Chemistry Reference
In-Depth Information
in
E
ʱ
when the difference between the maximum and minimum values of
E
ʱ
is more
than 20-30 % of the average
E
ʱ
value. In the case of calcium oxalate monohydrate,
this difference is close to 100 % of the average value.
Needless to say that performing integration over small temperature segments in
accord with Eq. 2.21:
T
−
E
RT
a
∫
α
IE T
(,
)
=
exp
d
T
(2.21)
αα
T
a
−
∆
α
would also eliminate the error associated with the variability of
E
ʱ
in the isocon-
versional method based on Eq. 2.17. Consistency of the resulting
E
ʱ
dependence
with the dependence produced by the Friedman method has been demonstrated by
Budrugeac [
38
], who also proposed a new nonlinear differential isoconversional
method that makes use of a numerical algorithm similar to that represented by
Eqs. 2.15 and 2.16.
In addition to the methods proposed by Vyazovkin, there are several other flex-
ible integral isoconversional methods [
39
-
46
]. Among them, a method proposed by
Popescu [
39
] deserves a special attention because it combines the flexibility with
computational simplicity. The method makes use of the so-called mean value theo-
rem to approximate the temperature integral. The theorem states that a definite inte-
gral of a continuous function
f
(
x
) on the closed interval [
a, b
] can be presented as:
b
(2.22)
∫
f
( )
x dx
=−
(
b
a f c
)
( ),
a
where
f
(
c
) is the mean value of
f
(
x
) over the interval [
a,b
]. The equality is exact
because the mean value is defined as:
b
1
∫
f c
()
=
f
() .
x dx
(2.23)
(
ba
−
)
a
The theorem does not say where the point
c
is located, it only proves that it is lo-
cated within the interval [
a, b
]. The application of the theorem to the temperature
integral defined on the temperature interval [
T
m
,T
n
] allows one to arrive at the fol-
lowing equation [
39
]:
T
n
−
E
−
E
=−
∫
exp
dT
(
T
T
) exp
.
(2.24)
n
m
RT
RT
ξ
T
m
The equation is exact only when the value of
T
ξ
is such that the respective value
of the exponential function equals the mean value of the function in the interval
[
T
m
,T
n
]. Unfortunately, there is no simple practical way to determine such tempera-
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