Biomedical Engineering Reference
In-Depth Information
The picture to this point then is the collision of hard spheres given in
Fig. 6.4
, with energies as
defined in
Eqn (6.23)
and probabilities as given in
Eqn (6.24)
. How do we turn this into a reac-
tion rate expression that is compatible with all the work done previously to get to
Eqn (6.19)
?
This can be achieved by defining yet a new quantity: the reactive cross-section. We define a reac-
tive cross-sectional area as
E
r
Þ¼
R
N
0
S
R
ð
d½
E
c
ð
b
Þ2pb
d
b
(6.25)
¼ 0;
E
r
<
E
a
;
E
a
E
r
¼ pd
2
AB
E
r
E
a
1
The energy dependence of S
R
is essentially the collision area that the interacting molecules
see as a function of the relative energy. Now in order to determine a reaction rate (or rate
constant) from the result of
Eqn (6.25)
, we need to integrate over the distribution of E
r
and
over the range of possible values of b
b
max
associated with each E
r
. After some rather
complex geometry, one comes up with a collision rate as a function of the relative energy
that says:
<
s
p
M
Z
N
E
r
Þ¼
2
C
C
þ
M
B
A
e
E
r
=ðRTÞ
S
R
ð
Z
ð
E
r
Þ
E
r
d
E
r
(6.26)
Þ
3=2
ðR
T
A
B
E
a
which yields
s
8
M
RT
p
þ
M
E
r
Þ¼pd
2
AB
e
E
a
=ðRTÞ
C
Z
ð
C
(6.27)
A
B
A
B
Thus, we have obtained the reactive collision rate.
Often a steric factor,
x
, is introduced to account for all additional factors affecting the
collision
e
reaction probability except for those directly associated with energy. Thus
s
8
M
RT
p
A
þ
M
ZE
r
¼ pxd
2
AB
e
E
a
=ðRTÞ
C
C
(6.28)
A
B
B
We may now compare
Eqn (6.28)
with the bimolecular rate law to find what a corresponding
rate constant would be. This is given by
s
8
M
RT
p
A
þ
M
¼ pxd
2
AB
e
E
a
=ðRTÞ
k
(6.29)
B
and is the rate constant of a bimolecular reaction as determined from collision theory. It is
essentially of the Arrhenius form, since the weak temperature dependence of the preexpo-
nential factor is generally negligible in comparison with the activation energy term. Thus,
the reaction rate law observed experimentally is explained, and this is the foundation for
the discussion of kinetics.
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