Some Basics En Route to Isoconversional Methodology - Isoconversional Kinetics of Thermally Stimulated Processes - page 10

Fig. 1.8 Isoconversional
method uses an individual
rate equation for each extent
conversion and a narrow tem-
perature interval ∆ T, related
to this conversion. The use of
different heating rates, ʲ 1 and
ʲ 2 , allows for determining
different rates

d
d

α d

d

α

= β

t T

at the same conversion.
(Reproduced from Vyazovkin
and Sbirrazzuoli [ 23 ] with
permission of Wiley)

where the subscript ʱ indicates isoconversional values, i.e., the values related to a
given extent of conversion. The second addend on the right-hand side of Eq. 1.12
is zero because at ʱ = const, f ( ʱ ) is constant. The first addend readily derives from
Eq. 1.2 so that Eq. 1.12 reduces to:

∂ ln( dd α / t ) E

R

α

=− .

(1.13)

− 1

∂ T

α

It follows from Eq. 1.13 that the temperature dependence of the isoconversional
rate can be utilized to determine the isoconversional values of the activation energy,
E ʱ without identifying or assuming any form of the reaction model. That is why
isoconversional methods are frequently termed as “model-free” methods. While
catchy, this term is not to be taken literally. It should be kept in mind that although
the methods do not have to identify explicitly the model, they still assume implicitly
that there is some f ( ʱ ) that defines the conversion dependence of the process rate.
The temperature dependence of the isoconversional rate is obtained experimen-
tally by performing a series of runs at different temperature programs. It usually
takes four to five runs at different heating rates or at different temperatures to de-
termine such dependence. Figure 1.8 [ 23 ] illustrates the idea of determining the
isoconversional rate from two nonisothermal runs conducted at the heating rates ʲ 1
and ʲ 2 . The conversion versus temperature plots can be estimated by scaling TGA
data in accord with Eq. 1.4 (Fig. 1.2 ). By selecting a certain conversion ʱ , one then
finds the temperature related to it at each heating rate, i.e., T ʱ, 1 and T ʱ, 2 . The conver-
sions need to be selected in a wide ʱ range, e.g., 0.05-0.95 with a step not larger
than 0.05. Since it is unlikely that the experimental ʱ versus T curves would contain
points exactly at selected values of ʱ one has to use interpolation to find the values
of T ʱ . Then the slope (numerical derivative) of the ʱ versus T curve at T ʱ would give
the values of

d
d

α

that can be converted to the isoconversional rate as follows:

T

α