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Kinetic analysis of thermogravimetric data. Comparison of integral discrimination methods
z%ermochimicu Actu, 146 (1989) 75-85 Elsevier Science Publishers B.V., Amsterdam
75 - Printed
in The Netherlands
KINETIC ANALYSIS OF THERMOGRAVIMETRIC DATA. COMPARISON OF INTEGRAL DISCRIMINATION METHODS
A. ROMERO
SALVADOR
Dpto. Ingenieria Quimica, Facultad de Ciencias Quimicas, Universidad Complutense, 28040 Madrid (Spain) E. GARCIA
CALVO
* and P. LETON
GARCIA
Ingenieria Quimica, Facultad de Ciencias, Universidad de Alcalrj, 28871 Alcaki de Henares (Spain) (Received
10 August
1988)
ABSTRACT The Coats-Redfern and the Romero et al. integral discrimination methods are compared through kinetic analysis of thermogravimetric data. Both are used to determine the kinetic model an obtain kinetic parameters from a - T data generated from a wide range of kinetic parameters and nine different kinetic models. The Romero et al. method allows determination of the kinetic model in all cases. The Coats-Redfern method does not allow discrimination between most of the models. When the kinetic model is known, both methods calculate kinetic parameters with accuracy.
INTRODUCTION
The practical importance of thermal decomposition kinetics in solids has led to a very extensive effort to understand the mechanisms which control these reactions. Thermogravimetric (TG) analysis is the experimental method most commonly used for determining the model which best describes the decomposition reaction. Several techniques [l-6] are used to analyse weight loss vs. temperature data obtained using TG. The most common of these techniques is the one developed by Coats and Redfern [l]. In this paper, conversion vs. temperature data are generated using different models and kinetic parameters at different heating rates. These data are
* Author
to whom all correspondence
0040-6031/89/$03.50
should be addressed.
0 1989 Elsevier Science Publishers
B.V.
76
used to compare the Coats-Redfern technique with the one proposed by Romero et al. [6] so that the advantages of each can be established. Coats-Redfern
method
The mathematical description of solid weight loss during heating can be represented with an Arrhenius-type reaction equation g
=A exp( -E/RT)f(a)
If the reaction is carried out at a constant heating rate j3, defined as T= T,+pt
(2)
we can use (Y and T to remove the time dependence in eqn. (1) da A do = p exp( - E/RT)f(
a)
and then integrate the equation
a da
J0 -f(a)
= g(a) = $/rrexp(
-E/RT)
We can then apply the (1 - 2RT/E) = 1, to obtain ln
go=lnAR_x,8E RT
dT
Coats-Redfern
approximation,
supposing
(5)
T2
A linear regression of this expression is required to obtain kinetic parameters. The A and E values can be estimated from the values of the slope and the intercept. Romero et al. method By using several heating rates, it is possible to split the temperature and conversion influences in a two-part analysis. The first is at constant temperature and the second at constant conversion [6]. Points that correspond to the same temperature for a set of a-T experimental curves at different heating rates are used to study the conversion influence. Each of these points corresponds to a time t obtained from eqn. (2). The a-t data have to fulfil the condition g(a) =
K*it
(6)
where &=
‘A exp( - E/RT) / To T- TO
dT (7)
77
Using the Coats-Redfem simplification to integrate eqn. (7), we obtain the relationship between the kinetic constant K and the slope of eqn. (6) K, K=
L(T-
TM
RT
(8)
(l-F)
Assuming (1 - 2RT/E) = 1, K, values can be used to obtain by taking into account the Arrhenius equation
A and E
(9)
ln$=ln$-&
The temperature influence is obtained by analysis at constant conversion, i.e. using temperature values for a given conversion in a set of a--T experimental curves. From eqn. (5) we obtain In-=
P RT2
In -A’E id4
It is possible constant (Y.
GENERATION
-- E RT
to obtain
OF a-T
00) values for the kinetic
parameters
for any given
DATA
In order to compare the Coats-Redfern and the Romero et al. methods, sets of a--T data were generated at five different heating rates (0.5, 1, 2, 5 and 10 K mm-‘). Simpson’s rule (AT = 0.1-0.25 K) was used to integrate the right side of eqn. (4) and to obtain the a-T data. With E = 30 kcal mol-’ and A = lo6 mm-‘, nine sets of experimental a-T data were obtained, corresponding to each of the models in Table 1. From these data it was possible to determine the influence of the conversion on the method of discrimination. In order to determine the influence of the kinetic parameters on the method of dicrimination, three sets of generated a-T data with different kinetic parameters were used with the R3 model. The initial kinetic parameters for each set were E = 10 kcal mol-’ and A = 100 mm-‘; E = 30 kcal mol-’ and A = lo6 mm’; and E = 70 kcal mol-’ and A = lo7 min-‘.
ANALYSIS
OF THE a-T
DATA
Analysis of the kinetic data by the Coats-Redfem method requires only one of the a-T curves from each set. In this paper the curves used were for /?=lKmin-‘, but curves for different heating rates yield similar results. Applying the Romero et al. method requires the set of five a-T curves.
