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H. Scott Fogler
Chapter 3 1/24/
1
Professional Reference Shelf
A. Collision Theory
Overview – Collision Theory
In Chapter 3, we presented a number of rate laws that depended on both
concentration and temperature. For the elementary reaction
A + B → C + D
the elementary rate law is
"r A
= kC A
C
B
= Ae
"E RT
C A
C
B
We want to provide at least a qualitative understanding of why the rate law takes this
form. We will first develop the collision rate, using collision theory for hard spheres of
cross section S r
,
AB
2
. When all collisions occur with the same relative velocity, U R
, the
number of collisions between A and B molecules,
Z
AB
, is
Z
AB
= S
r
U
R
C
A
C
B
[collisions/s/molecule]
Next, we will consider a distribution of relative velocities and only consider those
collisions that have an energy of E A
or greater in order to react to show
"r A
= Ae
"E A
RT
C A
C
B
where
A = "#
AB
2
8 k B
T
"μ AB
1 2
N
Avo
with σ AB
= collision radius, k B
= Boltzmann’s constant, μ AB
= reduced mass,
T = temperature, and N Avo
= Avogadro’s number. To obtain an estimate of E A
, we use
the Polyani Equation
E
A
= E
A
o
#H
Rx
Where ΔH Rx
is the heat of reaction and
E
A
o
and γ P
are the Polyani Parameters. With
these equations for A and E A
we can make a first approximation to the rate law
parameters without going to the lab.
2
HOT BUTTONS
I. Fundamentals of Collision Theory
II. Shortcomings of Collision Theory
III. Modifications of Collision Theory
A. Distribution of Velocities
B. Collisions That Result in Reaction
- Model 1 Pr = 0 or 1
- Model 2 Pr = 0 or Pr = (E–E A
)/E
IV. Other Definitions of Activation Energy
A. Tolman’s Theorem E a
= E
B. Fowler and Guggenheim
C. Energy Barrier
V. Estimation of Activation Energy from the Polyani Equation
A. Polyani Equation
B. Marcus Extension of the Polyani Equation
C. Blowers-Masel Relation
VI. Closure
References for Collision Theory, Transition State Theory, and Molecular Dynamics
P. Atkins, Physical Chemistry , 6th ed. (New York: Freeman, 1998)
P. Atkins, Physical Chemistry , 5th ed. (New York: Freeman, 1994).
G. D. Billing and K. V. Mikkelsen, Introduction to Molecular Dynamics and Chemical
Kinetics (New York: Wiley, 1996).
P.W. Atkins, The Elements of Physical Chemistry , 2nd ed. (Oxford: Oxford Press, 1996).
K. J. Laidler, Chemical Kinetics , 3rd ed. (New York: Harper Collins, 1987).
G. Odian, Principles of Polymerization , 3rd ed. (New York: Wiley 1991).
R. I. Masel, Chemical Kinetics and Catalysis (New York: Wiley Interscience, 2001).
As a shorthand notation, we will use the following references nomenclature:
A6p701 means Atkins, P. W., Physical Chemistry , 6th ed. (1998) page 701.
L3p208 means Laidler, K. J., Chemical Kinetics , 3rd,ed. (1987) page 208.
This nomenclature means that if you want background on the principle, topic,
postulate, or equation being discussed, go to the specified page of the referenced text.
I. FUNDAMENTALS OF COLLISION THEORY
The objective of this development is to give the reader insight into why the rate
laws depend on the concentration of the reacting species (i.e., – r A
= kC A
C B
) and
why the temperature dependence is in the form of the Arrhenius law, k=Ae
.
To achieve this goal, we consider the reaction of two molecules in the gas phase
A + B !! "C + D
We will model these molecules as rigid spheres.
4
m A
= mass of a molecule of species A (gm)
m B
= mass of a molecule of species B (gm)
μ AB
= reduced mass =
m A
m B
m A
(g), [Let μ ≡ μ AB
]
M A
= Molecular weight of A (Daltons)
N Avo
= Avogadro’s number 6.022 molecules/mol
R = Ideal gas constant 8.314 J/mol•K = 8.314 kg • m
2
/s
2
/mol/K
We note that R = N Avo
k B
and M A
= N Avo
, therefore we can write the ratio
(k B
/μ AB
) as
k B
μ AB
R
M
A
M
B
M
A
+ M
B
(R3.A-2)
An order of magnitude of the relative velocity at 300 K is U R
! 3000 km hr,
i.e., ten times the speed of an Indianapolis 500 Formula 1 car. The collision
diameter and velocities at 0°C are given in Table R3.A-1.
