The kinetic analysis of non-isothermal nickel oxide reduction in hydrogen atmosphere using the invariant kinetic parameters method

Article abstract of DOI:10.1016/j.tca.2007.01.033

The powder sample of nickel oxide was synthesized by sol-gel procedure. The non-isothermal reduction of nickel oxide using hydrogen, was investigated by temperature-programmed reduction (TPR) at four different heating rates (2.5, 5, 10 and 20 °C min^-1). The kinetics parameters (E_a, lnA) were determined using Friedman (FR) isoconversional method and the invariant kinetic parameters method (IKP). The IKP method was applied in the conversion range of 0.20 ≤ α ≤ 0.60. The activation energy value obtained by the IKP method (E_a = 96.4 kJ mol^-1) is somewhat higher than the value obtained by Friedman's method (E_a = 90.8 kJ mol^-1). The values of invariant kinetic parameters were used for numerical evaluation of conversion function, f(α). The kinetic model of the investigated reduction process was determined, and this model corresponds to the empirical two-parameter S?esta?k-Berggren equation, which gives a more quantitative description. The kinetic triplet obtained for the investigated process is: E_a = 96.4 kJ mol^-1; A = 1.04 × 10⁸ min^-1 and f(α) = α^0.63(1 - α)^1.39. In agreement with the obtained results, the mechanism of nickel oxide reduction by hydrogen was considered.

Full text of DOI:10.1016/j.tca.2007.01.033

Thermochimica Acta 456 (2007) 48–55
The kinetic analysis of non-isothermal nickel oxide reduction in hydrogen
atmosphere using the invariant kinetic parameters method
∗
´
´
B. Jankovic , B. Adnad¯evic, S. Mentus
Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12-16, P.O. Box 137, 11001 Belgrade, Serbia and Montenegro
Received 24 October 2006; received in revised form 22 December 2006; accepted 28 January 2007
Available online 3 February 2007
Abstract
The powder sample of nickel oxide was synthesized by sol–gel procedure. The non-isothermal reduction of nickel oxide using hydrogen, was
investigated by temperature-programmed reduction (TPR) at four different heating rates (2.5, 5, 10 and 20 ◦C min−1). The kinetics parameters
(Ea, lnA) were determined using Friedman (FR) isoconversional method and the invariant kinetic parameters method (IKP). The IKP method was
applied in the conversion range of 0.20 ≤ α ≤ 0.60. The activation energy value obtained by the IKP method (Ea = 96.4 kJ mol−1) is somewhat
higher than the value obtained by Friedman’s method (Ea = 90.8 kJ mol−1). The values of invariant kinetic parameters were used for numerical
evaluation of conversion function, f(α). The kinetic model of the investigated reduction process was determined, and this model corresponds to the
ˇ
empirical two-parameter Sesta´k–Berggren equation, which gives a more quantitative description. The kinetic triplet obtained for the investigated
process is: Ea = 96.4 kJ mol−1; A = 1.04 × 108 min−1 and f(α) = α0.63(1 − α)1.39. In agreement with the obtained results, the mechanism of nickel
oxide reduction by hydrogen was considered.
© 2007 Elsevier B.V. All rights reserved.
ˇ
Keywords: Invariant kinetic parameters method (IKP); Nickel oxide; Reduction; Activation energy; Sesta´k–Berggren kinetic model
1. Introduction
indication of the extent of reaction for a sample of NiO made
by heating nickel nitrate at 400 ◦C. They were approached to
some very important conclusions and these are: (a) reduction
occurs at the interface between NiO and previously reduced Ni;
(b) there is an “autocatalytic” effect; (c) there is an induction
(i.e. nucleation) period that depends on the nature of the sample
and temperature; and (d) added water reduces the reduction rate
and increases the induction period. Koga and Harrison, reviewed
general reactions between solids and hydrogen and represented
the induction process as generation of nickel atoms on the outer
surfaceofNiOgrains[9]. Followingnucleation, Niclustersgrow
two-dimensionallyacrossthesurfaceuntiltheyoverlap, atwhich
point hydrogen rapidly dissociates on Ni and the interface pro-
ceeds quikly into the grain [9]. Bandrowski et al. [10] measured
water produced during the reduction to generate sigmoidal con-
version curves they explained with a two-step kinetic model. The
firststep, reactionbetweenNiOandhydrogenatomsadsorbedon
NiO, predominates in the early part of the reduction and is pro-
portional to the square root of the hydrogen pressure. The second
step is reaction at the metal–oxide interface between NiO and
hydrogen atoms adsorbed on the previously reduced NiO. The
above models emphasized the chemical mechanism and kinetics
Metal oxides are used as catalysts in a large variety of
commercial processes for the conversion of hydrocarbons
[1,2]. Selective oxidation, ammoxidation, and selective dehy-
drogenation probably constitute the most important catalytic
applications of metal oxides [2]. In addition, due to their low
cost, metal oxides are also used as supports of many other cat-
alytic materials (metals, sulfides, carbides, nitrides, etc.) [1,2].