Phase boundary reaction (a) One-dimensional (zero order) (b) Two-dimensional (cylindrical geometry) (c) Three-dimensional (spherical geometry)
(c) Three-dimensional diffusion (spherical geometry)
Rl R2 R3
Dl D2 D3
Fl F2 F3
Nucleation and nuclei growth (a) Random nucleation (b) Two-dimensional nuclei growth (c) Three-dimensional nuclei growth
Diffusion (a) One-dimensional transport (b) Two-dimensional transport (cylindrical geometry)
Symbol
Rate mechanism
Conversion functions of different models
TABLE 1
Constant (1 - a)1’2 (l- a)2’3
[(l - a)-l’3 -11
2[1- (1 - a)l12] 3[1 -(la)“3]
a
+[1-2a/3-(1-a)3’2]
(l-a)ln(l-a)+a
-ln(la) - [ln(l - a)]“2 - [ln(l - a)]1’3 a2/2
a)]-’
ln(l - a)]‘12 ln(l - a)] 2’3
[-ln(l-
a)[ -
a)[ -
g(a)
a-l
l-a 2(1 3(1-
f(o)
2
270
125
100
155
173
180
263
276
282
Fl
F2
F3
RI
R2
R3
Dl
D2
D3
A=106min-’
Number Of a-T points
E=30kcalmoI-'
True model
A (min-‘) E (kcal mol-‘) r A (min-‘) E (kcal mol _ ’ ) r A f&n-‘) E (kcal mot _ ’ ) r A (min-‘) E (kcal mol-‘) r A (min-‘) E {kcai mot - ’ ) r A @in-‘) E (kcal mol-‘) r A @in-‘) E (kcal mol-‘) r * A (min-‘) E (kcal moi - ’ ) r A (min-‘) E (kcal mol-‘)
8.2 ES 29.8 0.999 8.1 El4 62.7 0.999 7.2 E21 88.2 0.999 3.1 E7 34.7 0.992 2.8 E7 33.6 0.995 2.4 E7 32.8 0.997 1.8 E2 16.1 0.991 1.9 E2 15.8 0.993 2.0 E2 15.6 0.994
Fl 16.1 13.3 0.999 7.9 E5 29.8 l.ooO 2.8 E9 42.5 0.999 1.2 E2 15.9 0.992 1.1 E2 15.3 0.994 1.0 E2 15.0 0.997 0.19 6.7 0.988 0.19 6.5 0.991 0.20 6.5 0.993
F2
Probed model
Coats and Redfem method: discrimination between models
TABLE
0.33 7.8 1.000 6.1 E2 18.8 0.999 1.6 ES 27.3 0.999 1.4 9.6 0.991 1.37 9.3 0993 1.3 9.0 0.996 1.3E-2 3.6 0.983 1.4E-2 3.5 0.986 1.4E-2 3.4 0.991
F3
1.000 1.9 E5 27.0 0.997 8.9 E4 25.6 0.993 19.4 13.6 1.000 13.5 12.8 0.999 11.6 12.4 0.997
3.4 E? 19.2 0.956 2.6 El0 41.3 0.982 2.4 El4 62.0 0.976 8.3 ES 29.9
Rl 2.8 E3 22.9 0.982 9.1 El1 53.7 0.994 7.3 El6 71.7 0.991 2.2 E6 32.1 0.998 8.4 ES 29.9 1.000 4.8 ES 28.6 0.999 27.2 14.8 0.998 22.2 14.1 0.999 20.6 13.8 0.999
R2 6.9 E3 24.8 0.990 3.8 El2 56.3 0.997 9.1 El? 76.1 0.995 2.7 E6 32.9 0.997 1.3 E6 31.0 0.999 8.3 E5 29.9 1.000 26.5 15.2 0.996 23.3 14.6 0.998 22.4 14.4 0.999
R3 2.4 E8 41.5 0.963 5.8 E23 97.8 0.983 4.4 E31 127.3 0.978 7.8 El4 62.7 0.999 4.6 El3 57.0 0.997 1.0 El3 54.0 0.994 8.5 E5 29.9 1.000 4.3 ES 28.3 0.999 3.2 ES 27.5 0.998
Dl 2.2 E9 45.6 0.975 3.7 E2S 105.1 0.990 2.2 E34 137.9 0.986 2.9 El5 65.4 0.999 2.8 El4 60.4 0.999 7.7 El3 57.6 0.998 1.4 E6 31.3 0.999 8.5 ES 29.9 1.000 6.8 E5 29.2 0.999
D2
6.9 EQ 47.7 0.981 2.5 E26 108.6 0.993 4.6 E35 143.3 0.989 4.4. El5 66.6 0.999 5.6 El4 61.8 0.999 1.7 El4 59.2 0.999 1.6 E6 31.9 0.999 1.0 E6 30.6 0.999 8.7 E5 29.9 1.000
