Influence of heating rate on kinetic quantities of solid phase thermal decomposition

Article abstract of DOI:10.1007/s10973-005-7378-0

Thermogravimetric analyses of thermal decomposition (pyrolysis, thermal dissociation and combustion) of 9 different samples were carried out in dynamic conditions at different heating rates. The kinetic parameters (E, A and k _m) of thermal decomposition were determined and interrelations between the parameters and heating rate q were analyzed. There were also relations between Arrhenius and Eyring equations analyzed for thermal decomposition of solid phase. It was concluded that Eyring theory is an element, which interconnects used thermokinetic equations containing Arrhenius law and suggests considering kinetic quantities in way relative to 3 kinetic constants (E, A and k_m). Analysis of quantities other than k_m (i.e. E, A, Δ⁺H, Δ⁺S) in relation to heating rate is an incomplete method and does not lead to unambiguous conclusions. It was ascertained that in ideal case, assuming constant values of kinetic parameters (E and A) towards heating rate and satisfying both Kissinger equations, reaction rate constant k_m should take on values intermediate between constants (k_m)₁ and (k_m)₂ determined from these equations. Whereas behavior of parameters E and A towards q were not subjected to any rule, then plotting relation k_m vs. q in the background of (k_m)₁ and (k_m)₂ made possible classification of differences between thermal decomposition processes taking place in oxidizing and oxygen-free atmosphere.

Full text of DOI:10.1007/s10973-005-7378-0

Journal of Thermal Analysis and Calorimetry, Vol. 84 (2006) 2, 453–465
INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES OF SOLID
PHASE THERMAL DECOMPOSITION
1
*
R. Bigda and A. Mianowski
2
1
Institute for Chemical Processing of Coal, Zamkowa 1, 41-803 Zabrze, Poland
2
Department of Inorganic Chemistry and Technology, Silesian University of Technology, Krzywoustego 6, 44-100 Gliwice, Poland
Thermogravimetric analyses of thermal decomposition (pyrolysis, thermal dissociation and combustion) of 9 different samples
were carried out in dynamic conditions at different heating rates. The kinetic parameters (E, A and k ) of thermal decomposition
m
were determined and interrelations between the parameters and heating rate q were analyzed. There were also relations between
Arrhenius and Eyring equations analyzed for thermal decomposition of solid phase. It was concluded that Eyring theory is an ele-
ment, which interconnects used thermokinetic equations containing Arrhenius law and suggests considering kinetic quantities in
+
+
way relative to 3 kinetic constants (E, A and k
is an incomplete method and does not lead to unambiguous conclusions. It was ascertained that in ideal case, assuming constant val-
ues of kinetic parameters (E and A) towards heating rate and satisfying both Kissinger equations, reaction rate constant k should
take on values intermediate between constants (k and (k determined from these equations. Whereas behavior of parameters E
and A towards q were not subjected to any rule, then plotting relation k vs. q in the background of (k and (k made possible
m m
). Analysis of quantities other than k (i.e. E, A, D H, D S) in relation to heating rate
m
m
)
1
m 2
)
m
m
)
1
m 2
)
classification of differences between thermal decomposition processes taking place in oxidizing and oxygen-free atmosphere.
Keywords: Arrhenius law, Eyring theory, heating rate, isokinetic effect, thermogravimetry, thermokinetic analysis
+
Introduction
free energy of activation (D G) in form of classical
formula [11]:
Heating rate, most frequently realized as dT/dt=q>0,
particularly in the newest works [1–10], may influ-
ence kinetic parameters included in thermokinetic
equations. In general, one looks for Arrhenius equa-
tion constants, i.e. activation energy (E) and pre-ex-
ponential factor (A):
+
+
+
+
D G = D H – TD S, ¥ > D S > – ¥
(5)
It is necessary to stress that while typical
Arrhenius equation is related in close way to reaction
kinetics (though in limited temperature range), Eyring
Eqs (3) and (4) are thermodynamic possibilities of in-
terpretation of absolute reaction rate constant con-
nected with formation of activated complex. One may
also notice that form Eq. (4) is mathematically similar
to other forms of Arrhenius equation, having totally
different interpretation.
–
E/ RT
k = Ae
(1)
or alternatively one uses temperature of maximal re-
action rate (Tm):
–
E/ RTm
km = Ae
(2)
Internal correlation between coefficients of law
Eq. (1), called compensation or isokinetic effect (IE),
expressed as:
On rare occasions one uses Eyring equation in
form [11–14] (assuming all activity coefficients equal
to 1):
E
lnA =
+ lnk , q = idem
(6)
+
iso
kB
–D G/ RT
k = Te
(3)
RTiso
h
is also translated by Eyring Eq. (4). It comes from the
fact of activation energy assigning to enthalpy part
and pre-exponential factor to entropy one. There is a
designation following from the fact, namely
enthalpy-entropy correlation [15]. Another quantity
connected with discussed issue is reaction rate con-
or in expanded form:
k = BTe
D+S/ R –D H/ RT
+
e
(
4)
1
1
–1 –1
B = k /h = 0.20836 ×1 0 K s
B
Searched coefficients in Eq. (4) are enthalpy
D H) and entropy (D S) of activation related to the
+
+
stant in temperature T
, according to Eq. (2). In
m
(
Eq. (6) for dynamic conditions [16–19]:
*
Author for correspondence: bigda@ichpw.zabrze.pl
1
388–6150/$20.00
Akadémiai Kiadó, Budapest, Hungary
Springer, Dordrecht, The Netherlands
©
2006 Akadémiai Kiadó, Budapest