Table R3.A- 1 Molecular Diameters
†
Molecule
Average Velocity,
(meters/second) Molecular Diameter (Å)
H 2
1687 2.
CO 453 3.
Xe 209 4.
He 1200 2.
N 2
450 3.
O 2
420 3.
H 2
O 560 3.
C 2
H 6
437 5.
C 6
H 6
270 3.
CH 4
593 4.
NH 3
518 4.
H 2 S 412 4.
CO 2
361 4.
N 2
O 361 4.
NO 437 3.
Consider a molecule A moving in space. In a time Δt, the volume ΔV swept out
by a molecule of A is
†
Courtesy of J. F. O’Hanlon, A User’s Guide to Vacuum Technology (New York: Wiley, 1980).
5
CD/CollisionTheory/ProfRef.doc
!V = U R
( !t)
!l 647 48
"# AB
2
!V
A
Figure R3.A- 3 Volume swept out by molecule A in time Δt.
The bends in the volume represent that even though molecule A may change
directions upon collision the volume sweep out is the same. The number of
collisions that will take place will be equal to the number of B molecules,
ΔV
˜ C B
, that are in the volume swept out by the A molecule:
˜ C B
!V = No. of B molecules in !V
[ ]
where
˜ C B
is in
molecules dm
3
[ ]
rather than [moles/dm
3
]
In a time Δt, the number of collisions of this one A molecule with many B
molecules is
U
R
C
B
AB
2
$t. The number of collisions of this one A molecule
with all the B molecules per unit time is
˜ Z 1A• B
= !" AB
2 ˜ C B
U R
(R3.A-3)
However, we have many A molecules present at a concentration,
C
A
,
(molecule/dm
3
). Adding up the collisions of all the A molecules per unit
volume,
˜ C A
, then the number of collisions
Z
AB
of all the A molecules with all
B molecules per time per unit volume is
Z
AB
AB
2
S r 678
U
R
C
A
C
B
= S
r
U
R
C
A
C
B
(R3.A-4)
Where S r
is the collision cross section (Å)
2
. Substituting for S r
and U R
Z
AB
AB
2
8 k B
T
"μ
1 2
C
A
C
B
[molecules/time/volume] (R3.A-5)
If we assume all collisions result in reactions, then
"˜r A
Z
AB
AB
2
8 k B
T
#μ
1 2
C
A
C
B
[molecules/time/volume] (R3.A-6)
Multiplying and dividing by Avogadrós number, N Avo
, we can put our
equation for the rate of reaction in terms of the number of moles/time/vol.
"˜r A
N
Avo
"r A
N
Avo
AB
2
8 k B
T
)μ
1 2
˜ C A
N
Avo
C A
C
B
N
Avo
C B
N
Avo
2
(R3.A-7)
7
k B
μ
= 8571 m
2
s
2
K
Calculate the relative velocity
U
R
273 K
m
2
s
2
K
1 2
= 2441 m s = 2.44 ( 10
13
Å s (R3.A-E-2)
S
r
AB
2
= " # A
B
[ ]
2
% 10
m +1.55 $ 10
% 10
m
[ ]
2
% 20
m
2
molecule
Calculate the frequency factor A
A =
20
m
2
molecule
[^2441 m^ s] 6.02^ "^10
23
molecule mol
[ ]
(R3.A-E-3)
A = 2.29 " 10
8
m
3
mol# s
11
dm
3
mol # s
(R3.A-E-4)
A = 2.29 " 10
6
m
3
mol# s
1 mol
23
molecule
10
Å
m
3
A = 3.81" 10
14
Å
3
molecule s (R3.A-E-5)
The value reported in Masel
†
from Wesley is
A =1.5 " 10
14
Å
3
molecule s
Close, but no cigar, as Groucho Marx would say.
For many simple reaction molecules, the calculated frequency factor A calc
, is in
good agreement with experiment. For other reactions, A calc
, can be an order of
magnitude too high or too low. In general, collision theory tends to overpredict the
frequency factor A
8
dm
3
mol•s
< A
calc
11
dm
3
mol • s
T erms of cubic angstroms per molecule per second the frequency factor is
12
Å
3
molecule s < A calc
15
Å
3
molecule s
There are a couple of things that are troubling about the rate of reaction given by
Equation (R3.A-10), i.e.