D3
100 10 (181) a
106 30 (180) a
10’ 60 (115) a
A (rnin-‘) E (kcal mol-‘) r
A (min-‘) E (kcal mol-‘) r
A (mm-‘) E (kcal mol-‘) r
number
3.9 E8 66.2 0.996
2.4 E7 32.8 0.997
8.2 E2 10.8 0.997
Fl
Model
of a-T
3.3. E2 30.5 0.995 points.
2.5 18.6 0.995
1.3 9.0 0.996 5.6 E5 51.0 0.993
8.9 E4 25.6 0.993
29.1 8.4 0.993
Rl
4.2 E6 57.2 0.999
4.8 E5 28.6 0.999
62.8 9.5 0.999
R2
using the various models
3.2 E-2 2.4 0.997
F3
calculated
1.0 E2 15.0 0.997
0.52 4.5 0.997
F2
kinetic parameters
represent
True values
Kinetic parameters
a Values in parentheses
method:
Coats and Redfern
TABLE 3
R3
8.7 E6 59.9 1.000
8.3 E5 29.9 1.000
74.4 9.9 1 .ooo
6.5 El4 107.1 0.994
1.0 El3 54.0 0.994
1.2 E6 18.6 0.995
Dl
6.9 El5 114.3 0.998
7.7 El3 57.6 0.998
3.4 E6 19.8 0.998
D2
1.9 El6 117.6 0.999
1.7 El4 59.2 0.999
4.6 E6 20.4 0.999
D3
81
Coats-Redfern
method
Table 2 shows the calculated kinetic parameters and the correlation coefficients obtained when each of the nine kinetic models is used to fit the a-T data generated from each of the nine kinetic models. The kinetic parameters obtained when the probed model is the best fitting one are very close to the initial values. In practice, however, the initial values and/or kinetic model are obviously unknown. The only way to discriminate between models is by the value of the correlation coefficient, but this is so high in all cases that choosing between models is impossible. Table 3 shows the kinetic parameters obtained for each of the nine models used to fit the a-T data generated with the R3 model and three different combinations of kinetic parameters. Again, discrimination between the models is difficult. Romero et al. method From an analysis at constant temperature, a--T data corresponding to a given set were fitted using eqn. (6). Table 4 shows the results of fitting data generated with E = 30 kcal mol-‘, A = lo6 rnin-’ and g(a) = 1 - (1 - (Y)*‘~. In all of these cases, discrimination between models is achieved by comparing the correlation coefficients. The kinetic parameters are obtained from the slopes generated with each model and using eqns. (8) and (9) (see Table 5). It is clear that the kinetic parameters that are most like the initial ones are the parameters obtained when the R3 model is used. However the value of the correlation coefficient is insufficient for discrimination between the nine models. An analysis at constant conversion is necessary to obtain the kinetic parameters and compare these with those obtained in the analysis at constant temperature. Table 6 shows the E and A/g( cx) values obtained when eqn. (10) is applied to fit data generated at five different constant conversions. The correlation coefficient is 1.000 in all cases. The E values are very close to those obtained in the analysis at constant temperature with g( CX)corresponding to the R3 model. Table 7 shows the A values at any given constant (Y and considering the nine possible models. All the A values are around lo6 and the influence of (Y is not pronounced. However, A decreases for the F, Rl and R2 models and increases for the D models with increasing (Y. The A values for R3 are similar to those obtained at constant temperature for any conversion level. The Rl and R2 models also generate A values which are close to the initial values but very different from those obtained in analysis at constant temperature (see Table 5).
Slope Intercept Coefficient
Slope Intercept Coefficient
Slope Intercept Coefficient
Slope Intercept Coefficient
Slope Intercept Coefficient
710
740
770
800
830
T (W
3.12 E-3 - 0.127 0.996
1.04 E-3 -4.22 E-2 0.998
4.10 E-4 - 6.33 E- 3 1.000
1.66 E-4 -9.24 E-4 1.000
6.48 E-5 -2.16 E-5 1.000
Fl
Model
1.62 E-3 0.319 0.997
7.07 E-4 0.250 0.991
4.46 E-4 0.171 0.987
2.88 E-4 0.111 0.985
1.84 E-4 6.93 E-2 0.984
F2
1.12 E-3 0.507 0.991
5.40 E-4 0.420 0.979
3.97 E-4 0.323 0.973
3.00 E-4 0.241 0.971
2.25 E-4 0.175 0.970
F3
1.11 E-3 0.113 0.986
5.33 E-4 4.94 E-2 0.992
3.12 E-4 1.04 E-2 0.999
1.49 E-4 1.87 E-3 1.000
6.22 E-5 3.82 E-4 1.000
Rl
8.94 E-4 2.07 E-2 0.999
3.65 E-4 8.02 E-3 0.999
1.78 E-4 1.49 E-3 1.000
7.85 E-5 2.65 E-4 1.000
3.18 E-5 9.18 E-5 1.000
R2
Romero et al. method: analysis at constant temperature, fitting using eqn. (6)
TABLE 4
7.11 E-4 2.08 E-5 1.ooo
2.73 E-4 3.11 E-5 1.000
