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©Hai Wang
Version 1.3
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AME599 Combustion Chemistry and Physics
Lecture 4
4. Bimolecular Reaction Rate Coefficients
In the last lecture, we learned qualitatively the reaction mechanisms of hydrocarbon
combustion.
To make this description quantitative, we will need to have a basic knowledge
of reaction rate theories.
While the rate coefficients of a large number of combustion
reactions are experimentally measured, reaction rate theories are often necessary to interpret
the experimental data.
We shall focus our discussion here to bimolecular reactions of the
type
A
+
B
→
C
+
D
and leave the discussion for unimolecular reactions to a later time.
4.1 Hard Sphere Collision
We discussed earlier that an elementary chemical reaction requires molecular collision.
The
rate coefficient of a bimolecular reaction is essentially the product of reaction probability of
two reactants upon collision at a given temperature
γ
(
T
) and the frequency of collision
Z
AB
,
k
(
T
) =
(
T
)
Z
AB
(4.1)
Here we assume that molecules may be described by rigid spheres.
Figure 4.1 shows several
scenarios of molecular encounter.
Starting from the headon collision, an offset of the two
axes of molecular motion leads to offcenter collision.
Suppose the diameter of
A
and
B
are
σ
A
and
B
, respectively.
The limiting offcenter collision would have a spacing equal to
(
A
+
B
)/2 between the two axes of molecular motions.
It may be concluded from this
simple analysis that two molecules with their axes of motions lie within a cylindrical volume
of cross section equal to
Figure 4.1
Various scenarios
of molecule collision.
AB
Headon collision
Offcenter collision
Limiting offcenter collision
A
B
AB
=(
A
+
B
)/2
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2
2
2
AB
AB
S
σσ
πσ
π
⎛⎞
+
⎟
⎜
==
⎟
⎜
⎟
⎜
⎝⎠
(4.2)
would collide with each other.
Here
AB
σ
is the collision diameter.
Suppose an
N
B
number of
B
molecules are at rest and confined to an arbitrary volume
V
.
An
A
molecule travels through the volume with a mean velocity equal to
A
v
.
The
cylindrical collision volume that
A
sweeps through per unit time is
2
AB A
v
πσ
(see, Figure
4.2).
The number density of
B
is
N
B
/
V
.
Therefore the number of collision per unit time is
()
12
s
B
BA
B
A
N
Zv
V
πσ
−
=
.
(4.3)
If we have
N
A
A
molecules in the same volume, the collision rate is equal to
31
2
cm
s
AB
AB A
NN
VV
πσ
−−
=
.
(4.4)
Figure 4.2
Schematic illustration of the cylindrical
collision volume in a total volume
V
.
The above derivation is simple, but it has one problem.
That is, it treats the molecules like
“ghost” particles since collision does not lead to changes in the direction of motion of the
A
molecule.
This problem is easily mended by realigning the cylindrical volume with the
velocity vector of the
A
molecule every time it undergo scattering with a
B
molecule.
Since
the number densities of
A
and
B
are independent of the orientation of the cylindrical
volume, the result is not affected by the “ghost” particle assumption.
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 Spring '08
 Wang
 Energy, Chemical reaction, University of Southern California, ©Hai Wang, Southern California Version

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