Chemistry Reference
In-Depth Information
k
k
IE T
E
RT
(
,
)
1
∑∑
αα
,
i
,
i
t
=
t
=
.
α
α
,
i
β
−
(2.45)
i
=
1
i
=
1
α
,
i
exp
0
The resulting Eq. 2.45 allows one to use nonisothermal constant heating-rate ex-
periments to predict the isothermal lifetimes while properly accounting for variation
of
E
ʱ
with
ʱ
. Similarly, the isothermal lifetimes can be predicted from data obtained
under arbitrary temperature programs,
T
(
t
). The respective equation is as follows:
k
k
JE Tt
E
RT
[
,
[(
)]
∑∑
α
,
i
α
,
i
t
=
t
=
,
α
α
,
i
−
(2.46)
i
=
1
i
=
1
α
,
i
exp
0
where
t
ʱ,i
is calculated as [
81
]
*
t
α
,
i
−
E
α
,
i
∫
exp
d
t
RT t
()
*
t
α
,1
i
−
(2.47)
t
=
.
α
,
i
−
E
RT
α
,
i
exp
0
In Eq. 2.47,
t
ʱ
*
is the experimentally estimated time to reach a given value of
ʱ
under the temperature program,
T
(
t
) =
T
*
(
t
). This is one of the several temperature
programs employed for evaluating the
E
ʱ
dependence.
The predictions made by Eqs. 2.43, 2.45, and 2.46 can be called “model-free
predictions,” because they get rid of the reaction model
g
(
ʱ
) in the numerator of
Eq. 2.39. The most important feature of the model-free predictions is that each value
of
t
ʱ
is predicted by using the corresponding value of
E
ʱ
. In other words, the model-
free predictive equations allow for using the actual
E
ʱ
dependence. This expands
the application area of these equations to both single-step (
E
ʱ
does not depend on
ʱ
) and multistep (
E
ʱ
depends on
ʱ
) processes. The model-free predictions provide
two obvious advantages over the ASTM methods. First, they are not limited to the
first-order kinetics or any other reaction model. Second, they do not require
E
ʱ
to
be invariable with
ʱ
. For this reason, they generally give rise to more reliable ki-
netic predictions than the ASTM methods. This fact is exemplified in Fig.
2.19
and
elsewhere [
73
,
79
].
Although most commonly one makes predictions of the lifetime at a given con-
stant temperature,
T
0
, by using a set of nonisothermal measurements, the kinetic
predictions can be made from kinetic data measured at temperature programs
T
*
(
t
)
to any temperature program of interest,
T
0
(
t
). Using the same assumption as in de-
riving Eq. 2.43, one can arrive at a model-free equation:
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