1.24 E-4 3.12 E-5 1.000
5.33 E-5 2.00 E-5 1.ooo
2.13 E-5 3.87 E-5 1.000
R3
1.19 E- 3 -7.93 E-2 0.999
4.43 E-4 -5.83 E-2 0.993
1.39 E-4 -2.16 E-2 0.981
2.98 E-5 -4.70 E-3 0.976
4.95 E- 6 -7.57 E-4 0.973
Dl
9.81 E-4 -9.04 E-2 0.994
3.19 E-4 -4.96 E-2 0.984
8.20 E-5 -1.34 E-2 0.977
1.59 E-5 -2.56 E-3 0.974
2.54 E-5 -3.91 E-4 0.972
D2
8.07 E-4 -8.18 E-2 0.990
2.45 E-4 -4.02 E-2 0.980
5.79 E-5 -9.60 E-3 0.975
1.09 E-5 -1.76 E-3 0.973
1.71 E-6 -2.63 E-4 0.972
D3
w
83 TABLE 5 Kinetic parameters from analysis at constant temperature, using eqns. (8) and (9) Model
E (kcal mol- ‘)
A (rnin-‘)
r
Fl F2 F3 Rl R2 R3 Dl D2 D3
35.64 19.04 13.53 21.50 27.43 29.87 43.75 49.49 52.29
1.65 1.62 26.65 5.14 2.09 8.33 9.80 3.15 1.57
0.9960 0.9874 0.9707 0.9878 0.9991 1.0000 0.9850 0.9916 0.9943
E8 E3 E3 E5 E5 E9 El1 El2
TABLE 6 Romero et al. method: analysis at constant conversion lx 0.1 0.3 0.5 0.7 0.9
&al mol-‘)
A/g(a) (n-tin-‘)
29.87 29.85 29.86 29.84 29.84
0.24 0.73 0.40 0.24 0.15
E8 El El El El
TABLE 7 Pre-exponential factor from an analysis at constant conversion Model
Fl F2 F3 Rl R2 R3 Dl D2 D3
A X 10e6 (rnin-‘) a = 0.1
a = 0.3
a = 0.5
a = 0.7
a = 0.9
2.53 7.79 11.30 2.40 1.23 0.83 0.24 0.12 0.08
2.6 4.36 5.18 2.20 1.19 0.82 0.66 0.37 0.25
2.77 3.33 3.54 2.00 1.17 0.82 1.00 0.61 0.44
2.89 2.63 2.55 1.68 1.08 0.79 1.17 0.81 0.61
3.45 2.28 1.98 1.35 1.03 0.80 1.21 1.00 0.83
Comparing the E values in Table 6 with those sufficient to indicate the true model. If this were not values would allow discrimination between models. parameters obtained from a-T data generated for
in Table 5 is usually possible, comparing A Table 8 shows kinetic nine different models
84 TABLE 8 Influence of the kinetic model on the calculated E = 30 kcal mol-‘, A = lo6 rnin-‘)
kinetic
Model
A X 10m6 (min-‘)
Fl F2 F3 Rl R2 R3 Dl D2 D3
E (kcal mol- ‘)
parameters
(initial
parameters:
Analysis with T constant
Analysis with a constant
Analysis with T constant
Analysis with a constant
28.67 27.04 27.35 29.58 29.65 29.70 29.83 29.84 29.87
29.87 29.84 29.89 29.87 29.87 29.89 29.87 29.88 29.88
0.38 0.14 0.17 0.69 0.72 0.75 0.81 0.82 0.83
0.82 0.81 0.88 0.82 0.82 0.84 0.83 0.83 0.83
TABLE 9 Influence of the initial kinetic parameters the R3 model)
on the calculated
E (kcal mol-‘)
A (min-‘)
Initial
Initial
10 30 60
Calculated Analysis with T constant
Analysis with a constant
9.80 29.70 59.46
9.85 29.89 59.82
lo2 lo6 10’
parameters
(data generated
with
Calculated Analysis with T constant
Analysis with a constant
0.67 x lo2 0.75 x lo6 0.75 x 10’
0.71 x 102 0.84 x lo6 0.85 x 10’
using E = 30 kcal mol-r and A = lo6 min-‘. In all cases, discrimination between models is easy and, as can be seen in Table 8, the kinetic parameters obtained are very close to the original ones. Table 9 shows the kinetic parameters obtained from a-T data corresponding to the R3 model and three different combinations of initial parameters. Discrimination between models is easy, and the kinetic parameters obtained from both analyses at constant temperature and at constant conversion are very similar. The lower the E and/or A value, the greater the error with respect to the true value. However, the errors are not excessive in any case.
CONCLUSIONS
The Coats-Redfern method is useful for calculating kinetic parameters when the kinetic model is known, but that is not always the case.
85
The Romero et al. method allows discrimination between models in all cases, and the calculated kinetic parameters are very close to the true values.
LIST OF SYMBOLS
A E
f(a) g(a) K R T t
= frequency factor (mu-‘) = activation energy (kcal mol-‘) = conversion influence on the rate of decomposition = /da/f(a) = reaction rate constant (mm’) = gas constant (kcal mol-’ (K-r) = temperature (K) = time (min)
Greek letters
= fractional conversion = heating rate (K mm’) Subscripts
0”’
= observed from non-isothermal = initial
data
REFERENCES 1 2 3 4 5 6
A.W. Coats and J.P. Redfern, Nature, 201 (1964) 68. E.S. Freeman and B. Carroll, J. Phys. Chem., 62 (1958) 394. H.H. Horowitz and G. Metzger, Anal. Chem., 35 (1963) 1464. L. Reich and S.S. StivaIa, Thermochim. Acta, 59 (1982) 247. D.W. van Krevelen, C. van Heerden and F.J. Hutgens, Fuel, 30 (1951) 253. A. Romero Salvador, E. Garcia CaIvo and A. Irabien Gulias, Thermochim. Acta, 73 (1984) 101.
