2406
J. ZSAK6
Kinetic Analysis of Thermogravimetric Data
by J. Zsako Depurtment of Chemistry, Babes-Bolyai University, Cluj, Rumania Aceepred and Transmitted by The Faraday Society
(February 1 , 1968)
The trial-and-error method of Doyle was modified. Instead of curve fitting, the calculation of standard deviation is suggested. The integrals of eq 11 are tabulated as a function of temperature and of activation energy. Testing of different kinetic equations and estimating of apparent activation energy, frequency factor, and activation entropy are discussed. The suggested method is compared with the methods given by Horowitz and Metzger and by Freeman and Carroll using experimental data obtained for two cobalt(II1)-dimethylglyoxime complexes.
Theoretical Section The thermal decomposition of solids is a very complex process even in the simple case expressed by the stoichiometric equation
A(4
=
B(s)
+ C(g)
This process takes place in several stages, e.g.: the chemical act of breaking of bonds; destruction of the initial crystal lattice; formation of the crystal lattice of the solid product B, consisting of the formation of crystallization centers and the growth of these centers; adsorption-desorption of the gaseous product C; diffusion of C ; heat transfer. The rate of the thermal decomposition is determined by the rate of one or more of these stages. Sometimes the rate-determining stage a t the beginning of the pyrolysis can lose its significance later and another stage can take its place. Thus pyrolysis rate depends not only upon the nature of the studied substance but also upon many other factors, such as particle size, weight of the sample, shape of the crucible, etc. When investigated under the conditions obtained in thermogravimetric analysis, the heating rate has a decisive part too. Taking into account the complexity of the process, there is little chance to find a general equation able to describe the kinetics of all thermal decomposition reactions, especially in the case when kinetics varies during the process. Many attempts have been made to obtain kinetic equations, mainly for cases where the process is governed by a unitary kinetic law. Using these equations, a considerable number of techniques for deriving kinetic data from thermogravimetric curves has been developed. The decomposition rate can be defined as daldt, where a stands for the fraction of the initial compound reacted. I n isothermal conditions, we can presume this reaction rate to depend only upon the fraction reacted The Journal of Physical Chemistry
da - = kf(ff) dt where f(a) is a certain function of a, and IC can be considered to be a rate constant. When pyrolysis takes place under the conditions of thermogravimetric analysis, k cannot remain constant and will depend upon temperature. The validity of the Arrhenius equation can be presumed for this dependence
Ic = z exp(
-
2)
where Z is the frequency factor and E , is the activation energy. Since thermogravimetric analysis is carried out with a constant heating rate, q = dT/dt, the substitution dt = dT/q can be made, and from eq 1 and 2, the following differential equation is obtained
(3) The equation of thermogravimetric curves could be obtained by integration of eq 3, but here two main difficulties arise: the analytical form of function f ( a ) is generally unknown, and the right side of eq 3 cannot be integrated in finite form. The analytical form of the function f(a) depends upon the mechanism of the thermal decomposition. Considerable efforts have been made to deduce this function using various models. One of the most general equations able to describe various solid-state reactions is f(a) = aa(l -
a)b
(4)
with a and b constants, called frequently homogeneity factors. A large number of cases were discussed by different (1) J. Sestak, Silikaty, 11, 163 (1967).
KINETICANALYSIS OF THERMOQRAVIMETRIC DATA authors on the basis of both theoretical and experimental investigations. The most simple forms of eq 4 are those with a = 0. I n this case, eq 1 has the same form as the kinetic equation of a b-order homogeneous reaction, and, consequently, b can be considered an apparent reaction order f ( a ) = (1 - a)*
da - = k(1 - a)b dt
Equations of the same form as eq 5 are considered to be the most simple laws of the dissociation of solids. Equation 5 with b = was found theoretically for the first time by Roginsky and Schulzla and later by many other authors, both theoretically and ex~ e r i m e n t a l l y . ~ -It~ is considered to be the fundamental law of topochemical reactions.’ Different authors have considered also the following b values: 0, l / 3 , 1/2, 1, and 2.* For a group of reactions, kinetic equations with b = 0, i.e. f(a) = aa
