Isoconversional Methodology - Isoconversional Kinetics of Thermally Stimulated Processes - page 37

Fig. 2.8 Illustration to the
Popescu method (Eq. 2.26)

β 1

β 2

α n

α m

T n,1

T n,2

T m,1

T m,2

ture. For practical purposes, it can be approximated as the mean value of the interval
[ T m , T n ], i.e., T ξ = ( T m + T n )/2. This approximation is accurate only for integration of
a linear function. For any other function, the accuracy of this approximation would
depend on the size of the interval [ T m , T n ] so that the smaller the interval, the better
the accuracy. Subject to Eq. 2.24, one can write Eq. 2.8 for two arbitrary conver-
sions ʱ m and ʱ n , as follows:

A TT E

RT

−

g ()() ( α − g α = − ) exp ,

(2.25)

n m n m

β

ξ

where T m and T n are the temperatures that correspond, respectively, to the conver-
sions ʱ m and ʱ n on the ʱ versus T curve at a given heating rate ʲ (Fig. 2.8 ). Assum-
ing that the form of the reaction model does not change with the heating rate, the
left-hand side of Eq. 2.25 is constant for a set of different heating rates, ʲ i . Then
Eq. 2.25 can be rearranged to:

β E

RT

i

ln = Const −

(2.26)

TT −

ni mi , , ξ , i

The activation energy can be determined from the slope of a linear plot of the left-
hand side of Eq. 2.26 against 1/ T ξ,i .
Although in the original publication [ 39 ] by Popescu, the intervals ∆ ʱ = ʱ m − ʱ n
used are relatively wide (0.3-0.8), the relative errors in the estimated E values re-
main within the tenths of percent. However, the primary advantage of the method is
in the flexibility to choose the integration limits by selecting any desired values of
ʱ m − ʱ n . In particular, by making the ∆ ʱ intervals smaller (e.g., 0.01-0.02), one can
practically eliminate the integration error that arises from the variability of E ʱ with
ʱ . This opportunity has been explored efficiently by Ortega [ 42 ], who applied the
mean-value theorem to derive isoconversional equations for the conditions when