75 - Printed
in The Netherlands
KINETIC ANALYSIS OF THERMOGRAVIMETRIC DATA. COMPARISON OF INTEGRAL DISCRIMINATION METHODS
A. ROMERO
SALVADOR
Dpto. Ingenieria Quimica, Facultad de Ciencias Quimicas, Universidad Complutense, 28040 Madrid (Spain) E. GARCIA
CALVO
* and P. LETON
GARCIA
Ingenieria Quimica, Facultad de Ciencias, Universidad de Alcalrj, 28871 Alcaki de Henares (Spain) (Received
10 August
1988)
ABSTRACT The Coats-Redfern and the Romero et al. integral discrimination methods are compared through kinetic analysis of thermogravimetric data. Both are used to determine the kinetic model an obtain kinetic parameters from a - T data generated from a wide range of kinetic parameters and nine different kinetic models. The Romero et al. method allows determination of the kinetic model in all cases. The Coats-Redfern method does not allow discrimination between most of the models. When the kinetic model is known, both methods calculate kinetic parameters with accuracy.
INTRODUCTION
The practical importance of thermal decomposition kinetics in solids has led to a very extensive effort to understand the mechanisms which control these reactions. Thermogravimetric (TG) analysis is the experimental method most commonly used for determining the model which best describes the decomposition reaction. Several techniques [l-6] are used to analyse weight loss vs. temperature data obtained using TG. The most common of these techniques is the one developed by Coats and Redfern [l]. In this paper, conversion vs. temperature data are generated using different models and kinetic parameters at different heating rates. These data are
* Author
to whom all correspondence
0040-6031/89/$03.50
should be addressed.
0 1989 Elsevier Science Publishers
B.V.
76
used to compare the Coats-Redfern technique with the one proposed by Romero et al. [6] so that the advantages of each can be established. Coats-Redfern
method
The mathematical description of solid weight loss during heating can be represented with an Arrhenius-type reaction equation g
=A exp( -E/RT)f(a)
If the reaction is carried out at a constant heating rate j3, defined as T= T,+pt
(2)
we can use (Y and T to remove the time dependence in eqn. (1) da A do = p exp( - E/RT)f(
a)
and then integrate the equation
a da
J0 -f(a)
= g(a) = $/rrexp(
-E/RT)
We can then apply the (1 - 2RT/E) = 1, to obtain ln
go=lnAR_x,8E RT
dT
Coats-Redfern
approximation,
supposing
(5)
T2
A linear regression of this expression is required to obtain kinetic parameters. The A and E values can be estimated from the values of the slope and the intercept. Romero et al. method By using several heating rates, it is possible to split the temperature and conversion influences in a two-part analysis. The first is at constant temperature and the second at constant conversion [6]. Points that correspond to the same temperature for a set of a-T experimental curves at different heating rates are used to study the conversion influence. Each of these points corresponds to a time t obtained from eqn. (2). The a-t data have to fulfil the condition g(a) =
K*it
(6)
where &=
‘A exp( - E/RT) / To T- TO
dT (7)
77
Using the Coats-Redfem simplification to integrate eqn. (7), we obtain the relationship between the kinetic constant K and the slope of eqn. (6) K, K=
L(T-
TM
RT
(8)
(l-F)
Assuming (1 - 2RT/E) = 1, K, values can be used to obtain by taking into account the Arrhenius equation
A and E
(9)
ln$=ln$-&
The temperature influence is obtained by analysis at constant conversion, i.e. using temperature values for a given conversion in a set of a--T experimental curves. From eqn. (5) we obtain In-=
P RT2
In -A’E id4
It is possible constant (Y.
GENERATION
-- E RT
to obtain
OF a-T
00) values for the kinetic
parameters
for any given
DATA
In order to compare the Coats-Redfern and the Romero et al. methods, sets of a--T data were generated at five different heating rates (0.5, 1, 2, 5 and 10 K mm-‘). Simpson’s rule (AT = 0.1-0.25 K) was used to integrate the right side of eqn. (4) and to obtain the a-T data. With E = 30 kcal mol-’ and A = lo6 mm-‘, nine sets of experimental a-T data were obtained, corresponding to each of the models in Table 1. From these data it was possible to determine the influence of the conversion on the method of discrimination. In order to determine the influence of the kinetic parameters on the method of dicrimination, three sets of generated a-T data with different kinetic parameters were used with the R3 model. The initial kinetic parameters for each set were E = 10 kcal mol-’ and A = 100 mm-‘; E = 30 kcal mol-’ and A = lo6 mm’; and E = 70 kcal mol-’ and A = lo7 min-‘.
ANALYSIS
OF THE a-T
DATA
Analysis of the kinetic data by the Coats-Redfem method requires only one of the a-T curves from each set. In this paper the curves used were for /?=lKmin-‘, but curves for different heating rates yield similar results. Applying the Romero et al. method requires the set of five a-T curves.
