 10 
- 6 
and 4.8 respectively.
Before we consider  being larger than , we will consider the dependence of  on energy or velocity of
molecules. When the steric factor is small, very few collisions occur with suitable relative orientations of
reactants.
 
The above discussion brings out the central role of reaction cross sections (  ) in chemical dynamics. In
addition to the dependence of the reaction cross section on the relative orientations of reactants,  depends on
the relative velocity (or relative kinetic energy) as well. For smaller relative velocities, the reactants often have
more time to feel the presence of one another.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 32.3 Dependence of  (reaction cross section) on relative velocity (  ) .
 
 
To calculate the collision frequency, consider the motion of the molecule labeled A. In time  t, the
molecule moves a distance  t where  is the average speed of the molecule. Any other molecule which
is within the volume of the cylinder   t d 
2 
spanned by molecule A, will collide with molecule A. Here 
d 
2
 is the area of the cylinder spanned by A. This collision cylinder is shown in Fig 32.1. All the molecules
that collide with A can be easily identified in the figure. During collisions, the directions of molecules
change and the cylinder is not a rigid space but the effectively available space for the molecules. Since all
molecules are moving, we need to use the relative speeds between molecules. From kinetic theory of
gases, the average relative speed between molecules A and B is
 
 
rel
 = (8 k
B
T /  )
1/ 2
, 
-1
 = m 
A
-1
 + m 
B
-1
 
(32.2)
Where  is the reduced mass between A and B. For identical molecules,  
rel 
=  , where the latter
is calculated with m in the denominator of Eq. (32.2) in place of the reduced mass. The factor 2
1/2
 is
because        = m / 2. The area of the cylinder is called the collision cross section. Molecules outside this
area do not collide with A. If N
A
 is the number of molecules of A in volume V, the collision frequency is
given by
 
z =    N
A
 / V
(32.3)
This z is the number of collisions encounter by a single molecule. Since there are N
A
 molecules in volume
V, the number of collisions between molecules of A in unit time and unit volume, Z
A
A
  
Z
AA
 = (1/2) z N
A
 / V
(32.4)
Substituting the values of all the factors, we have
Z
AA
 =  (4 k
B
T /  m )
1/ 2
 ( N
A
 / V )
 2
If n
A
 is the number of moles of A in volume V, then, [A] = molar concentration of A = n
A
 / V . Since N
A
 =
n
A
L where L = Avogadro number, we have for Z 
AA
 
Z
AA
 =  (4 k
B
T /  m )
1/2
 L
2
 [A]
 2
(32.5)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Since the cross section depends on relative energy, we need to integrate over the distribution function of
energy to get the rate constant, i.e.,
 
d [ A] / dt = - (  )  
rel
 L [ A] [B] becomes 
(32.12)
 
d [ A] / dt = - {  (  ) 
rel 
f ( ) d  } L [A] [B]
(32.13)
 
and with 
rel 
= (2 /  )
1/2
 from  = 1/2 
2
rel
(32.14)
Substituting velocities by energies by using the above substitutions, we have
k 
c
 = L   (  ) ( 2  /  )
1/2
 f ( ) d 
(32.15)
 
The primary reason for converting from 
rel
 to relative kinetic energy  is that experimentally cross
sections are often tabulated as functions of energy. Let us first convert the Maxwell Boltzmann velocity
distribution into an energy distribution.
f (  ) d  = 4 ( m / 2  k
B
T )
3/2
 
2
 exp ( - m 
2 
/ k
B
T ) d (32.16)