BIGDA, MIANOWSKI
Assumptions for analytical part
Tiso = Tm,
Comparing derivatives dlnk/dT for Eqs (1) and
4) we obtain:
g(a)=var
(7)
•
E and A are constant for q=var, what is expressed
(
by Kissinger law in Wendtland’s version [22]:
+
E = D H + RT
(8)
q
AR
E
E
ln
= ln
2
–
, E = E ,A = AI
(11)
I
and from comparison of Eq. (1) with Eq. (4), taking
Eqs (8) and (5) into consideration:
Tm
RTm
+
and from comparison of formula (11) with Arrhenius
law (2) it comes:
D S
lnA =1+ lnBT +
(9)
R
qE1
–E / RT
I
m
(
k ) =
= AI e
(12)
(13)
(14)
Because in general one determines adequate
thermodynamic functions of activation from coeffi-
cients of Eqs (1), (9) may be expressed as:
m
1
2
RTm
as well as Kissinger law in form [16, 17]:
æ
A
ö
+
q
E
D S = Rç ln – lnB –1÷
(10)
ln
= lnA –
, E = E , A = A
II II
T
è
ø
Tm
RTm
–
1
when A in s , then lnB = 23.76.
from which:
It was proved in works [12, 14, 19] that in
Eqs (3)–(5) and (8)–(10) one may use T as tempera-
q
–
E
II / RT
m
m
(
k ) =
= AII e
m
2
ture of activation.
Tm
Thus, there is following relation between both
reaction rate constants (12) and (14):
The aim of the work
E1
The aim of the work is searching for dependencies be-
(k
m
)
1
= (k
m
)
2
, (k ) >(k ) because
m 1 m 2
+
+
RTm
tween E, A, T D H, D S and heating rate q, taking in-
m
(
15)
ternal correlations between kinetic quantities into
consideration. Constant km may be recognized as
privileged quantity with regard of its meaning con-
nected with formation of activated complex according
to Eyring theory Eq. (4). We may assume that using
Eyring theory presented in works [11–14] for thermal
decomposition processes makes possible carrying out
similar considerations also for combustion. The more,
one may use mentioned theory for reaction systems, in
which it is hard to imagine existence of activated com-
plex (tetraline+coal®naphthalene+H-coal [20, 21]).
We would like to emphasize differences between
thermal decomposition processes of solids in inert
and oxidizing atmosphere. The inert atmosphere pro-
cesses are understood by two types of reactions, de-
fined by the use of stoichiometric equations within
following categories:
E1
>
> 1
RTm
where values of k are calculated from Eq. (2) or (4)
substituting T=T ), of which parameters are interre-
m
lated by relationships (8) and (9), what leads to ob-
taining the same values of rate constant. T increases
m
(
m
together with q.
•
From literature data [12] follows that experimental
values of E and A decrease linearly with increase of
heating rate q (Fig. 1a) or strong deviations from
this relationship appears at low heating rates, be-
–1
low q=10 K min (Figs 1b–d).
+
•
•
Free energy of activation (D G) is approximately
constant regardless of assumed functions g(a) [14]
as well as heating rate [12, 14] (compare with Ap-
pendix A).
Values of thermodynamic functions of activation
D H and D S decrease together with heating rate in-
•
strictly chemical and thermodynamic – thermal dis-
sociation (usually applied to inorganic compounds
and salts of organic acids),
+
+
crease [12].
•
conventional and symbolic – pyrolysis (usually ap-
plied to chemically complicated substances – e.g.
coal, polymers – or mixtures).
Analysis of thermokinetic equations
We are proving in present article that there are
differences in behavior of kinetic rate constant (km)
towards heating rate (q) between reactions taking
place in different atmospheres (inert and oxidizing).
Similarly to works [16, 17], two most popular
thermokinetic equations in integral forms (g(a)~T
and g(a)~T Table 1, No. 1 and 4) have been taken in
order to search for relationships of reaction rate con-
2
stants from condition g(a ) = 1, introducing equa-
m
454
J. Therm. Anal. Cal., 84, 2006

INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES
Fig. 1 Relationship of activation energy on heating rate for: a – dehydration reaction: MnSO
model F1) [12], b – dehydration reaction: CuSO ·3H O®CuSO ·H O+2H O (model F1) [23, 24], c – decomposition of
,3,5,7- tetranitro-1,3,5,7-tetrazocine (HMX) (power law) [25], d – dehydration reaction: MnC ·2H O®MnC +2H
model A) [26]
4 2 4 2 2
·5H O®MnSO ·3.25H O+1.75H O
(
4
2
4
2
2
1
2
O
4
2
2
O
4
2
O
(
tions, which contain Eyring Eq. (4) instead of
Arrhenius one (1) (Table 1, No. 6 and 7). From suit-
able functions presented in Table 1 emerge searched
relationships (12) and (14) connected by Eq. (15). It is
necessary to stress that equation (Table 1, No. 6):
2
as well as (12) and (14). Their analysis makes possi-
ble continuation of discussion presented in own
works [16, 17] where authors concluded that using
only kinetic parameters (E and A) is unreliable and
does not bring unambiguous results. Only rate con-
stant k connects together values of mentioned pa-
m
BT
D+G/ RT
–
g(a )=
e
(16)
rameters and unambiguously informs about changes
in reaction kinetics towards heating rate q. This is be-
cause km in version (12) or (14) follows from
Kissinger law, which involves influence of heating
q
is a result of special case of integration of function
containing temperature according to BÓaüejowski
rate q only by T without influence of E and A, which
m
[
13]; in other cases one obtains solution given in Ta-
both behave different towards heating rate (Fig.1),
what was proved in mentioned works [16, 17].
ble 1, No. 8, which follows from chosen approxima-
tion of Euler integral (Appendix B).
From Table 1 follows that relationship (12) is
more connected with typical thermokinetic equation
g(a)~T and Arrhenius and Kissinger laws and is mul-
2
Arrhenius law and Eyring equation
tiplicity of Eq. (14), which is connected as well with
thermokinetic equation g(a)~T, as with Eyring equa-
Table 2 contains comparison of adequate kinetic (E
+
+
+
and A) and thermodynamic (D H, D S and D G) quan-
tities, from which follows that, with regard to equality
of free energy, two couples of alternative dependen-
cies exist:
tion. Forms of k depend on form of initial equation
m
g(a) as well as from assumptions for differential
thermokinetic equations da/dT.
Analysis of equations in Table 1 allows to con-
clude that reaction constants in temperature of maxi-
mal reaction rate are presented by 3 relationships: (2)
æ
A
ö
+
+
D H = E –RT , D S = Rç ln –24.76÷ (17)
m
ç
÷
Tm
è
ø
J. Therm. Anal. Cal., 84, 2006
455

BIGDA, MIANOWSKI
Table 1 Interrelations between form of thermokinetic equation and reaction rate constant
da
Function g(a) or
No.
1
Condition
Reaction rate constant Notes
dT
2
ART
qE
–
e
E/ RT
g(a) =
1
) g(a
m
) =1, T=T
m
2
RT
q
E
No. (1) is
approximation
of solution No. 3
g(a) = k
da
(k ) =
m 1
2
qE
Tm RTm
2
2
A
2) d a/dT = 0, T=T
d =1
m
,
–E/ RT
=
f (a) q
e
3
dT
(
Eq. (27) in [16])
AT
–
E/ RT
g(a) =
e
4
5
1) g(a
m
) =1, T=T
m
–
q
2
2
T
q
2
) d a/dT =0, T=T
m
,
g(a) = k
Eqs (40), (41) in [16]
m
E/(E+RT ) @1
q
(k ) =
m 2
2
Tm
g(a) acc. [13]
when k= C Te
C
BT
q
+
+
D S/ R -D H/ RT
–C2 / T
g(a) =
e
e
g(a ) =1, T=T
6
m
m
1
1
, C
2
=const
T
g(a)=k
g(a) =
7
8
9
–
q
2
BT
+
+
D S/ R -D H/ RT
e
e
g(a ) =1, T=T
m
m
–
2
q
2
q
(
k ) =
m
2
Tm
T
g(a)= k
–
2
q
Table 2 Interrelations between enthalpy, entropy and free energy of activation
Method of
comparison
+
+
a
a)
+
a
No.
1
D H=
E–RT , Eq. (8)
D S
D G
æ
A ö
÷
æ
A
ö
from derivative
E+RT çln B–ln
Rçln –ln B–1÷, Eq. (10)
m
ç
÷
m
ç
÷
Tm
dlnk/dT
Tm
è
ø
è
ø
æ
A ö
æ
A
ö
Eqs (2) and (4)
with (5)
E+RT çln B–ln
÷
Rçln –ln B÷
m
ç
÷
2
E
ç
÷
Tm
Tm
è
ø
è
ø
æ
ARö
÷
æ
AR
E
ö
E+RT çln B-ln
Rçln
-ln B÷
m
è
E ø
è
ø
from
relationship
g(a)
æ
ç
è
A
ö
@ +
E
RT çln B– ln ÷ +T (22+ R)@
m
m
÷
3
ln
2
E
T
m
ø
T
æ
ö
A
b)
b
Rçln –ln B÷–(22+R)
(Table 1,
æ
A
ö
÷
÷
ø
ç
÷
Tm
@E+ RT çln B– ln
è
ø
m
No. 1 and 6)
ç
è
Tm
+
m
D G>>T (22 + R)
a
lnB=23.76
æ
A
ö
æ
A
ö
b
from correlation Rç ln – ln B – 1÷ = 1.03Rç ln – ln B ÷ + 22
ç
÷
ç
÷
è
Tm
ø
è
Tm
ø
or
Third considered case leads to approximated val-
ues (Table 2, No. 3) and we may pass it over. Thus,
energy of activation is directly enthalpy of activation,
taking assumptions given in the aim of the work and
T=Tm.
æ
A
ö
+
+
D H =E, D S =Rç ln –23.76÷
(18)
ç
÷
Tm
è
ø
+
because in both cases one obtains equality of D G ac-
cording to Eq. (5).
456
J. Therm. Anal. Cal., 84, 2006

INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES
Experimental
Thermogravimetric measures of thermal dissociation
of CuSO ·5H O and CaC O ·H O, as well as pyroly-
4
2
2
4
2
sis of PVC and high rank bituminous coal have been
performed in dynamic nitrogen atmosphere and com-
bustion of PVC and mixtures of electrographite with
3
chosen sodium silicates with different molar ratios
have been performed in air. The inorganic compounds
CuSO ·5H O and CaC ·H O) have been chosen as
(
4
2
2
O
4
2
model as thermal dissociation examples, while coal
and PVC have been used as pyrolysis examples.
Electrographite and its mixtures with sodium silicates
have been chosen for investigations, because reactiv-
ity of graphite in the presence of inorganic com-
pounds towards oxygen is very crucial from thermo-
kinetic and technological point of view.
Thermogravimetric measures of all samples
have been carried out on Mettler TG-50 thermo-
balance in TA-4000 thermoanalytic system. Samples
have been heated in open Pt crucibles. Table 3 pres-
ents experimental conditions for each sample and Ta-
ble 4 – characteristics of coal, electrographite and so-
dium silicate samples.
Fig. 2 Relationship of dehydration reaction rate:
CaC ·H O®CaC +H O acc. to Eq. (2) for differ-
ent thermokinetic models and values of (k
iso on heating rate
O
2 4
2
2
O
4
2
m
)
1
, (k
m
)
2
and
k
(k ) ³ k ³ (k ) for q=var
(19)
m
1
m
m
2
Eqs (12), (2) and (14), respectively.
The values of (k ) , (k ) and k have been cal-
m 1
m 2
iso
culated from Eqs (12), (14) and (6), respectively. The
values of kinetic parameters (E and A) used in these
equations have been determined from following rela-
tionships, respectively:
Verification of experimental data
There is graphical presentation of reaction rate con-
stant vs. heating rate relationships presented on Fig. 2
for thermal dissociation of CaC O ·H O (dehydra-
•
•
•
Kissinger law in version (11) (for (km)1),
Kissinger law in version (13) (for ((km)2),
Arrhenius law (2), where values of E and A formed
an IE, with the use of which the value of kiso has
been determined.
2
4
2
tion), described in work [27] and used again in work
16]. The described dehydration process fulfills as
[
well relationship (11) as (13) and IE (6) for each heat-
ing rate used in thermogravimetric measures. Taking
The relationship between (k ) and q (Eq. (12))
m 1
mentioned facts into consideration, values of k
termined with help of Eq. (2) should fulfill inequality:
m
de-
are in most cases rectilinear, but in a few cases the re-
lationship has been found to be slightly curvilinear.
Table 3 Experimental conditions for each of analyzed sample
Sample mass, m/
Temperature range/
°C
Furnace
atmosphere
Gaz flow rate/
mL min
Heating rate, q/
Sample
–1
–1
mg
K min
CuSO
4
·5H
·H
2
O
25–1100
25–300
200
2
0
0
CaC
2
O
4
2
O
N
2
1.5–3–6–12–24
PVC
PVC
1
2
80–380
5–1100
electrographite
electographite
1
100
+
30% Na
electographite
30% Na O·2SiO
electographite
30% Na O·3.5SiO
bituminous coal
2
O·SiO
2
air
6–12–24
+
2
2
+
2
2
15
N
2
5–10–15–20–50
J. Therm. Anal. Cal., 84, 2006
457