Phase boundary reaction (a) One-dimensional (zero order) (b) Two-dimensional (cylindrical geometry) (c) Three-dimensional (spherical geometry)
(c) Three-dimensional diffusion (spherical geometry)
Rl R2 R3
Dl D2 D3
Fl F2 F3
Nucleation and nuclei growth (a) Random nucleation (b) Two-dimensional nuclei growth (c) Three-dimensional nuclei growth
Diffusion (a) One-dimensional transport (b) Two-dimensional transport (cylindrical geometry)
Symbol
Rate mechanism
Conversion functions of different models
TABLE 1
Constant (1 - a)1’2 (l- a)2’3
[(l - a)-l’3 -11
2[1- (1 - a)l12] 3[1 -(la)“3]
a
+[1-2a/3-(1-a)3’2]
(l-a)ln(l-a)+a
-ln(la) - [ln(l - a)]“2 - [ln(l - a)]1’3 a2/2
a)]-’
ln(l - a)]‘12 ln(l - a)] 2’3
[-ln(l-
a)[ -
a)[ -
g(a)
a-l
l-a 2(1 3(1-
f(o)
2
270
125
100
155
173
180
263
276
282
Fl
F2
F3
RI
R2
R3
Dl
D2
D3
A=106min-’
Number Of a-T points
E=30kcalmoI-'
True model
A (min-‘) E (kcal mol-‘) r A (min-‘) E (kcal mol _ ’ ) r A f&n-‘) E (kcal mot _ ’ ) r A (min-‘) E (kcal mol-‘) r A (min-‘) E {kcai mot - ’ ) r A @in-‘) E (kcal mol-‘) r A @in-‘) E (kcal mol-‘) r * A (min-‘) E (kcal moi - ’ ) r A (min-‘) E (kcal mol-‘)
8.2 ES 29.8 0.999 8.1 El4 62.7 0.999 7.2 E21 88.2 0.999 3.1 E7 34.7 0.992 2.8 E7 33.6 0.995 2.4 E7 32.8 0.997 1.8 E2 16.1 0.991 1.9 E2 15.8 0.993 2.0 E2 15.6 0.994
Fl 16.1 13.3 0.999 7.9 E5 29.8 l.ooO 2.8 E9 42.5 0.999 1.2 E2 15.9 0.992 1.1 E2 15.3 0.994 1.0 E2 15.0 0.997 0.19 6.7 0.988 0.19 6.5 0.991 0.20 6.5 0.993
F2
Probed model
Coats and Redfem method: discrimination between models
TABLE
0.33 7.8 1.000 6.1 E2 18.8 0.999 1.6 ES 27.3 0.999 1.4 9.6 0.991 1.37 9.3 0993 1.3 9.0 0.996 1.3E-2 3.6 0.983 1.4E-2 3.5 0.986 1.4E-2 3.4 0.991
F3
1.000 1.9 E5 27.0 0.997 8.9 E4 25.6 0.993 19.4 13.6 1.000 13.5 12.8 0.999 11.6 12.4 0.997
3.4 E? 19.2 0.956 2.6 El0 41.3 0.982 2.4 El4 62.0 0.976 8.3 ES 29.9
Rl 2.8 E3 22.9 0.982 9.1 El1 53.7 0.994 7.3 El6 71.7 0.991 2.2 E6 32.1 0.998 8.4 ES 29.9 1.000 4.8 ES 28.6 0.999 27.2 14.8 0.998 22.2 14.1 0.999 20.6 13.8 0.999
R2 6.9 E3 24.8 0.990 3.8 El2 56.3 0.997 9.1 El? 76.1 0.995 2.7 E6 32.9 0.997 1.3 E6 31.0 0.999 8.3 E5 29.9 1.000 26.5 15.2 0.996 23.3 14.6 0.998 22.4 14.4 0.999
R3 2.4 E8 41.5 0.963 5.8 E23 97.8 0.983 4.4 E31 127.3 0.978 7.8 El4 62.7 0.999 4.6 El3 57.0 0.997 1.0 El3 54.0 0.994 8.5 E5 29.9 1.000 4.3 ES 28.3 0.999 3.2 ES 27.5 0.998
Dl 2.2 E9 45.6 0.975 3.7 E2S 105.1 0.990 2.2 E34 137.9 0.986 2.9 El5 65.4 0.999 2.8 El4 60.4 0.999 7.7 El3 57.6 0.998 1.4 E6 31.3 0.999 8.5 ES 29.9 1.000 6.8 E5 29.2 0.999
D2
6.9 EQ 47.7 0.981 2.5 E26 108.6 0.993 4.6 E35 143.3 0.989 4.4. El5 66.6 0.999 5.6 El4 61.8 0.999 1.7 El4 59.2 0.999 1.6 E6 31.9 0.999 1.0 E6 30.6 0.999 8.7 E5 29.9 1.000