† M1p367.
8
"r A
= AC
A
C
B
(R3.A-10)
First and most obvious is the temperature dependence. A is proportional to the
square root of temperature and, therefore, is – r A
:
"r A
~ A ~ T
However we know that the temperature dependence of the rate of chemical
reaction on temperature is given by the Arrhenius equation
"r A
= Ae
"E RT
C A
C
B
(R3.A-11)
or
k = Ae
"E RT
(R3.A-12)
Next, we will discuss this shortcoming of collision theory, along with the
assumption that all collisions result in reaction.
II. SHORTCOMINGS OF COLLISION THEORY
A. The collision theory outlined above does not account for orientation of the
collision, front-to-back and along the line-of-centers. That is, molecules need to
collide in the correct orientation for reaction to occur. Figure R3.A-4 shows
molecules colliding whose centers are offset by a distance b.
b
B
b = impact parameter
A
Figure R3.A- 4 Grazing collisions.
B. Collision theory does not explain activation barriers. Activation barriers occur
because bonds need to be stretched or distorted in order to react and these
processes require energy. Molecules must overcome electron-electron repulsion
in order to come close together
†
C. The collision theory does not explain the observed temperature dependence
given by Arrhenius equation
k = Ae
E RT
D. Collision theory assumes all A molecules have the same relative velocity, the
average one.
U
R
8 k B
T
"μ AB
1 2
(R3.A-1)
However, there is a distribution of velocities f(U,T). One distribution most used
is the Maxwell-Boltzmann distribution.
† Masel, 1p
10
We now let
k(U) be the specific reaction rate for a collision and reaction
of A-B molecules with a velocity U.
k (U ) = S
r
(U )U^ m^
3
molecule s
[ ]
(R3.A-15)
Equation (R3.A-15) will give the specific reaction rate and hence the reaction
rate for only those collisions with velocity U. We need to sum up the collisions
of all velocities. We will use the Maxwell-Boltzmann distribution for f(U,T) and
integrate over all relative velocities.
k (T ) = k
0
"
(U )f(U,T)dU^ =^ f^ (U ,T)
0
"
S
r
(U ) UdU^ (R3.A-16)
Maxwell distribution function of velocities for the A/B pair of reduced mass
μ AB
is
†
f (U ,T) = 4 "
μ
2 "k B
T
3 2
U
2
e
)
μU
2
2 k B
T
(R3.A-17)
Combining Equations (16) and (17)
k T
= S
r 0
"
U 4 $
μ
2 $k B
T
3 2
U
2
e
μU
2
2 k B
T
dU (R3.A-18)
For brevity, we let S r
=S r
(U), we will now express the distribution function in
terms of the translational energy ε T
.
We are now going to express the equation for
k (T)in terms of kinetic
energy rather than velocity. Relating the differential translational kinetic
energy, ε
, to the velocity U:
! t
=
μU
2
2
Multiplying and dividing by
2
μ
and μ, we obtain
d" t
= μ UdU
and hence, the reaction rate
k (T ) = 4 "
μ
2 "k B
T
3 2
S
r 0
)
μ
μU
2
e
μU
2
2
1
k B
T
$
%
&
'
(
1
μ
μUdU
dμU
2
2
d,t
123
Simplifying
μ
2 "k B
T
3 2
μ
2
S
r 0
)
t
e
,
t
k B
T
d+ t
† 2p185, A5p
11
k T
"μ k B
( T)
3
1 2
S
r 0
)
t
e
,
t
k B
T
d+ t
m
3
s molecule
[ ]
(R3.A-19)
k (T ) =
"μ k B
( T)
3
1 2
S
r
t
0
t
e
,) t
k B
T
d) t
Multiplying and dividing by k B
T and noting
t
k B
( T) = (E RT), we obtain
k (T ) =
8 k B
T
"μ
1 2
S
r
(E )
0
)
Ee
+E RT
RT
dE
RT
1 (R3.A-20)
Again, recall the tilde, e.g.,
k (T), denotes that the specific reaction rate is per
molecule (dm
3
/molecule/s). The only thing left to do is to specify the reaction
cross-section, S r
(E), as a function of kinetic energy E for the A/B pair of
molecules.
B. Collisions that Result in Reaction
We now modify the hard sphere collision cross section to account for the fact
that not all collisions result in reaction. Now we define S r
to be the reaction
cross section defined as
S r
= P r
!" AB
2
where P r
is the probability of reaction. In the first model we say the probability
is either 0 or 1. In the second model P r
varies from 0 to 1 continuously. We will
now insert each of these modules into Equation (R3.A-20).