BIGDA, MIANOWSKI
Table 4 Characteristics of coal, electrographite and sodium silicate samples
–3
Sample
Molar ratio(SiO
1
2
/Na
2
O)
Bulk density/g cm
1.02
Water insoluble/%
0.3
2 3
Fe O /%
sodium metasilicate
0.02
0.05
0.08
Na
2
O·SiO
sodium disilicate
Na O·2SiO
sodium silicate
O·3.5SiO
2
2
0.97
–
0.3
0.2
2
2
3
.5
Na
2
2
d
daf
Total moisture, W
t
/ %
Ash, A /%
Volatile matter, V /%
Swelling Index, SI
electrographite
0.2
8.2
0.05
6.8
0.71
21.3
–
bituminous coal*
6.5
*
Bituminous coal from Polish coal mine. Type 35.2B according to Polish Standard Classification of Coal
Values of kinetic parameters (E and A) have
been determined with the use of Coats-Redfern rela-
tionship (Table 1, No. 1 and Table 5).
‘three-model’ cases (where 3 models have been se-
lected) – Figs 3a, d, e (exception: Fig. 3c) and Fig. 4c,
c) values of k are greater than (k ) : k >(k ) –
m
m 1
m
m 1
Analyzing Fig. 2 one may notice that not every
kinetic model fulfills said condition for each of ana-
lyzed heating rates, except for models A3 and D4. Us-
Fig. 3f and Figs 4b, e,
d) one secluded case for thermal dissociation,
where k <<(k ) – Fig.3 b;
m
m 2
ing both criterion g(a )=1 [16, 17] and statistical one
m
2
(
r ), the preferred model is R2, which still does not
B. We have 2 cases of kiso value behavior towards
(k ) and (k ) :
fulfill condition (19). Values of k increase together
with heating rate, but slower than (k ) , what leads to
fulfillment of condition (19) for greater and greater
m
m 1
m 2
m 1
a) for processes of thermal dissociation (Figs 3c, e, f),
pyrolysis (Figs 3g, h) and combustion (Fig. 4 – all
cases) kiso satisfies condition (k ) ³k ³(k )
–
1
amount of models (for q=1.5 K min the condition
19) is fulfilled only by model A3, whereas for
m 1
iso
m 2
(
–
1
b) for thermal dissociation processes, for which 3
q=24 K min – models A2, A3, D1, D3, D4 and R1).
Such a behavior of constant k is not met in every
case. However, using only models, which fulfill both
criteria, statistical one and g(a )=1, causes that de-
pendencies k vs. q for selected models arrange them-
selves in various way towards straight-lines (12) and
(
models have been selected, then: k >(k ) (Figs 3a, d).
iso
m 1
m
Such different behavior of k and k towards
m
iso
rectilinear relationships (12) and (14) follows from
mutual relations between kinetic parameters deter-
mined from Kissinger law (11) and (13) and IE (6) (or
more specifically: the point of intersection of
straight-lines lnA vs. E for individual heating rates
m
m
14), but according to some rule, which we are going
to discuss about further.
(
Fig. 4). For all analyzed cases, points P
(lnA ,E ) –
I I I
Analysis of all cases of thermal decomposition
discussed in present work is reflected on Fig. 3, where
distribution of reaction rate constants vs. heating rate
according to Eqs (2), (12) and (14) are presented for
Eq. (11) and P (lnA ,E ) – Eq. (13) are located
II
II II
around the intersection point, P. Comparing depend-
encies k vs. q (Figs 3a-h and Figs 4a-e) with corre-
m
sponding IE relationships (Figs 3i–p and Figs 4f–j),
one may conclude:
models satisfying both criteria: g(a–
m
)=1 and statisti-
1
cal one, for q=6, 12 and 24 K min . One may draw
following conclusions from Fig. 3:
case A-a corresponds to the following situations (EP
–
value of activation energy in point P):
A. We have 4 cases of k value behavior towards
m
(
k ) and (k ) :
m 1 m 2
• E >E , E – for inert atmosphere processes: thermal
p
I
II
dissociation (Fig. 3k) and pyrolysis (Figs 3o, p),
a) condition (19) is satisfied for all selected models –
thermal dissociation (Fig. 3c), pyrolysis (Figs 3g, h)
and combustion (Figs a, d); as we can see, in case of
pyrolysis we have only one model satisfying criterion
•
E <E , E – for combustion – Figs 4f, i,
p
I
II
case A-b (area 1 on Fig. 5)
E »E , E – thermal dissociation (Figs 3i, l, m) and
g(a
)=1 (Figs 3g, h), while dissociation of
m
•
p
I
II
CuSO ·H O (Fig. 3c) – three ones,
4
2
combustion (Fig. 4h),
b) values of k oscillate around (k ) , reaching
m
m 1
greater or lower g(a )=1 values: k »(k ) – for
m
m
m 1
458
J. Therm. Anal. Cal., 84, 2006

INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES
Table 5 Comparison between relationships by means of which determined kinetic parameters are dependent or independent on
heating rate – examples from literature
Equation/method
Kissinger law:
Obtained kinetic parameters
Literature
q
AR
E
[
16, 17, 22, 25]
ln = ln E
–
I I
, E = E , A = A
2
Tm
RTm
E, A=const. for q=var
I I
E ¹ EII, A ¹ AII
Kissinger law in modified form:
q
E
[
16, 17]
ln = ln A–
, E = EII, A = AII
Tm
RTm
Kissinger-Akahira-Sunose (KAS) method:
q
2
Ta
AR
E
[
29]
ln = ln
–
a a
, E = E , A = A
Eg(a) RTa
Isoconversional methods
Flynn-Wall-Ozawa (FWO) method in version:
AE
E
E, A = const. for q = var and a = const.
kinetic parameters do not depend on
heating rate, but depend on a; E varies
with a;
A is determined e.g. by the use of IE or
Kinetic Compensation Effect (KCE) [35]
lnq= ln
–5.331–1.052
Rg(a)
RTa
[29, 37]
E
for 20 <
< 60,
RT
, A = A
a
E = E
a
Friedman’s method:
æ da ö
E
[
29, 36, 37]
lnçq ÷= ln Af (a)–
a a
, E = E , A = A
è dT ø
RT
Coads-Redfern method (comp. Table 1, No.1 and Table 2,
No. 3):
[
16, 17]
g(a)
AR
= ln qE
E
[
29, 38–40]
ln
–
2
T
RT
Model-fitting methods
E, A = var for q = var and g(a) = var
‘
General method’ (comp. Table 1, No.4):
g(a)
A
E
[
16, 17]
ln
= ln q
–
T
RT
Kinetic parameters obtained by
model-fitting methods are intercorrelated,
forming rectilinear relationships for each
heating rate. The relationships intersect at
(nearly) one point that corresponds to true
values of A and E – the so-called invariant
Kinetic compensation effect (KCE)/isokinetic effect
IE)/invariant kinetic parameters method (IKP):
lnA=a+bE, q = var
(
[16, 17, 28, 29]
kinetic parameters (Einv, Ainv or E
which are independent on heating rate:
, A = const. for q = var
P P
, A ),
E
inv, Ainv or E
P
P
case A-c
Analysis of location of the points, which coordi-
nates are values of kinetic parameters E and A deter-
mined from Eqs (11) and (13), towards IE allows cap-
turing significant differences in kinetics of processes
taking place in inert (pyrolysis and thermal dissocia-
tion) and oxidizing (combustion) atmosphere. While
•
•
E <E , E – thermal dissociation (Fig. 3f)
p
I
II
E >E , E – combustion (Figs 4g, j),
p
I
II
case A-d (area 2 on Fig. 5):
P »P , and E »0, Fig. 3j,
k (Eq. (2)) approaches (k ) , then for oxygen-free
•
m m 2
I
II
P
processes we may notice decreasing of value of acti-
vation energy (E ) and its approaching towards values
P
case B-a:
E , E ¹E – Figs 3k, n-p (exception: Fig. 3m) and
of E (Eq.(11)) or E (Eq. (13)), while for combustion
I
II
•
– increasing of E is observed. This tendency is also
I
II
p
P
Figs 4f, g, i, j (exception: Fig. 4c),
valid in opposite situation. Decrease of k value be-
m
low (k ) causes further decreasing of E for thermal
m 2
P
dissociation and pyrolysis and increasing for combus-
case B-b:
E , E »E – Figs 3i, l.
tion. On the other hand, when k reaches (k ) (for
m
m 1
•
I
II
p
‘
three-model’ cases – Figs 3a, d, e and Fig. 4c), then
E =E or E . Reaching minimal values of k (Fig. 3b)
P
I
II
m
J. Therm. Anal. Cal., 84, 2006
459