In most cases, pure stoichiometric oxides do not exhibit high
catalytic activity [2]. For the preparation of active oxide cat-
alysts, the partial reduction of nickel oxide under hydrogen at
elevated temperatures is the effective method for this preparation
[3–5]. In general terms, two different kinetic models have been
proposed for the reduction of oxides: the “nucleation model”
[4,6] and the “interface-controlled model” [7]. The first reported
systematic measurement of bulk NiO reduction was by Benton
and Emmett [8]. These authors measured water formation as an
∗
Corresponding author. Tel.: +381 11 187 133; fax: +381 11 187 133.
E-mail address: bojanjan@ffh.bg.ac.yu (B. Jankovic´).
0040-6031/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.tca.2007.01.033

B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
49
of the reduction process, but there was a little uniformity in the
physical properties of NiO grains and pellets used. Nevertheless,
it was suspected that morphological factors were just as impor-
tant as topological properties in the determining the course of
the reduction. These factors were demonstrated by Moriyama
and Yamaguchi [11], who found that reduction rate constants
were inversely proportional to grain size above a diameter of
about 10 ␮m. Such dependence is predicted by the shrinking
core model [12]. Richardson et al. [13] conducted a series of
experiments using isothermal H2 consumption and magnetiza-
tion measurements to determine the Ni O bond rupture (NiO
conversion) and the growth of nucleated Ni atoms, respectively.
They found the growth process lagged NiO conversion by a
time interval that increased with decreasing temperature, lower
gas flow rates and the presence of H2O added to the reducing
gas. Rodriguez et al. [14] showed that in experiments with NiO
(1 0 0) crystal and NiO powders, oxide reduction is observed at
atmospheric pressures and elevated temperatures (250–350 ◦C),
but only after an induction period. These authors showed that
the presence of O vacancies lead to an increase in the adsorp-
tion energy of H2 and substantially lowers the energy barrier
associated with the cleavage of the H H bond. Richardson et al.
[15] studied the hydrogen reduction of porous bulk NiO particles
with in situ hot-stage X-ray diffraction (XRD) in the tempera-
ture range 175–300 ◦C. The results obtained by these authors
indicated that reduction in the absence of water added to the
reducing gas followed several steps: (1) an induction period
associated with the initial reduction of NiO and the appearance
of Ni metal clusters; (2) acceleration of the reduction rate as
the size of the clusters increase; and (3) a pseudo-first-order
process in which NiO disappeared and Ni appeared in con-
cert until reduction slowed at a fractional conversion of about
0.8. When 2.2 × 10−2 atm of H2O was added to the reduc-
ing gas, induction time increased by approximately a factor of
two and the reduction rate decreased, with an apparent activa-
tion energy of 126 27 kJ mol−1 compared to 85 6 kJ mol−1
without added water [15]. Utigard et al. [16] investigated the
reduction kinetics of NiO granules formed by vapour deposi-
tion from a chloride solution, using thermal gravimetry. They
reported that in the temperature range from 400 to 600 ◦C,
the rate of reduction increased with increasing temperature and
increasing hydrogen pressure. Microscopic analysis showed that
in this temperature range the reaction followed the shrinking
core model [16]. The same authors found that the activation
energy for reduction process have a value of 90 kJ mol−1, in
the above temperature range. Based on the facts given above, in
this paper, the kinetics and mechanism of powder nickel oxide
reduction, which was obtained by gel-combustion procedure
with hydrogen at the atmospheric pressure was investigated by
temperature-programmed reduction (TPR).
obtained by drying an aqueous solution of nickel nitrate hexahy-
drate (Fluka, 99.5%) and citric acid (Fluka, 99.5%), dissolved
in a mole ratio of 1.8:1. This gel further underwent a self-
ignition by heating in air up to 300 ◦C, and by an additional
heating up to 500 ◦C which produce a very fine nickel oxide
powders.
2.2. Thermogravimetric measurements
The experiments were carried out in a TA SDT 2960 device,
capable of simultaneous TGA-DTA analysis in the temper-
ature range from 25 to 1500 ◦C. The nickel oxide samples
were reduced directly within the thermobalance, in korund
pans, under (99.9995 vol.%) hydrogen flowing at a rate of
100 mL min−1, and using various heating rates: 2.5, 5, 10 and
20 ◦C min−1, in the temperature range from an ambient one up
to 500 ◦C. The sample mass used for thermogravimetric inves-
tigations was about 25 0.5 mg.
3. Kinetic analysis
Experimental data for the kinetic analysis of heterogeneous
solid–gas reactions can be obtained under different conditions.