B.1 Model 1
In this model, we say only those hard collisions that have kinetic energy E A
or
greater will react. Let E ≡ ε t
. That is, below this energy, E A
, the molecules do not
have sufficient energy to react so the reaction cross section is zero, S r
=0. Above
this kinetic energy all the molecules that collide react and the reaction cross-
section is
S
r
AB
2
P
r
= 0 " S
r
[ E,T] =^0 for^ E^ <^ E^
A
P
r
= 1 S
r
[ E,T] =^ #$
AB
2
for E % E A
(R3.A-21)
(R3.A-22)
13
(Click Back 1 cont’d)
udv "
= uv # vdu "
k (^) (T ) = "# AB
2
8 k B
T
μ"
1 2
X
u
E A
RT
e
,X
dX
dv
= ,Xe
,X
E A
RT
, e
,X
dX
E A
RT
AB
2
8 k B
T
μ"
1 2
E
A
RT
e
*E A
RT
*E A
RT
AB
2
8 k B
T
μ"
1 2
E
A
RT
e
0 E A
RT
E
A
RT
k (^) (T ) = N Avo
k (^) (T ) = "# AB
2
8 k B
T
μ"
1 2
N
Avo
E
A
RT
= A
E
A
RT
Over predicts the
frequency factor
S
r
= P
r
AB
2
Generally,
E
A
RT
1 , so
k =
U
R
AB
2
E A
RT
A $
e
%E A
RT
Converting
kto a per mole basis rather than a per molecular basis we have
A^ " =
E
A
RT
AB
2
T
μ AB
1 2
N
Avo
A
E
A
RT
A
k = A" e
#E A
RT
=
E
A
RT
Ae
#E A
RT
We have good news and bad news. This model gives the correct temperature
dependence but predicted frequency factor A′ is even greater than A given by
Equation (R3.A-9) (which itself is often too large) by a factor (E A
/RT). So we
have solved one problem, the correct temperature dependence, but created
another problem, too large a frequency factor. Let’s try Model 2.
14
B.2 Model 2
In this model, we again assume that the colliding molecules must have an
energy E A
or greater to react. However, we now assume that only the kinetic
energy directed along the line of centers E <<
is important. So below E A
the
reaction cross section is zero, S r
=0. The kinetic energy of approach of A toward
B with a velocity U R
is E = μ AB
U
R
2
. However, this model assumes that only
the kinetic energy directly along the line of centers contributes to the reaction.
(Click Back 2)
Here, as E increases above E A
the number of collisions that result in reaction
increases. The probability for a reaction to occur is
†
P
r
E " E
A
E
for E > E A
(R3.A-24)
and
S
r
(E, T) =^0 for^ E^ "^ E^
A
S
r
AB
2
E % E
A
E
for E > E A
(Click Back 2)
A
U R
U LC
B
b
The impact parameter, b, is the off-set distance of the centers as they approach
one another. The velocity component along the lines of centers, U LC
, can be
obtained by resolving the approach velocity into components.
At the point of collision, the center of B is within the distance σ AB
.
U R
b
! AB
"
†
Courtesy of J. I. Steinfeld, J. S. Francisco, and W. L. Hayes, Chemical Kinetics and Dynamics ,
(Englewood Cliffs NJ: Prentice Hall, 1989, p.250); Mp483.
(R3.A-25)
(R3.A-26)
16
is shown in Figure R3.A-7.
Model 2
Model 1
S r
E A E
AB
2
Figure R3.A- 7 Reaction cross section for Models 1 and 2.