BIGDA, MIANOWSKI
Fig. 3 Reactions of thermal dissociation analyzed in present work. a–h: Relationship of rate constant k and kiso on heating rate on
m
the background of straight-lines (12) and (14). Key the same as on Fig. 2
finds its reflection in minimal values of E (Fig. 3j),
parameters determined from two forms of Kissinger
law (11) and (13), which result from some kind aver-
aging of kinetic parameters for q=var. The points PI,
PII and P depend only on values of kinetic parameters
P
however this conclusion is based only on one case of
thermal dissociation. Basing on said analyses, one
may suppose that for combustion we can expect oppo-
site situation, i.e. maximal values. Figure 5 recapitu-
lates above discussion.
E and A, without influence of temperature T
cause T increases together with q), which is included
in reaction rate constant equations. The fact manifests
itself in similar behavior of k vs. q relationship (e.g.
(be-
m
m
The reasons of such location of pairs of kinetic
parameters forming coordinates of points P
I
, PII and P
m
follow from the following fact: straight-lines of IE for
all heating rates intersect in (nearly) one point P, in
which kinetic parameters are the same for each heat-
ing rate, thus they are independent on q. The fact was
also mentioned, among others, by Lesnikovich and
Levchik [28] as well as by Budrugeac and Segal [29].
Independence on q is also characteristic for kinetic
Fig. 3h and Fig. 4a), while in lnA vs. E system we can
see opposite behavior (e.g. Fig. 3p and Fig. 4f). As we
can see the obtained kinetic parameters depend or are
independent on heating rate. There are some exam-
ples of non-isothermal kinetic methods in Table 5,
which represent dependence or independence of eval-
uated kinetic parameters on heating rate.
460
J. Therm. Anal. Cal., 84, 2006

INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES
Fig. 3 cont. Reactions of thermal dissociation analyzed in present work. i–p: Location of point P [+] of intersection of IE (6)
straight-lines for different heating rates towards points P (m) and PII (l)
I
Therefore, scattering of km values determined
with the use of Eq. (2) depending on heating rate, fol-
lows from the fact of unfulfillment of assumption of
kinetic parameters (E and A) invariability towards
heating rate. On the other hand, obtained values of E
and A for different thermokinetic models are charac-
terized by their large scattering, oscillating around av-
erage and, in spite of the fact that they satisfy IE, the
latter is blurred (as it was presented previously
¶lnA
1
=
,
T
m
= f(q)
(20)
¶
E
RTm
We can not precisely say on the basis of above
analyses, at what heating rate thermogravimetric mea-
sures of thermal decomposition of solids should be
carried out, because precision in heating rate values
partition does not exist. Instead, we may distinguish
two ranges of heating rate values, for which the
partition can be made:
[
18, 26]). Heating rate variability influences changes
–
1
•
q<10 K min , where behavior of kinetic quantities
E, A, k ) is chaotic and unpredictable (Figs 1b
of slope of lnA vs. E rectilinear relationship in very
subtle way or rather imperceptible in categories of
statistical analyses, what is undoubtedly connected
with relationship [16]:
(
m
and 2),
–
1
•
q>10 K min , where relationship between q and E,
A, k is either rectilinear or invariable towards q ac-
m
cording to condition (11) or (13).
J. Therm. Anal. Cal., 84, 2006
461