We here analyse the data obtained under non-isothermal con-
ditions, with a linear regime of temperature increase in time
(β = dT/dt = const., where β is the heating rate, T is the tem-
perature, and t is the time). Under such conditions, for a
heterogeneous solid–gas reaction, occurring in a single step, the
reaction rate is expressed by the well-known general equation
[18]:
ꢀ
ꢁ
dα
dt
dα
dt
Ea
≡ β
= Af (α) exp
−
(1)
RT
where α is the degree of conversion, A is the pre-exponential fac-
tor, Ea is the activation energy, f(α) is the differential conversion
function and R is the gas constant. The use of Eq. (1) supposes
that a kinetic triplet (Ea, A, f(α)) describes the time evolution of
a physical or chemical change.
3.1. The methods used to evaluate the kinetic parameters
3.1.1. Friedman method [19] (FR method)
The differential isoconversional method suggested by Fried-
man [19] is based on the general form of rate equation, written
in its logarithmic form:
dα
dt
dα
dT
Ea
ln
≡ ln β
= ln Af (α) =
(2)
RT
For α = const., the plot of ln(dα/dt) versus 1/T, obtained from
thermograms recorded at several heating rates, should be a
straight line whose slope allows an evaluation of the activation
energy.
2. Experimental procedure
2.1. Materials and methods
3.1.2. The invariant kinetic parameters method [20,21]
(IKP method)
The NiO samples were obtained by gel-combustion method
described elsewhere [17]. A green-colored transparent gel was
The IKP method is based on the observation [22,23] that
the same experimental curve α = α(T) can be described rela-

50
B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
Table 1
Expressions for f(α) and g(α) functions for some of the common mechanisms operating in the solid state reactions
No.
Symbol
Reaction model
f(α)
g(α)
1
2
3
4
5
6
P1
P2
P3
P4
R1
R2
Power law
Power law
Power law
Power law
4α3/4
α1/4
α1/3
α1/2
α3/2
3α2/3
2α1/2
2/3α−1/2
1
Zero-order (Polany–Winger equation)
Phase-boundary controlled reaction (contracting
area, i.e. bidimensional shape)
Phase-boundary controlled reaction (contracting
volume, i.e. tridimensional shape)
First-order (Mampel)
Three-halves order
Second-order
Third-order
Avrami–Erofe’ev (r = 0.5)
Avrami–Erofe’ev (r = 1.5)
Avrami–Erofe’ev (r = 2)
Avrami–Erofe’ev (r = 3)
Avrami–Erofe’ev (r = 4)
One-dimensional diffusion
Two-dimensional diffusion (bidimensional
particle shape) Valensi equation
Three-dimensional diffusion (tridimensional
particle shape) Jander equation
Three-dimensional diffusion (tridimensional
particle shape) Ginstling–Brounshtein
α
2(1 −α )1/2
[1 − (1 − α)1/2
]
]
7
R3
3(1 − α)2/3
[1 − (1 − α)1/3
−ln(1 − α)
8
9
F1
F3/2
F2
(1 − α)
(1 − α)3/2
2[(1 − α)−1/2 − 1]
(1 − α)−1 − 1
(1/2)[(1 − α)−2 − 1]
[−ln(1 − α)]2
[−ln(1 − α)]2/3
[−ln(1 − α)]1/2
[−ln (1 − α)]1/3
[−ln(1 − α)]1/4
α2
10
11
12
13
14
15
16
17
18
(1 − α)2
F3
(1 − α)3
A1/2
A3/2
A2
A3
A4
(1/2)(1 − α)[−ln(1 − α)]−1
(3/2)(1 − α)[−ln(1 − α)]1/3
2(1 − α)[−ln(1 − α)]1/2
3(1 − α)[−ln(1 − α)]2/3
4(1 − α)[−ln(1 − α)]3/4
1/2α
D1
D2
1/[−ln(1 − α)]
(1 − α)ln(1 − α) + α
2
19
20
D3
D4
3(1 − α)1/3/2[(1 − α)−1/3 − 1]
3/2[(1 − α)−1/3 − 1]
[1 − (1 − α)1/3
]
(1 − 2α/3) − (1 − α)2/3
tively correct by several functions of conversion and, for a single
α = α(T) curve, the values of the activation parameters, obtained
for various analytical forms of g(α), are correlated through an
apparent compensation effect.
exponential factors (Avj) are calculated using the Coats and
Redfern [24] method.
Using the relation of the apparent compensation effect, the
compensation parameters (α∗v, βv∗) are determined for each heat-
ing rate. The straight lines ln Av versus Ea,v for several heating
rates should intersect themselves in a point which corresponds
to the true values of A and Ea. These were called by Lesnikovich
ln A = α∗ + β∗ Ea
(3)
where α* and β* are constants (the compensation effect param-
eters).
and Levchik [20,21], the invariant activation parameters (Ainv
,
Ea,inv). Certain variations of the experimental conditions actu-
ally determine a region of intersection in the lnA, Ea space. For
this reason, the evaluation of the invariant activation parameters
is performed using the relation:
In order to apply this method for a given heterogeneous
reaction, α = α(T) curves for several heating rates (βv, v =
1, 2, 3, . . .) should be recorded. A set of conversion functions,
gj, j = 1,2,3, . . ., is also considered. The differential and integral
functions of conversion used in this work are listed in Table 1.