D3
100 10 (181) a
106 30 (180) a
10’ 60 (115) a
A (rnin-‘) E (kcal mol-‘) r
A (min-‘) E (kcal mol-‘) r
A (mm-‘) E (kcal mol-‘) r
number
3.9 E8 66.2 0.996
2.4 E7 32.8 0.997
8.2 E2 10.8 0.997
Fl
Model
of a-T
3.3. E2 30.5 0.995 points.
2.5 18.6 0.995
1.3 9.0 0.996 5.6 E5 51.0 0.993
8.9 E4 25.6 0.993
29.1 8.4 0.993
Rl
4.2 E6 57.2 0.999
4.8 E5 28.6 0.999
62.8 9.5 0.999
R2
using the various models
3.2 E-2 2.4 0.997
F3
calculated
1.0 E2 15.0 0.997
0.52 4.5 0.997
F2
kinetic parameters
represent
True values
Kinetic parameters
a Values in parentheses
method:
Coats and Redfern
TABLE 3
R3
8.7 E6 59.9 1.000
8.3 E5 29.9 1.000
74.4 9.9 1 .ooo
6.5 El4 107.1 0.994
1.0 El3 54.0 0.994
1.2 E6 18.6 0.995
Dl
6.9 El5 114.3 0.998
7.7 El3 57.6 0.998
3.4 E6 19.8 0.998
D2
1.9 El6 117.6 0.999
1.7 El4 59.2 0.999
4.6 E6 20.4 0.999
D3
81
Coats-Redfern
method
Table 2 shows the calculated kinetic parameters and the correlation coefficients obtained when each of the nine kinetic models is used to fit the a-T data generated from each of the nine kinetic models. The kinetic parameters obtained when the probed model is the best fitting one are very close to the initial values. In practice, however, the initial values and/or kinetic model are obviously unknown. The only way to discriminate between models is by the value of the correlation coefficient, but this is so high in all cases that choosing between models is impossible. Table 3 shows the kinetic parameters obtained for each of the nine models used to fit the a-T data generated with the R3 model and three different combinations of kinetic parameters. Again, discrimination between the models is difficult. Romero et al. method From an analysis at constant temperature, a--T data corresponding to a given set were fitted using eqn. (6). Table 4 shows the results of fitting data generated with E = 30 kcal mol-‘, A = lo6 rnin-’ and g(a) = 1 - (1 - (Y)*‘~. In all of these cases, discrimination between models is achieved by comparing the correlation coefficients. The kinetic parameters are obtained from the slopes generated with each model and using eqns. (8) and (9) (see Table 5). It is clear that the kinetic parameters that are most like the initial ones are the parameters obtained when the R3 model is used. However the value of the correlation coefficient is insufficient for discrimination between the nine models. An analysis at constant conversion is necessary to obtain the kinetic parameters and compare these with those obtained in the analysis at constant temperature. Table 6 shows the E and A/g( cx) values obtained when eqn. (10) is applied to fit data generated at five different constant conversions. The correlation coefficient is 1.000 in all cases. The E values are very close to those obtained in the analysis at constant temperature with g( CX)corresponding to the R3 model. Table 7 shows the A values at any given constant (Y and considering the nine possible models. All the A values are around lo6 and the influence of (Y is not pronounced. However, A decreases for the F, Rl and R2 models and increases for the D models with increasing (Y. The A values for R3 are similar to those obtained at constant temperature for any conversion level. The Rl and R2 models also generate A values which are close to the initial values but very different from those obtained in analysis at constant temperature (see Table 5).
Slope Intercept Coefficient
Slope Intercept Coefficient
Slope Intercept Coefficient
Slope Intercept Coefficient
Slope Intercept Coefficient
710
740
770
800
830
T (W
3.12 E-3 - 0.127 0.996
1.04 E-3 -4.22 E-2 0.998
4.10 E-4 - 6.33 E- 3 1.000
1.66 E-4 -9.24 E-4 1.000
6.48 E-5 -2.16 E-5 1.000
Fl
Model
1.62 E-3 0.319 0.997
7.07 E-4 0.250 0.991
4.46 E-4 0.171 0.987
2.88 E-4 0.111 0.985
1.84 E-4 6.93 E-2 0.984
F2
1.12 E-3 0.507 0.991
5.40 E-4 0.420 0.979
3.97 E-4 0.323 0.973
3.00 E-4 0.241 0.971
2.25 E-4 0.175 0.970
F3
1.11 E-3 0.113 0.986
5.33 E-4 4.94 E-2 0.992
3.12 E-4 1.04 E-2 0.999
1.49 E-4 1.87 E-3 1.000
6.22 E-5 3.82 E-4 1.000
Rl
8.94 E-4 2.07 E-2 0.999
3.65 E-4 8.02 E-3 0.999
1.78 E-4 1.49 E-3 1.000
7.85 E-5 2.65 E-4 1.000
3.18 E-5 9.18 E-5 1.000
R2
Romero et al. method: analysis at constant temperature, fitting using eqn. (6)
TABLE 4
7.11 E-4 2.08 E-5 1.ooo
2.73 E-4 3.11 E-5 1.000