2407 direction too. Introducing the notation u = E,/RT, eq 3 can be written as da - = - f(d
ZE, e-u - du Rq u2
(9)
Integration of this equation from the absolute-zero temperature up to the actual one of the sample leads to the equation
where x means the value of u at the temperature of the sample. The values of the integral
have been calculated and tabulated by Doyle12 for x values covering the range from 10 to 50. The equation of the therniogravimetric curves given by Doyle is
(6)
were found to be valid. The a values, proposed by different authors6 are 1 and 2/3. An equation the same type as (4) with a # 0 and b # 0, namely
was found for the first time by Lewisgand was obtained theoretically by Prout and Tompkins. lo Equations of this type have been discussed also for the cases a < 1, b = 1 ; and a = 2/3, b = 2/a.6 Concerning the integration of the right side of eq 3, some authors have tried to approximate the exponential factor by integrable functions. Thus Horowitz and Metzger” have made an asymptotic expansion of 1/T at a conveniently chosen temperature, and neglecting all the terms, except the first two, they obtained an integrable form for eq 3. Taking the logarithm of the function
The main difficulty in applying eq 12 consists in the dependence of p(x) on both temperature and activation energy. Doyle has suggested a trial-and-error curve-fitting method for the determination of activation energy. He discussed reactions for which function f(a) was known, and thus g(a) values could be computed from thermogravimetric data. Under such conditions this author obtains the approximate value of Ea from the slope of the thermogram and calculates the theoretical curve by means of eq 12. By modifying the presumed E, value, the agreement between the theoretical and the experimental curves can be imporved. The activation energy value which ensures the best consistency will be the required one, Ozawa13 has suggested a method based on the eq 12 for determining the activation energy. This author has shown that the activation energy can be graphically
they have found this to vary linearly with temperature. From the slope of the corresponding straight line, the activation energy, E,, can be computed. Since a thermal decomposition step takes place frequently over a temperature range of 60-80”,this asymptotic expansion is not quite justifiable. Our calculations showed indeed that log g(a) does not vary linearly over such a large temperature range. The slope of the theoretical curve log g(a) us. T shows a variation of about 25-30%, which introduces a large error in the activation energy data. Thus the accurate integration of eq 3 is absolutely necessary. Some attempts have been made in this
(2) M. T. Trambouae and B. Imelik, J . Chim. Phys., 57, 656 (1960). (3) S. Roginsky and E. Schula, 2.Phys. Chem., A138, 21 (1928). (4) K.L. Mampel, ibid., A187, 43, 235 (1940). (5) P.W.M. Jacobs and F. C. Tompkins in “Chemistry of the Solid State,” W. E. Garner, Ed., Butterworth and Co. Ltd., London, 1955,p 184. (6) G. F. Huttig, Monatsh. Chem., 85, 365 (1954). (7) W.E. Garner, Chem. Ind. (London), 1010 (1951). (8) E. A. Prodan and M. M. Pavlyuchenko in “Heterogeneous Chemical Reactions,” M. M. Pavlyuchenko and E. A. Prodan, Ed., Nauka i tekhnika, Minsk, U.S.S.R., 1966,p 20. (9) G. M.Lewis, 2.Phys. Chem., 52, 310 (1905). (10) E. G. Prout and F. C. Tompkins, Trans. Faraday Soc., 40, 488 (1944). (11) H. H. Horowita and G. Metzger, Anal. Chem., 35, 1464 (1963). (12) C. D.Doyle, J . Appl. Polym. Sci., 15, 285 (1961). (13) T.Oaawa, Bull. Chenz. Soc. Jap., 38, 1881 (1965).
f(a) = a(l - a)
(7)
Volume 7.2, Number 7 July IS68
J. Z S A K 6
2408 Table I : -Log p(z) Values Corresponding to Different Temperatures and Activation Energies
100 100 112 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430
8.175 8.002 7.833 7.678 7.523 7.383 7.241 7.110 6.984 6.862 6.744 6.632 6.524 6.417
... ... ...
*..
... ...
...
*..
... ... ...
... ... ...
.*.
...
*.. ..,
,..
9.498 9.294 9,099 8.908 8.731 8.565 8.402 8.244 8.096 7.953 7.818 7.683 7.558 7.438 7.317 7.206 7.095 6.989 6.887 6.790 6.693 6.601 6.509 6.422 .
I
.
... ...
... ...
... ...
... ... ...
10.798 10 559 10.338 10,121 9.917 9.725 9.537 9.357 9.186 9.025 8.864 8.717 8.569 8.432 8.294 8.165 8.042 7.918 7.803 7.688 7.583 7.473 7.372 7.271 7.175 7.085 6.996 6.908 6.821 6.739 6.657 6,581 6.504 6.427 t
12.083 11.812 11,561 11.318 11.090 10.865 10.689 10.458 10.261 10.078 9,900 9.730 9.566 9.411 9.259 9.113 8.972 8.834 8.702 8.574 8.452 8.333 8.220 8.106 7.997 7.893 7.793 7.693 7.598 7.503 7.413 7.327 7,241 7.155
13.348 13.052 12.770 12.503 12.243 11.997 11.765 11.542 11.328 11.118 10.923 10.731 10.549 10.377 10.208 10.044 9.885 9.735 9.590 9.450 9.313 9,181 9.055 8.928 8.810 8.692 8.579 8.471 8.363 8.259 8.160 8.061 7.968 7.873