For each heating rate βv, using an integral or differential
method, the pairs (Avj, Ea,vj), characteristic for each conversion
function, are determined. In this work, the Coats and Redfern
[24] method was used to integrate Eq. (1). It leads to the follow-
ing relation:
ln Ainv = α∗v + βv∗ Ea,inv
(5)
which leads to the super correlation relation:
α∗v = ln Ain − βv∗ Ea,inv
(6)
Thus, a plot α∗v versus βv∗ is actually a straight line whose param-
eters allow evaluation of the invariant activation parameters.
The IKP method can be only used if Ea is independent of α
[25]. On the other hand, the IKP method can be used for numer-
ical evaluation of f(α), by introducing in Eq. (1) the value of the
invariant kinetic parameters. Consequently, the values of f(α)
determined by IKP method are proportional to the real f(α) val-
ues. In order to discriminate the kinetic model, the dependence
of f(α) versus α obtained by IKP method and the dependence
f(α) versus α from theoretical kinetics models can be compared.
From these dependences, conclusion about the mechanism of
the process can be brought, and exact expression for f(α) can be
evaluated.
ꢂ
ꢃ
ꢀ
ꢁ
gj(αvi)
Tv2i
Avj R
Ea,vj
RTvi
ln
= ln
dα
=
with
βvEa,vj
ꢄ
α
gj(α) =
(4)
fj(α)
0
where i is data point, j is the numbers of the conversion function
listed in Table 1.
A plot ln[gj(αvi)/Tv2i] versus (1/Tvi) for a given analytical
form of g(␣) should be a straight line whose slope and intercept
allows the evaluation of activation energy and pre-exponential
factor, respectively. Twenty activation energies (Ea,vj) and pre-

B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
51
Fig. 1. TG curves for nickel oxide reduction process in hydrogen atmosphere at
different heating rates.
Fig. 2. The degree of conversion (α) as a function of temperature (T) for the
reduction of nickel oxide under linearly rising temperature, in hydrogen atmo-
sphere, at the following heating rates: 2.5, 5, 10 and 20 ◦C min−1 (from left to
right).
4. Results and discussion
The values of activation energies (Ea) of the investigated
reduction process, at different constant values of α, which were
determined by Friedman (FR) method are shown in Fig. 4.
Within the limits of the experimental errors in determination
of Ea by Friedman’s method, the activation energy of the reduc-
tionprocessofNiOintherangeof0.20 ≤ α ≤ 0.60istheconstant
value, which is independent on α values (Ea = 90.8 kJ mol−1).
In other words, the calculated values of errors in determination
of Ea vary in the range from 5.8% ( 5.3 kJ mol−1) to 6.0%
( 5.5 kJ mol−1) (Fig. 4) in considered range of α, until for
α ≤ 0.20 and α ≥ 0.60, the calculated errors in Ea determina-
tion are extensively higher (Fig. 4). These results enable us to
conclude that the values of Ea for 0.20 ≤ α ≤ 0.60 were con-
stant and independent on α values. For α ≥ 0.60, the values of
Ea increased with increasing of α values.
The thermogravimetric (TG) curves of the reduction process
of NiO by hydrogen obtained at different heating rates (2.5, 5,
10 and 20 ◦C min−1) are shown in Fig. 1.
The total mass-change of NiO samples at all considered heat-
ing rates is about 21.4%. The degree of conversion (α), was
calculated from the TGA data, thus:
m0 − mt
α =
(7)
m0 − mf
where mt represents the mass of the sample at arbitrary time
t (or temperature T), whereas m0 and mf is the mass of
the sample at the beginning and at the end of the process,
respectively.
The α–T curves at different heating rates for reduction of
nickel oxide in hydrogen atmosphere are shown in Fig. 2.
Table 2 shows the influence of heating rate on the character-
istic temperatures of reduction process (initial temperature (Ti),
inflection temperature (Tp), final temperature (Tf) and tempera-
ture differences (ꢀT = Tf − Ti)) (Fig. 1).
The increasing of heating rate leads to the increasing of Ti,
Tp, Tf and ꢀT values. Dependence of dα/dt versus T at different
heating rates of the considered system is shown in Fig. 3.
With increasing of heating rate (β), the rate of reduction
process of nickel oxide by hydrogen increased.
Table 2
The influence of heating rate on the characteristic temperatures of the reduction
process of nickel oxide using hydrogen
␤/◦C min−1
Ti/◦C
Tp/◦C
Tf/◦C
ꢀT/◦C
2.5
5
10
20
260
275
285
300
285
300
315
340
395
455
465
485
135
180
180
185
Fig. 3. Experimental (lines) reaction rates curves (dα/dt vs. T) for the reduction
of nickel oxide by hydrogen at different heating rates.

52
B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
Fig. 4. Dependence of the activation energy (Ea) on the degree of conversion
(α) determined using the Friedman’s method for the reduction of nickel oxide
by hydrogen.
