Chemistry Reference
In-Depth Information
α
d
α
∫
g
()
α
=
(2.33)
m
n
αα
(1
−
)
0
does not have an analytical solution. However, the values of
n
and
m
can still be
determined by fitting the right-hand side of Eq. 2.33 to the numerical values of
g
(
ʱ
)
estimated by Eq. 2.30.
2.2.3 The Use of the y(α) or z(α) Master Plots
The use of the
y
(
ʱ
) or
z
(
ʱ
) master plots is another popular method of estimating
the reaction models and preexponential factors. The method is contingent on
E
ʱ
being practically invariable with respect to
ʱ
. Therefore, as the first step in using
this method, one needs to apply an isoconversional method for estimating
E
ʱ
as a
function of
ʱ
and making sure that there is no significant variation. Then
E
ʱ
can be
replaced with the mean value,
E
0
. The
y
(
ʱ
) function [
69
] has the following form:
d
d
α
E
RT
=
=
0
y
()
α
exp
Af
(
α
).
(2.34)
t
α
α
Equation 2.34 is derived by rearranging Eq. 2.2. The values of
y
(
ʱ
) are calculated
by using the experimental dependence of
(
)
d /d
t
α
on
T
ʱ
and multiplying it by the
exponential term containing the
E
0
value estimated by an isoconversional method.
The resulting numerical values of
y
(
ʱ
) are then plotted against
ʱ
and matched with
the theoretical
y
(
ʱ
) master plots. The best match identifies a suitable model. Each
heating rate gives rise to one experimental dependence of
d /d
t
α
on
T
ʱ
and, those,
to one
y
(
ʱ
) plot. However, the resulting
y
(
ʱ
) plots should not demonstrate any sig-
nificant variation with
ʲ
producing a single
y
(
ʱ
) plot.
The fact that
A
is constant in 2.34 suggests that the shape of the
y
(
ʱ
) master plot
is defined exclusively by the shape of the
f
(
ʱ
) functions (Fig. 1.4) that represent
the differential form of the reaction model. Since the preexponential factor is yet to
be estimated, the experimental and theoretical
y
(
ʱ
) plots are matched in a normal-
ized form that sets their range of variation from 0 to 1. Examples of some normal-
ized theoretical
y
(
ʱ
) plots derived from the
f
(
ʱ
) models (Table 1.1) are depicted
in Fig.
2.15
. The shape of the experimental
y
(
ʱ
) plot provides the first clue about
the type of the reaction model. A convex decreasing dependence of
y
(
ʱ
) on
ʱ
is an
indication of the contracting geometry models (coded R in Table 1.1). A concave
decreasing plot is indicative of the diffusion models (code D). A dependence with a
maximum is representative of either the Avrami-Erofeev (code A) or truncated SB
models. The position of the maximum,
ʱ
m
, depends on the model (Table
2.1
) that
can help with identifying a particular one.
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