Recalling Equation (20)
k T ( )
8 k B
T
"μ
1 2
ES
r
(E )e
)E RT
dE
(^ RT )
0 2
+^ (R3.A-20)
Substituting for S r
in Model 2
k (^) (T ) =
8 k B
T
"μ
1 2
AB
2
E * E A
( )e
*E RT
dE
RT
( )
E 2 A
,
(R3.A-27)
Integrating gives
k (^) (T ) = "# AB
2
8 k B
T
"μ
1 2
e
*E A
RT
Derive
(Click Back 3)
(Click Back 3)
E " E
A
P
r
= 0 S
r
E > E
A
P
r
S
r
AB
2
1 #
E
A
E
k (^) (T ) =
"μ k B
T
( )
3
2
AB
2
1 *
E A
2
3
*+ kT
d+
AB
2
"μ k B
( T)
3
1 2
kT ( )
kT
**
,
e
-* RT
d* - ** e
-* kT
**
,
d*
kT
17
(Click Back 3 cont’d)
k B
T
E
RT
AB
2
8 k B
T
"μ
1 2
E
A
RT
E
A
RT
e
*E RT
k T ( )
AB
2
8 k B
T
"μ
1 2
e
*E A
RT
Multiplying both sides by N Avo
k (^) ( T) = Ae
"E A
RT
Multiplying by Avogodro’s number
k (^) (T ) =
k (^) (T )N Avo
"r A
AB
2
U R
N
Avo
e
"E A
RT
C A
C
B
(R3.A-28)
This is similar to the equation for hard sphere collisions except for the term
e
"E A
RT
k (^) (T ) = " AB
2
8 #k B
T
μ
1 2
N
Avo
e
*E A
RT
(R3.A-29)
This equation gives the correct Arrhenius dependence and the correct order of
magnitude for A.
"r A
= Ae
"E RT
C A
C
B
(R3.A-11)
Effect of Temperature on Fraction of Molecules Having Sufficient Energy to React
Now we will manipulate and plot the distribution function to obtain a
qualitative understanding of how temperature increases the number of reacting
molecules. Figure R3.A-8 shows a plot of the distribution function given by
Equation (R3.A-17) after it has been converted to an energy distribution.
We can write the Maxwell-Boltzmann distribution of velocities
f (^) (U ,T)dU = 4 "
μ
2 "k B
T
3 2
U
2
e
)
μU
2
k B
T
dU
in terms of energy by letting
μU
2
to obtain
Bingo!
19
This integral is shown by the shaded area on Figure R3.A-8.
Figure R3.A- 8 Boltzmann distribution of energies.
As we just saw, only those collisions that have an energy E A
or greater
result in reaction. We see from Figure R3.A- 10 that the higher the temperature
the greater number of collision result in reaction.
However, this Equations (R3.A-9) and (R3.A-11) cannot be used to
calculate A for a number pf reactions because of steric factors and because the
molecular orientation upon collision need to be considered. For example,
consider a collision in which the oxygen atom, O, hits the middle carbon in the
reaction to form the free radical on the middle carbon atom CH 3
C
HCH 3
CH
2
O " CH
3
" CH
3
CH CH
3
+ • OH
CH
2
Otherwise, if it hits anywhere else (say the end carbon)
CH
3
CH CH
3
will not
be formed
†
O " CH
3
CH
2
CH
3
" CH
3
CHCH
3
" • CH
3
CH
2
CH
2
+ • OH
Consequently, collision theory predicts a rate 2 orders of magnitude too high
for the formation of CH 3
C
HCH 3
.
IV. OTHER DEFINITIONS OF ACTIVATION ENERGY
We will only state other definitions in passing, except for the energy barrier
concept, which will be discussed in transition state theory.
† M1p.36.
20
A. Tolman’s Theorem E a
= E
E
! e
( )
! a
=
Average Energy
of Molecules
Undergoing
Reaction
"
$
$
%
&
'
'
(
Average Energy
of Colloiding
Molecules
"
$
%
&
'
1
2
kT
The average transitional energy of a reactant molecule is
3
2
kT.
B. Fowler and Guggenheim
! a
=
Average Energy
of Molecules
Undergoing
Reaction
"
$
$
%
&
'
'
(
Average Energy
of Reactant
Molecules
"
$
%
&
'
C. Energy Barrier
The energy barrier concept is discussed in transition state theory, Ch3,
Profession Reference Shelf B.
A + BC " AB+ C
E
A, BC
E A
AB, C
A – B – C
Products
Figure R3.A- 9 Reaction coordinate diagram.
For simple reactions, the energy, E A
, can be estimated from computational
chemistry programs such as Cerius
2
or Spartan, as the heat of reaction between
reactants and the transition state
E
A
= H
ABC
" H A
o
" H BC
o
V. ESTIMATION OF ACTIVATION ENERGY FROM THE POLYANI EQUATION
A. Polyani Equation
The Polyani equation correlates activation energy with heat of reaction. This
correlation
E
A
P
#H
Rx
works well for families of reactions. For the reactions
R + H R "# RH + R"
where R = OH, H, CH3, the relationship is shown in Figure R3.A- 10.