BIGDA, MIANOWSKI
Fig. 4 Reactions of combustion analyzed in present work. a–e: Relationship of rate constant k and kiso on heating rate on the
m
background of straight-lines (12) and (14). Key the same as on Fig. 2
Fig. 4 cont. Reactions of combustion analyzed in present work. f–j: Location of point P [+] of intersection of IE (6) straight-lines
for different heating rates towards points P (m)and PII (l)
I
462
J. Therm. Anal. Cal., 84, 2006

INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES
of k towards q (Fig. 2). Therefore, without using sta-
m
tistical criterion and g(a )=1 one can not say, which
m
model is appropriate for analyses of thermal pro-
cesses. Analysis of k vs. q with the use of both men-
m
tioned criteria and Kissinger law ((k ) and (k ) ) al-
m 1
m 2
low to show differences between kinetics of inert and
oxidizing atmosphere processes (Figs 3a-h and
Figs 4a-e). The differences are consistent with IE for
different heating rates towards Kissinger law
(
Figs 3i-p and Figs 4f-j). As well kinetic rate constant
as isokinetic behavior are interdependent, what is pre-
sented on Fig. 5.
Analysis of quantities other than k , i.e. E, A,
m
+
+
D H, D S towards heating rate is an incomplete method
and does not lead to unambiguous conclusions, which
make possible answering the question, at what heating
rate thermokinetic analysis is most justified.
Fig. 5 Simplified scheme recapitulating relations between be-
havior of k towards (k and (k and E towards E
m
m
)
1
m
)
2
P
I
Values of k according to Eqs (2) and (4) (iden-
m
and EII. Comment in the text. (C – combustion, I – inert
atmosphere processes, TD – thermal dissociation)
tity) should be located within range defined by recti-
linear relationships (12) – maximum, and (14) – mini-
mum, for thermal processes carried out in dynamic
conditions (q=var), assuming that E and A are invari-
able towards q and satisfy Kissinger law in two ver-
sions: (11) and (13). However, there might be some
deviation caused by:
•
Reaction rate constant kiso in Eq. (6) is invariant in
sense of statistical defining of free term in equation
reduced to linear form. But in practice, using both
Eq. (2) and assumption (7), pairs of parameters E
and A for particular models g(a) lead to variable
numerical values, which oscillate around average
(
Fig. 2). IE is very often blurred (comp. [18, 30]).
•
Invariability of E and A towards q is not conserved.
It was concluded that thermogravimetric measure-
ments carried out at different heating rates is justified
from point of view of kinetic analysis of thermal decom-
Fig. 6 Course of integrated function in Euler’s integral
–1
position. At low heating rates (below 10 K min ) quan-
tities developed from thermokinetic calculations behave
somewhat different from those obtained at higher heat-
–1
ing rates (over 10 K min ). In result, we get two sets of
information about process kinetics. In case of
isoconversional methods at least three heating rates are
enough in order to present the full kinetic picture of the
process [31]. At the same time, using classic model-fit-
ting kinetics methods two thermogravimetric measure-
ments at low and high heating rate are the minimum,
which are necessary for this purpose.
Conclusions
In thermal decomposition of solids Eyring equation is
an element connecting used thermokinetic equations
g(a) for two most justified versions (No. 1 and 4 in
Table 1), which contains Arrhenius law. In the same
time Eyring law points to validity of consideration of
thermokinetic quantities (in dynamic conditions) in
way relative for 3 kinetic rate constants (k
km)2) according to Eq. (2) and Eqs (12) and (14).
Kinetic rate constants k , (k ) and (k ) result
m m 1
, (k ) and
(
Appendix A
m
m 1
m 2
from different laws (Arrhenius’ and both
Kissinger’s), therefore we may expect their different
arrangement towards q. Using values of kinetic pa-
rameters (resulted from different thermokinetic mod-
els) in Arrhenius equation (2) gave chaotic behavior
In Eq. (16) using g(a
m
) = 1 for T = T
m
we obtain:
2
BTm
+
D G =RT ln
(A1)
463
m
q
J. Therm. Anal. Cal., 84, 2006