Fig. 5. Compensation effect observed between apparent activation energy and
pre-exponential factor for the reduction of nickel oxide by hydrogen at different
heating rates.
The obtained average value of the apparent activation energy
using Friedman’s method is in good agreement with values of
Ea, which were determined in papers of Richardson et al. [15]
(for porous powder in loosely packed samples) and Nakajima et
al. [26] (for thin slab samples), respectively. From the existing
of “plateau” in Fig. 4, in wide range of α values, the proof of
existing of single-step reaction is evident.
vation parameters (invariant pre-exponential factor Ainv and
invariant activation energy Einv). As α∗v and βv∗ and
are correlated by supercorrelation relation, one obtains:
Ainv = 1.04 × 108 min−1 and Ea,inv = 96.4 kJ mol−1
.
The obtained value of pre-exponential factor and correspond-
ing temperature range of reduction process were considerable
different from the values of pre-exponential factor (A) and reduc-
tion temperature ranges obtained by Richardson et al. [15] and
Nakajima et al. [26]. These facts designated on the different
kinetical model for reduction process of our investigated NiO
samples in relation to the NiO samples, which were used in
reduction experiments by the above-cited investigators.
The IKP method gives an exact value of activation energy
of the investigated process, but although not one of the con-
sidered kinetic models (Table 1) does not completely kinetically
described the experimental data. The values of the invariant acti-
vation parameters were used further for the numerical evaluation
of f(α) function.
Fig. 6 shows the curves f(α) versus α obtained from Eq. (1) at
different heating rates, with the experimental values of β(dα/dT)
and the values of the invariant activation parameters.
As shown in this figure, for all observed heating rates the
curves f(α) versus α exhibits the same shape which designates
to the same reaction mechanism and shows the dependences
from the heating rate (β).
The increase of Ea with increasing of α for α ≥ 0.60 (Fig. 4)
designatedontheexistenceofparallelindependentreaction[27],
which possessed the higher value of activation energy and whose
part in the overall process in this range of α increasing.
In order to apply the IKP method, the conversion range in
which the value of Ea is practically constant has to be determined
andthathasbeendonethroughtheabovemethods. Fordetermin-
ing the compensation effect parameters (α*, β*) Coats–Redfern
method has been applied in the range of conversion where Ea is
independent of the degree of conversion. The values of kinetics
parameters (lnA, Ea) obtained by means of Coats and Redfern
method for the reduction of nickel oxide by hydrogen at different
heating rates (2.5, 5, 10 and 20 ◦C min−1) are listed in Table 3.
Calculated values of activation energies, Ea (see Table 3) are
not in agreement with the values calculated by Friedman isocon-
versional method, what means that not one of the tested kinetic
models does not correspond to the real kinetic model of reduc-
tion of nickel oxide by hydrogen. The apparent compensation
effect is observed for each heating rate for the process under
investigation and presented in Fig. 5.
A compensation effect of this kind is classified as a false or
superficial compensation effect [28] resulting from parameter
distortion by an inappropriate kinetic model function [29]. The
values of α* and β* were calculated from the intercepts and the
slopes of the straight lines obtained in the Fig. 5. The values of
the apparent compensation effect (α*, β*) obtained at different
heating rates (β) are presented in Table 4.
It is clear from Fig. 6 that the curve of f(α) versus α exhibits
a maximum, suggesting that the reduction of NiO by hydrogen
occurs through the reaction mechanism which is similar to the
mechanism represented by the Avrami–Erofe’ev equations (with
r > 1). An inspection of the conversion functions included in
Table 1 clearly confirms that only in these cases do the curves of
f(α) against α have a maximum at fꢀ(αm) = 0. The values reported
forthedegreeofconversionatthemaximumreactionratechange
from αm = 0.393 for r = 2 to αm = 0.527 for r = 4 [30]. However,
A linear regression from the straight-line plot of α∗v =
ln Ainv − βv∗ Ea,inv allows computation of the invariant acti-

B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
53
Table 3
