Environmental Engineering Reference
In-Depth Information
involved. Generally, simplifications are made to visualize the potential energy surface
as a three-dimensional plot.
The ACT presumes that an equilibrium between the reactants (A and B) and an
activatedcomplex(AB)
†
isestablished.Thiscomplexfurtherundergoesunimolecular
decay into product P:
B
k
†
(
AB
)
†
K
†
A
+
−→
P.
(5.64)
The ACT tacitly assumes that even when the reactants and products are not at
equilibrium, the activated complex is always in equilibrium with the reactants.
The rate of the reaction is the rate of decomposition of the activated complex.
Hence
k
†
P
AB
†
.
r
=
(5.65)
However, since equilibrium exists between A, B, and (AB)
†
P
AB
†
P
A
P
B
K
†
=
(5.66)
with units of pressure
−
1
since the reaction occurs in the gas phase. Thus
k
†
K
†
P
A
P
B
=
k
∗
P
A
P
B
,
r
=
(5.67)
where
k
∗
denotes the second-order reaction rate constant and is equal to
k
†
K
†
. The
rate constant for the decay of the activated complex is proportional to the frequency
of vibration of the activated complex along the reaction coordinate. The rate constant
k
†
is therefore given by
k
†
= κ · ν
,
(5.68)
where
is the
frequency of vibration and is equal to
k
B
T/h
, where
k
B
is the Boltzmann constant
and
h
is the Planck's constant.
κ
is called the transmission coefficient, which in most cases is
≈
1.
ν
10
12
s
−
1
at 300 K.
Concepts from statistical thermodynamics (which is beyond the scope of this
textbook) can be used to obtain
K
†
. A general expression is (Laidler, 1965)
ν
has a value of 6
×
Q
AB
†
Q
A
Q
B
K
†
e
−
(E
0
/RT)
,
=
(5.69)
where
Q
isthe
partitionfunction
,whichisobtaineddirectlyfrommolecularproperties
(vibration, rotation, and translation energies).
E
0
is the difference between the zero-
point energy of the activated complex and the reactants. It is the energy to be attained
by the reactants at 0 K to react, and hence is the activation energy at 0 K. Thus
k
B
T
h
Q
AB
†
Q
A
Q
B
k
∗
= κ
e
−
(E
0
/RT)
.
(5.70)
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