1
04
D. Zhan et al. / Thermochimica Acta 430 (2005) 101–105
as following, respectively.
Table 4
Comparison of the results obtained from the model-fit and the different
calculation methods
ꢂꢀ ꢁ ꢃ
dα
Ea
ln
β = ln[Af (α)] −
(3)
(4)
dT
RT
First step
Second step
ꢂ AEa
ꢃ
Ea
Activation energy (kJ/mol)
Friedman method
FWO method
ln β =
− 5.3305 − 1.0516
148.8 ± 10
177.7 ± 23
171.1
165.0 ± 12
182.3 ± 5.2
174.4
RG(α)
RT
Model-fitting method
These two methods are usually used to calculate activation
energy. Friedman method is very sensitive to experimental
noise, and tends to be numerically unstable because of em-
ploying instantaneous rate value. But FWO method produces
a systematic error in Ea when varies with α. This error does
notappearinFriedmanmethod[10]. Inourpresentstudy, they
were used to calculate activation energy listed in Table 3.
It is shown that the values including activation energy cal-
culated by FWO method are higher than that of Friedman
method (in the range 0.2 ≤ α ≤ 0.8). It is due to that Fried-
man method is very sensitive to experimental noise, but FWO
method leads to meaningful result assuming that Ea invaries
with α. It also can be seen that activation energies change
little with α, which is just for FWO method. It also suggested
that each reaction correspond to the two steps probably obey
a single kinetic mechanism.
log A (S 1)
−
15.1
CnB
1.746
0.7923
0.998596
f(α) = (1 − α)
(1 + Kcatα)
11.6
C1B
1
Reaction type
Reaction order
log Kcata
Correlation coefficient
Function
1.5186
0.997371
f(α) = (1 − α)(1 + Kcatα)
n
a
Logarithm of the balance constant for autocatalysis reaction.
According to Table 4, it was seen that the value calculated
by FWO method was close to the optimized value. That
is because the FWO kinetic equation was obtained by
integrating the Arrhenius equation after assuming that the
kinetic model and the kinetic parameters were invariant
all over the process, which nearly accord with the fact in
the range 0.2 ≤ α ≤ 0.8. We can conclude that the kinetic
model for the dehydration is CnB, and the corresponding
3
.4.2. Determination of kinetic model
n
function is f(α) = (1 − α) (1 + Kcatα). The correlated kinetic
For one-step reaction, the determination of kinetic param-
parameters are Ea = 171.1 kJ/mol, log A = 15.1, n = 1.75,
respectively. The kinetic model for the second reaction
is C1B. The function is f(α) = (1 − α)(1 + Kcatα). The
optimized activation parameters are Ea = 174.4 kJ/mol,
log A = 11.6, n = 1.0.
eters can be turned into a multiple linear regression problem
through suitable transformations and simultaneous conver-
sion of Eq. (1).
First, the experimental values (such as TG data) are dif-
ferential or integrated and then transformed into the degree
of conversion. Application of Friedman equation yields Eq.
In previous works, L’vov also studied the kinetics of ther-
mal decomposition of nickel oxalate using a model of dis-
sociative evaporation of the reactant with simultaneous con-
densation of the low-volatility product. In his opinion, the
average activation energy for the steady-state decomposi-
tion of nickel oxalate in vacuum is 220 ± 5 kJ/mol, which
was mainly used for the selection of appropriate schemes
of thermal decomposition of nickel oxalate. In our study,
the kinetics of the thermal decomposition of nickel ox-
alate was investigated under non-isothermal conditions in
air using multiple linear regression method. The optimized
value of activation energy was 171.1 kJ/mol (dehydration),
(5) for reaction type Fn, Eq. (6) for nucleation process of
the nth dimension (An), described by Avrami–Erofeev equa-
tion [11–13], Eq. (7) for the complex reaction type, Bna
(Prout–Tompkins nth-order, ath autocatalysis) [14] as well
as the modified Friedman Eq. (8) for the remaining reaction
type.
dα
dt
E
ln
ln
= ln A −
+ n ln(1 − α)
(5)
(6)
RT
dα/dt
E
n − 1
= ln(nA) −
+
1
− α
RT
n ln[− ln(1 − α)]
1
74.4 kJ/mol (decomposition). Obviously, the value of ac-
dα
E
tivation energy calculated by L’vov is higher than that cal-
culated by us. We attempted to explain this as due to the
difference between the true scheme decomposition of nickel
oxalate. These differences consisted of in primary gasifica-
tion of all decomposition products and the methods that were
used to calculate Ea and log A to obtain the topochemical
equation.
ln
ln
= ln A −
+ n ln(1 − α) + a ln α
(7)
(8)
dt
RT
dα/dt
f(α)
E
= ln A −
RT
Replaced the data of experiments into the equations cor-
responding to different kinetic models [15], then we used
multiple linear regression method to determine the best-fit
kinetic model. Generally speaking, the one with highest cor-
relation coefficient (>0.99) is the best-fit kinetic model. The
optimized value is the data of activation energy and ln A,
which was calculated with the best equation. Experimental
results are shown in Table 4.
4. Conclusions
By properly choosing the primary product composition,
we have succeeded in getting the final data of Ea and log A,