Isoconversional Methodology - Isoconversional Kinetics of Thermally Stimulated Processes - page 35

contain a systematic error whose size increases with increasing range of E ʱ vari-
ability. The error can be eliminated by using a flexible integral method. All it takes
is to perform either time or temperature integration by segments that correspond to
small intervals of conversion, ∆ ʱ . Vyazovkin has proposed a method [ 36 ] that ef-
fectively eliminates this error by performing integration over small time segments.
The method is based on Eq. 2.18, whereas Eq. 2.19 is modified as follows:

t

E
RT t

−

a

∫

JE Tt [, ( ] = exp

α

d t .

(2.20)

α α

()

t

a − ∆ α

When integration is carried out over small segments, the E ʱ value is assumed to
be constant only within a small interval of ∆α that is typically taken to be 0.01-
0.02. The efficiency of integration by segments can be demonstrated by comparing
the resulting E ʱ versus ʱ dependence with that estimated by the Friedman method
(Eq. 2.5). Because the latter is a differential method, it does not introduce the afore-
mentioned integration error into the E ʱ values. The application of the integral meth-
od with integration by segments and the differential Friedman method to the same
set of simulated data demonstrates [ 36 ] that the resulting E ʱ versus ʱ dependencies
are virtually identical.
As already mentioned, in the integral methods, the systematic error introduced
by non-constancy of E ʱ is proportional to the magnitude of E ʱ variation. In the case
of significant variation of E ʱ with ʱ, the relative error in E ʱ can be as large as 20-
30 %. A process that is well known [ 2 ] to demonstrate significant variability of E ʱ is
the thermal dehydration of calcium oxalate monohydrate. The E ʱ dependencies ob-
tained by respectively neglecting and accounting for variability of E ʱ are displayed
in Fig. 2.7 which also shows the size of the error introduced when the variability is
neglected. It is generally recommended [ 37 ] that one should account for variability

Fig. 2.7 E ʱ dependen-
cies estimated for thermal
dehydration of CaC 2 O 4 ﾷH 2 O
by isoconversional method
(Eq. 2.18). Squares represent
the E ʱ values obtained when
using regular integration
(Eq. 2.19). Circles represent
the E ʱ values estimated via
integration by segments
(Eq. 2.20). The dotted line
shows the size of the relative
error in E ʱ caused by integra-
tion that does not account for
variability of E ʱ . (Repro-
duced from Vyazovkin [ 36 ]
with permission of Wiley)