Apparent kinetic parameters for the reduction of nickel oxide in hydrogen atmosphere obtained for each kinetic function vs. the heating rates
Modela β = 2.5 ◦C min−1 β = 5 ◦C min−1 β = 10◦ C min−1 β = 20 ◦C min−1
ln Av ln Av ln Av ln Av
Ea,v (kJ mol−1 Ea,v (kJ mol−1 Ea,v (kJ mol−1 Ea,v (kJ mol−1
9.38 5.55 5.51 6.56
)
R
)
R
)
R
)
R
1
2
57.6
79.8
0.9942
0.9946
0.9950
0.9955
0.9954
0.9973
0.9978
0.9986
0.9995
0.9999
0.9998
39.8
56.2
89.1
0.9905
0.9915
0.9924
0.9934
0.9932
0.9964
0.9973
0.9985
0.9996
0.9999
0.9991
37.7
53.5
85.1
0.9892
0.9905
0.9916
0.9927
0.9925
0.9963
0.9973
0.9987
0.9998
0.9999
0.9986
41.3
58.3
92.7
0.9921
0.9930
0.9938
0.9946
0.9944
0.9970
0.9976
0.9987
0.9996
0.9999
0.9995
0.9987
0.9986
0.9985
0.9984
0.9982
0.9947
0.9963
0.9977
0.9969
14.43
9.26
16.51
9.03
15.86
10.19
17.25
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
24.40 124.4
82.54 391.6
53.63 258.0
60.05 289.7
62.17 301.0
68.51 324.4
76.93 362.0
85.99 402.6
105.96 492.3
140.88 658.0
44.23 213.3
31.96 157.6
19.56 101.9
58.31 286.4
37.57 187.8
43.32 217.0
45.23 227.6
51.19 249.7
59.11 286.0
67.76 325.6
87.15 414.9
55.17 274.9
35.68 180.0
42.17 213.5
44.38 225.7
51.02 251.6
60.16 294.4
70.26 342.0
93.14 450.1
57.96 298.4
37.77 195.5
43.05 224.5
44.77 234.9
50.33 256.6
57.56 291.9
65.41 330.4
82.89 416.4
0.9987 105.86 508.9
0.9986 104.88 512.9
0.9987 102.88 523.4
0.9986
0.9986
0.9985
0.9984
0.9955
0.9968
0.9979
0.9972
32.79 163.4
23.47 120.1
14.02
9.21
78.92 385.0
86.26 421.9
94.03 464.6
87.83 436.1
0.9985
0.9984
0.9983
0.9981
0.9935
0.9956
0.9974
0.9963
32.90 164.6
23.72 120.9
14.39
9.65
74.54 369.8
82.73 411.6
91.71 461.1
84.70 427.9
0.9986
0.9985
0.9984
0.9983
0.9929
0.9953
0.9974
0.9961
32.65 167.8
23.68 123.2
14.58
9.95
78.02 401.2
84.81 437.9
91.84 479.9
86.14 451.8
76.8
55.3
77.3
55.6
78.7
56.5
13.30
74.2
111.33 525.2
119.66 565.7
128.27 611.2
121.51 580.8
a
Enumeration of the models is given in Table 1.
the values of f(α) determined for the reduction of nickel oxide
by hydrogen from the IKP method and shown in Fig. 6 could
not be fitted exactly by any of the ideal Avrami–Erofe’ev kinetic
laws included in Table 1. This behaviour can be understood if we
take into account that these kinetic equations have been derived
by assuming “ideal” models conditions (homogeneous particle
size, shape, etc.), which are not always satisfied by real samples.
In some cases, it is convenient to use more flexible empirical
kinetic model functions, f(α), for the description of the crystal-
lization processes. The advantage of such an approach is that it is
possible to describe experimental thermogravimetric data even
in the case when some of the basic assumptions for the validity
of the Avrami–Erofe’ev model are no longer held. These models
can also be understood in terms of the accommodation function
ˇ
´
introduced by Sestak [31], expressing a deviation from the ideal
case due to a more complex process. In this respect, it is very
ˇ
´
convenient to use the Sestak–Berggren (SB) function [32]:
f(α) = αm(1 − α)n
(8)
and this is the most popular form of f(α) for examination of
auto-catalyzed chemical reactions. This flexible two-parameter
(m, n) SB function, in fact, also includes the Avrami–Erofe’ev
kinetic model. It can be shown [33] that there are combinations
of parameters m and n corresponding to a given value of kinetic
exponent for the Avrami–Erofe’ev model (for r ≥ 1). Therefore,
the SB kinetic model can be used for a quantitative description of
more complex crystallization processes involving partially over-
lapping nucleation and growth phases. The kinetic parameters,
m and n in Eq. (8), are characteristic of a particular crystalliza-
tion process although it is rather problematic to find their real
physical meaning. It was shown [34], however, that physically
meaningful values of the parameter m (in Eq. (8)) should be con-
fined in the range of 0 < m < 1. From the complex dependence
Ea on α, which is indication of the complex mechanism of nickel
Table 4
Values of compensation effect parameters at four different heating rates for the
reduction of nickel oxide in hydrogen atmosphere
␤/◦C min−1
α∗v (min−1
)
βv∗ (mol kJ−1
)
R
2.5
5
10
20
−2.4140
−1.8628
−1.4555
−0.5092
0.2165
0.2109
0.2065
0.1968
0.9998
0.9996
0.9996
0.9996
Fig. 6. Dependence of f(α) vs. α for the reduction of nickel oxide by hydrogen,
established using invariant activation parameters at four different heating rates
(β = 2.5, 5, 10 and 20 ◦C min−1).