BIGDA, MIANOWSKI
¥
– x
– x
¥
– x
From Kissinger law in version (11) we have:
dx
e
e
1 e
–
e
x
=
–
+
dx
(B6)
ò
3
2
ò
2
Tm
x
2x
2x
2
x
E
x
x
E/ RTm
e
=
(A2)
q
AR
Next, we may treat the Euler’s integral in following way
Fig. 6):
(
and
¥
Ei(–x)=ò
x
– x
e
æ
E
ö
dx@0
(B7)
(B8)
+
D G =E+ RT çln B+ ln
÷
(A3)
(A4)
(A5)
m
x
è
ARø
or [33, 34]
Ei(–x) =
Using modified version (13) we obtain:
¥
– x
– x
E/ RTm
e
e
e
1! 2! 3!
æ
ö
Tm
dx =
ç1– + x
+
+...÷ =
=
ò
2
3
x
x è
x
x
ø
q
A
x
–
x
e
æ
1ö
and
=
ç1– ÷
x è xø
æ
Tm
ö
A ø
+
D G =E+ RT çln B+ ln
÷
or
m
è
–
x
e
According to Table 2 relationship (A3) is approximation of
Ei(–x)@ x
(B9)
(A5). For obvious relation E>RT
m
one may obtain inequality:
+
+
+
Finally, the Euler’s integral may be approximated in
three ways
D G(A1)@D G(A3)>D G(A5)
(A6)
+
From relationship (A1) follows that D G depends on in-
creasing values of q and T
A=idem, then D G depends on A and T in following way:
–
x
– x
e
æ
1ö
e
m
. According to (A5), while E and
Ei(–x)@0, Ei(–x)@ ç1– ÷, and Ei(–x)@
x è xø
(B10)
+
x
m
æ
B
ö
m
Aø
Conclusively, (B5) is presented as:
+
D G =E+ çRln ÷T + RT lnT
(A7)
m
m
è
+
2
–x
B
D+S/ R
æ D H ö e æ 1
ö
g(a)= q
e
ç
ç
è
÷
÷
ø
ç
–1÷£0 (x³1) (B11)
However, to all intents and purposes, even for relation
B<A the second term is much lower than the third one, thus
R
2x è x
ø
+
2
D G increases with increasing T
m
, what also has been proved
+
B
D+S/ R
æ D H ö
+
g(a)= q e
ç
ç
è
÷ ×0=0
(B12)
(B13)
using correlation analysis [12]. Variability of D G vs. q is in-
÷
R
ø
significant.
2
+
– x
B
D+S/ R
æ D H ö e
g(a)= q
e
ç
ç
è
÷
÷
ø
2
R
2x
Appendix B
and after all
Let’s introduce formula (3) into equation:
2
BT
D+S/ R – D H/ RT
+
da
dT
k
g(a)=
e
e
(B14)
=
f (a)
(B1)
(B2)
2q
q
The solution (B14) differ from Eq. (16) in value of coef-
ficient 1/2.
to obtain
da BT
D+G/ RT
–
e
= q
f (a)
dT
Nomenclature
After integration with using Eq. (5) we obtain:
–
1
T
A
a
pre-exponential factor in Arrhenius equation (s ),
intercept of lnA vs. E relationship (Table 5),
conversion degree, 0£a£1,
B
D+S/ R
+
– D H/ RT
g(a)= q
e
Te
dT
(B3)
(B4)
(B5)
ò
0
a
b
Let’s substitute:
slope of lnA vs. E relationship (Table 5),
+
–1
D G
D H
D S
E
free energy of activation (J mol ),
+
+
+
D H
RT
D H
D H dx
+
–1
=
x, T = Rx , dT =-
enthalpy of activation (J mol ),
2
R
x
+
–1
–1
entropy of activation (J mol K ),
–
1
–1
thus
activation energy (kJ mol or J mol ),
mass integral,
g(a)
h
2
+
¥
–
34
B
D+S/ R
æ D H ö
dx
– x
Planck’s constant, h = 6.626169·10 (J s),
g(a)= q
e
ç
ç
è
R
÷
÷
ø
e
x
ò
3
IE
k
isokinetic effect,
x
–
1
reaction rate constant (s ),
Boltzmann’s constant, k = 1.380622·10 (J K ),
We solve the integral in (B5) using fragmentary integra-
tion [32]
–23
–1
k
B
B
464
J. Therm. Anal. Cal., 84, 2006

INFLUENCE OF HEATING RATE ON KINETIC QUANTITIES
m
q
sample mass (mg),
heating rate (K s or K min ),
gas constant, R = 8.314 J mol K ,
15 F. Tudos and P. K. David, Polym. Degrad. Stab.,
50 (1995) 159.
–
1
–1
–
1
–1
16 A. Mianowski, J. Therm. Anal. Cal., 74 (2003) 953.
R
2
17 A. Mianowski and R. Bigda, J. Therm. Anal. Cal.,
75 (2004) 355.
r
determination coefficient for linear
2
regression; 0 £ r £ 1,
1
8 A. Mianowski and T. Radko, Pol. J. Appl. Chem.,
8 (1994) 395.
T
temperature (K),
time (s).
3
t
19 L. T. Vlaev, I. G. Markovska and L. A. Lyubchev,
Thermochim. Acta, 406 (2003) 1.
Subscripts:
20 J. Paj·k and ˜. Socha, Fuel Process. Technol.,
7–78 (2002) 131.
1 J. Paj·k and ˜. Socha, Energy Fuels, 19 (2005) 348.
7
1
, 2
m
denotes k determined from relationship (11)
2
or (13), respectively,
22 W. W. Wentlandt, Anal. Chim. Acta, 27 (1962) 315.
23 A. Mianowski and R. Bigda, J. Therm. Anal. Cal.,
74 (2003) 423.
a
denotes relationship on conversion degree in
isoconversional methods (Table 5),
denotes E and A for Kissinger law in version (11),
denotes E and A for Kissinger law in version (13),
isokinetic point,
denotes invariant kinetic parameter (Table 5),
maximum rate of the process,
denotes invariant kinetic parameter.
I
II
iso
inv
m
P
24 A. Mianowski and R. Bigda, J. Therm. Anal. Cal.,
7
4 (2003) 433.
5 S. Vyazovkin and C. A. Wight, Thermochim. Acta,
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6 X. Gao, D. Chen and D. Dollimore, Thermochim. Acta,
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8 A. I. Lesnikovich and S. V. Levchik, J. Thermal Anal.,
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9 P. Budrugeac and E. Segal, Int. J. Chem. Kinet., 33 (2001)
64.
2
2
2
2
2
3
3
2
2
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