1.24 E-4 3.12 E-5 1.000
5.33 E-5 2.00 E-5 1.ooo
2.13 E-5 3.87 E-5 1.000
R3
1.19 E- 3 -7.93 E-2 0.999
4.43 E-4 -5.83 E-2 0.993
1.39 E-4 -2.16 E-2 0.981
2.98 E-5 -4.70 E-3 0.976
4.95 E- 6 -7.57 E-4 0.973
Dl
9.81 E-4 -9.04 E-2 0.994
3.19 E-4 -4.96 E-2 0.984
8.20 E-5 -1.34 E-2 0.977
1.59 E-5 -2.56 E-3 0.974
2.54 E-5 -3.91 E-4 0.972
D2
8.07 E-4 -8.18 E-2 0.990
2.45 E-4 -4.02 E-2 0.980
5.79 E-5 -9.60 E-3 0.975
1.09 E-5 -1.76 E-3 0.973
1.71 E-6 -2.63 E-4 0.972
D3
w
83 TABLE 5 Kinetic parameters from analysis at constant temperature, using eqns. (8) and (9) Model
E (kcal mol- ‘)
A (rnin-‘)
r
Fl F2 F3 Rl R2 R3 Dl D2 D3
35.64 19.04 13.53 21.50 27.43 29.87 43.75 49.49 52.29
1.65 1.62 26.65 5.14 2.09 8.33 9.80 3.15 1.57
0.9960 0.9874 0.9707 0.9878 0.9991 1.0000 0.9850 0.9916 0.9943
E8 E3 E3 E5 E5 E9 El1 El2
TABLE 6 Romero et al. method: analysis at constant conversion lx 0.1 0.3 0.5 0.7 0.9
&al mol-‘)
A/g(a) (n-tin-‘)
29.87 29.85 29.86 29.84 29.84
0.24 0.73 0.40 0.24 0.15
E8 El El El El
TABLE 7 Pre-exponential factor from an analysis at constant conversion Model
Fl F2 F3 Rl R2 R3 Dl D2 D3
A X 10e6 (rnin-‘) a = 0.1
a = 0.3
a = 0.5
a = 0.7
a = 0.9
2.53 7.79 11.30 2.40 1.23 0.83 0.24 0.12 0.08
2.6 4.36 5.18 2.20 1.19 0.82 0.66 0.37 0.25
2.77 3.33 3.54 2.00 1.17 0.82 1.00 0.61 0.44
2.89 2.63 2.55 1.68 1.08 0.79 1.17 0.81 0.61
3.45 2.28 1.98 1.35 1.03 0.80 1.21 1.00 0.83
Comparing the E values in Table 6 with those sufficient to indicate the true model. If this were not values would allow discrimination between models. parameters obtained from a-T data generated for
in Table 5 is usually possible, comparing A Table 8 shows kinetic nine different models
84 TABLE 8 Influence of the kinetic model on the calculated E = 30 kcal mol-‘, A = lo6 rnin-‘)
kinetic
Model
A X 10m6 (min-‘)
Fl F2 F3 Rl R2 R3 Dl D2 D3
E (kcal mol- ‘)
parameters
(initial
parameters:
Analysis with T constant
Analysis with a constant
Analysis with T constant
Analysis with a constant
28.67 27.04 27.35 29.58 29.65 29.70 29.83 29.84 29.87
29.87 29.84 29.89 29.87 29.87 29.89 29.87 29.88 29.88
0.38 0.14 0.17 0.69 0.72 0.75 0.81 0.82 0.83
0.82 0.81 0.88 0.82 0.82 0.84 0.83 0.83 0.83
TABLE 9 Influence of the initial kinetic parameters the R3 model)
on the calculated
E (kcal mol-‘)
A (min-‘)
Initial
Initial
10 30 60
Calculated Analysis with T constant
Analysis with a constant
9.80 29.70 59.46
9.85 29.89 59.82
lo2 lo6 10’
parameters
(data generated
with
Calculated Analysis with T constant
Analysis with a constant
0.67 x lo2 0.75 x lo6 0.75 x 10’
0.71 x 102 0.84 x lo6 0.85 x 10’
using E = 30 kcal mol-r and A = lo6 min-‘. In all cases, discrimination between models is easy and, as can be seen in Table 8, the kinetic parameters obtained are very close to the original ones. Table 9 shows the kinetic parameters obtained from a-T data corresponding to the R3 model and three different combinations of initial parameters. Discrimination between models is easy, and the kinetic parameters obtained from both analyses at constant temperature and at constant conversion are very similar. The lower the E and/or A value, the greater the error with respect to the true value. However, the errors are not excessive in any case.
CONCLUSIONS
The Coats-Redfern method is useful for calculating kinetic parameters when the kinetic model is known, but that is not always the case.
85
The Romero et al. method allows discrimination between models in all cases, and the calculated kinetic parameters are very close to the true values.
LIST OF SYMBOLS
A E
f(a) g(a) K R T t
= frequency factor (mu-‘) = activation energy (kcal mol-‘) = conversion influence on the rate of decomposition = /da/f(a) = reaction rate constant (mm’) = gas constant (kcal mol-’ (K-r) = temperature (K) = time (min)
Greek letters
= fractional conversion = heating rate (K mm’) Subscripts
0”’
= observed from non-isothermal = initial
data
REFERENCES 1 2 3 4 5 6
A.W. Coats and J.P. Redfern, Nature, 201 (1964) 68. E.S. Freeman and B. Carroll, J. Phys. Chem., 62 (1958) 394. H.H. Horowitz and G. Metzger, Anal. Chem., 35 (1963) 1464. L. Reich and S.S. StivaIa, Thermochim. Acta, 59 (1982) 247. D.W. van Krevelen, C. van Heerden and F.J. Hutgens, Fuel, 30 (1951) 253. A. Romero Salvador, E. Garcia CaIvo and A. Irabien Gulias, Thermochim. Acta, 73 (1984) 101.