54
B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
Table 5
Bearing in mind previously exhibited kinetical models and
mechanisms for NiO reduction process using hydrogen (Section
1), it can be concluded with high reliability, that the reduction
process occurs at the interface between NiO and previously
reduced Ni without any well-ordered intermediate phase. The
constant value of Ea is a consequence of constant activation
energy for crystal growing of Ni crystallites, which were induced
in nucleation stage of the process whose duration period depends
on the heating rate of the system. The increasing of reduction
rate without changing of Ea value, which is interrelationship
with existing of “autocatalytic effect”, i.e. it is a consequence of
being proportionality between the reduction rate and area size of
boundary phase of interaction. The decreasing of reduction rate
and increasing of Ea values was probably caused by mutually
overlapping of Ni growing crystals which lead to the decreas-
ing of area size of boundary phase and the rate of process. The
The parameters of the conversion function f(α) = ␣m(1 − ␣)n (Eq. (8)) as well as
the lnA value resulting from Eq. (10), for the reduction of nickel oxide samples
in hydrogen atmosphere at four different heating rates
β/◦C min−1
m
n
αmax
lnA, A/min−1a
2.5
5
10
20
0.77
0.46
0.60
0.69
1.48
1.22
1.49
1.39
0.34
0.27
0.29
0.33
19.72
19.10
19.74
19.44
Average
0.63
1.39
0.31
19.50
a
Activation energy (by IKP method): 96.4 kJ mol−1
.
oxide reduction, the kinetic model of investigated process was
determined by applying the Malek’s procedure.
The kinetic exponents m, n and αmax satisfy the following
relationship [35,36]:
´
ˇ
´
obtained values of Sestak–Berggren parameters (m, n) shows,
that the increasing value of the kinetic parameter m indicates a
moreimportantroleofthecrystallizedphaseontheoverallkinet-
ics. On the other hand, the higher value of the kinetic parameter
n (n > 1) shows the increasing complexity of the investigated
reduction process. Accordingly, the proposed kinetical model
for the NiO reduction process (SB model) is more complex than
simple shrinking core model proposed by the authors in [12] and
[16].
m
n
αmax
s =
=
(9)
1 − αmax
where αmax is the value of α corresponding to the maximum of
f(α).
Once the value of s (ratio m/n) is known, the values of the
kinetic exponents m and n can be obtained using the following
relation, which results from Eqs. (1) and (8):
ꢀ
ꢀ
ꢁꢁ
dα
dt
Ea
ln
exp
= ln A + n ln[αs(1 − α)].
(10)
RT
5. Conclusions
The parameters of the straight lines ln(dα/dt·exp(Ea/RT)) ver-
sus ln[␣s(1 − ␣)] are lnA and n. Table 5 lists the values of kinetic
exponents m and n, the values of αmax as well as the values of lnA
(with their average values) obtained by the procedure described
above, for the reduction of nickel oxide in hydrogen atmosphere
at different heating rates.
The kinetic process for the temperature-programmed reduc-
tion (TPR) of nickel oxide, obtained by sol–gel procedure using
hydrogen has been studied. The Friedman’s isoconversional
method is used to yield dependency of the activation energy
of reduction process on the degree of conversion. The use of
the IKP method for the range where Ea is independent of α, led
to the invariant activation parameters, which were used for the
numerical evaluation of the conversion function (f(α)). The acti-
vation energy (Ea) value obtained by the IKP method displays
a good agreement with the value of Ea obtained by Friedman’s
method. It is established that the reduction of NiO using hydro-
gen is a multi-step mechanism and can be described by the
The values of kinetic exponents of SB model (m and n) as well
asthevaluesofαmax dependontheheatingrates, whilethevalues
of lnA do not show the significant variation with heating rates
(Table 5). The values of αmax correspond of the maximums in
Fig. 6 for SB model with kinetic exponents are given in Table 5.
The obtained values of A (calculated from Table 5) at different
heating rates, are in good agreement by order of magnitude with
the value of A obtained using IKP method (×108).
ˇ
´
Sestak–Berggren (SB) model.
On the other hand, the relationship between the values of m, n
and Avrami–Erofe’ev parameter r may be obtained by equating
α’s from the two equations at the peak of the reaction curve
where d(dα/dt)/dt = 0 [37]. This results in the relationship:
For the temperature-programmed reduction of nickel
oxide in hydrogen atmosphere, the following kinetic triplet
was obtained: Ea = 96.4 kJ mol−1, A = 1.04 × 108 min−1 and
f(␣) = α0.63(1 − α)1.39
.
1
r =
.
(11)
[1 + ln(n) − ln(n + m)]
Acknowledgements
The values of parameter r calculated from Eq. (11) for the
investigated reduction process of NiO by hydrogen at 2.5, 5, 10
and 20 ◦C min−1 (with SB kinetics exponents from Table 5) are:
1.72, 1.47, 1.51 and 1.67, respectively.
The investigation was partially supported by the Ministry
of Science and Environmental Protection of Serbia, under the
following Projects 142025 and 142047 (S. Mentus).
Based on the results showed above, it was decided that indeed
f(α) = α0.63(1 − α)1.39 is the best conversion function because
References
ˇ
´
according to Sestak et al. [38], in case of a system that con-
tains metallic particles the reduction processes take place mainly
along the border between the oxide and metallic particles.
[1] J.M. Thomas, W.J. Thomas, Principles and Practice of Heterogeneous
Catalysis, New York, VCH, 1997.

B. Jankovic´ et al. / Thermochimica Acta 456 (2007) 48–55
55
[2] H.H. Kung, Transition Metal Oxides: Surface Chemistry and Catalysis,
Elsevier, New York, 1989.
[18] See for example: F. Paulik, Special trends in thermal analysis, Chichester:
Wiley, 1995 (Chapter 10).
[3] B. Delmon, in: G. Ertl, H. Kno¨zinger, J. Weitkamp (Eds.), Handbook of
Heterogeneous Catalysis, Wiley-VCH, New York, 1997, p. 264.
[4] R.P. Furstenau, G. McDougall, M.A. Langell, Surf. Sci. 150 (1985) 55.
[5] B. Lescop, J.-Ph. Jay, G. Fanjoux, Surf. Sci. 548 (2004) 83.
[6] M.A. Langell, Surf. Sci. 164 (1985) 543.
[7] B. Delmon, Introduction a´ la cine´tique he´te´roge´ne, Technip, Paris, 1969
(Chapter 11).
[8] A.F. Benton, P.H. Emmett, J. Am. Chem. Soc. 46 (1924) 2728.
[9] Y. Koga, L.G. Harrison, in: C.H. Bamford, C.F.H. Tipper, R.G. Compton
(Eds.), Comprehensive Chemical Kinetics, vol. 21, Elsevier, Amsterdam,
1984, p. 120.
[19] H.L. Friedman, J. Polym. Sci. C 6 (1964) 183.
[20] A.I. Lesnikovich, S.V. Levchik, J. Therm. Anal. 27 (1983) 83.
[21] A.I. Lesnikovich, S.V. Levchik, J. Therm. Anal. 30 (1985) 667.
[22] J.M. Criado, J. Morales, Thermochim. Acta 19 (1977) 305.
[23] J.M. Criado, J. Morales, V. Rives, J. Therm. Anal. 14 (1978) 221.
[24] A.W. Coats, J.P. Redfern, Nature 201 (1964) 68.
[25] P. Budrugeac, Polym. Degrad. Stab. 71 (2001) 185.
[26] M. Nakajima, S. Shimizu, K. Onuki, Y. Ikezoe, S. Sato, Chem. Soc. Jpn. 4
(1989) 681.
[27] S.V. Vyazovkin, V. Goryachko, A.I. Lesnikovich, Thermochim. Acta 197
(1992) 41.
ˇ
[10] J. Bandrowski, C.R. Bickling, K.H. Yang, O.A. Hougen, Chem. Eng. Sci.
17 (1962) 379.
[28] N. Koga, J. Sesta´k, J. Therm. Anal. 37 (1991) 1103.
ˇ
[29] N. Koga, J. Sesta´k, J. Ma´lek, Thermochim. Acta 182 (1991) 333.
[11] J. Moriyama, A. Yamaguchi, Nippon Kinzoku Gakkaishi 28 (1964) 831.
[12] S. Yagi, D. Kunii, Proceedings of the 5th International Symposium on
Combustion, Reinhold, New York, 1955, p. 231.
[30] J.M. Criado, J. Therm. Anal. 21 (1981) 155.
ˇ
[31] J. Sesta´k, J. Therm. Anal. 37 (1991) 111.
ˇ
[32] J. Sesta´k, G. Berggren, Thermochim. Acta 3 (1971) 1.
[13] J.T. Richardson, M. Lei, B. Turk, K. Forster, M.V. Twigg, Appl. Catal. A
110 (1994) 217.
[33] J. Ma´lek, Thermochim. Acta 138 (1989) 337.
[34] J. Ma´lek, J.M. Criado, J. Sesta´k, J. Milittky´, Thermochim. Acta 153 (1989)
ˇ
[14] J.A. Rodriguez, J.C. Hanson, A.I. Frenkel, J.Y. Kim, M. Pe´rez, J. Am.
Chem. Soc. 124 (2002) 346.
429.
[35] V.M. Gorbachev, J. Therm. Anal. 18 (1980) 13.
[15] J.T. Richardson, R. Scates, M.V. Twigg, Appl. Catal. A 246 (2003) 137.
[16] T.A. Utigard, M. Wu, G. Plascencia, T. Marin, Chem. Eng. Sci. 60 (2005)
2061.
[17] S. Mentus, D. Majstorovic´, B. Tomic´, R. Dimitrijevic´, Mater. Sci. Forum
494 (2005) 345.
[36] J. Ma´lek, Y. Messaddeq, S. Inoue, T. Mitsuhashi, J. Mater. Sci. 30 (1995)
3082.
[37] J. Sesta´k, J. Ma´lek, Solid State Ionics. 63–65 (1993) 245.
ˇ ˇ
[38] J. Sesta´k, V. Satava, W. Wendlandt, Thermochim. Acta 7 (1973) 333.
ˇ