14.610 14.283 13.999 13.672 13.391 13.118 12.860 12.615 12.380 12.153 11.935 11.727 11.523 11.333 11.147 10.965 10.794 10.631 10.468 10,314 IO. 160 10,015 9.875 9.740 9.609 9.483 9.357 9.235 9.118 9,006 8,893 8.790 8.682 8,584
15.862 15.505 15.163 14.838 14.530 14,232 13.952 13,681 13.423 13.175 12,940 12,709 12.493 12,281 12.077 11.883 11.693 11.513 11.337 11.166 11.004 10.846 IO. 693 10.544 10,401 10,261 10.126 9.996 9.866 9.745 9.624 9.508 9,395 9.284
obtained by following thermogravimetrically the decomposition at different heating rates. In the present paper we have tried to simplify Doyle's trial-and-error method and to find new applications of eq 12 in the kinetic analysis of thermogravimetric curves. If the logarithm of eq 12 is found, the following expression is obtained
where B depends only upon the nature of the compound studied and upon the heating rate, but not upon the temperature. The value of g(a) for a given temperature can be calculated from the experimental data if f(a) is known. Similarly, p(z) for the same temperature can be found if the activation energy is known. The constancy of the difference log g(a) - log p(z) enables us to suggest a quantitative method of testing the validity of different kinetic equations of the same type as eq 3 and of determining the apparent activation energy, consistent with a given function f ( a ) . This method consists of the following. By preThe Journal of Physical Chemistry
17,108 16.716 16.347 15.996 15.658 15,341 15,033 14.740 14.461 14.190 13.934 13.691 13.451 13.222 13,001 12.789 12.587 12.388 12.200 12.016 11.841 11.670 11.504 11,342 11.185 11.037 10.889 10.746 10.661 IO. 477 10.348 10.218 10.097 9.977
18.349 17.926 17.527 17.149 16,785 16.440 16.112 15.792 15.491 15,204 14.926 14.661 14.405 14.158 13.919 13.695 13.475 13.264 13.057 12.860 12.667 12.483 12.304 12.134 11.963 11.803 11.646 11.494 11.347 11.199 11.061 10.927 10.793 10.664
19.584 19.132 18,701 18.294 17,908 17,536 17.182 16.845 16.518 16,209 15.913 15.626 15.353 15.089 14,838 14.591 14.358 14.130 13,910 13.700 13,494 13.297 13.104 12.921 12.742 12.568 12.398 12.233 12.077 11.921 11.774 11.627 11.485 11.347
20.811 20.333 19.871 19.438 19.023 18.628 18.252 17.890 17.545 17.214 16.896 16.592 16.296 16.015 15,746 15.487 15.232 14.991 14.754 14.535 14,316 14.106 13.901 13.705 13.512 13.325 13.147 12.973 12.803 12.639 12.479 12.323 12.171 12.025
22.038 21.530 21.043 20.579 20.136 19.716 19.315 18.930 18.564 18.211 17 876 17.550 17.237 16.938 16.652 16.375 16.107 15.853 15.603 15.362 15,130 14.907 14.693 14.484 14.279 14.083 13.891 13.705 13 527 13.353 13.184 13.020 12.860 12.705 I
23.262 22.719 22,206 21.716 21.248 20.802 20.376 19.947 19.579 19.205 18.848 18.504 18.179 17.862 17.555 17.260 16.979 16.707 16.444 16.190 15,945 15.709 15.478 15.255 15,042 14.833 14.633 14.437 14 246 14.064 13.882 13.709 13.541 13.376 I
24.784 23.910 23,366 22.845 22 351 21.885 21.434 21.002 20.592 20.200 19.821 19.461 19.100 18.779 18.454 18.147 17.848 17.559 17.283 17,017 16.758 16.504 16.264 16.028 15.802 15.584 15.371 15.163 14,963 14.768 14,582 14.395 14.218 14.046 I
suming the validity of a function f(a) and using thermogravimetric data, g(a) values are calculated for different temperatures. By means of a trial-and-error method, the apparent activation energy can be estimated by finding the E , value which ensures the maximum constancy of B = log g(a) - log p(z). The agreement between experimental data and presumed E, can he characterized quantitatively by the standard deviation of individual Bi values from their arithmetical mean B. This will be defined as
where r is the number of experimental data used for the calculation of B. The minimum of S indicates the best E , value. At the same time, this least value, &,in, is a measure of the consistency of the decomposition process with the presumed function f(a). By presuming other kinetic equations and by calculating the corresponding amin values, the minimum of Smin will indicate that function f(a), among the tested ones, which ensures the maximum consistency with ex-
KINETICANALYSIS OF THERMOGRAVIMETRIC DATA
----
-------
2409
--.
E,, koal/mol------------
38
40
42
44
46
48
50
52
54
56
58
60
62
64
88
25.702 25.097 24.524 23.978 23.457 22.963 22.487 22.038 21.602 21.189 20.793 20.410 20.044 19.693 19.556 19.026 18.715 18.413 18.128 17.839 17.564 17.302 17.048 16.799 16.564 16.329 16.107 15.890 15.677 15.473 15.274 15.084 14.893 14.712
26.919 26.284 25.679 25.107 24.560 24,041 23.543 23,067 22.610 22.174 21.761 21.361 20.975 20,606 20.250 19.907 19.579 19.260 18.953 18.655 18.371 18.096 17.825 17.568 17.320 17.076 16.841 16.615 16.393 16.176 15.968 15.765 15,670 15.376
28.129 27.464 26.834 26.235 25.661 25.115 24.592 24.095 23.620 23.162 22.723 22.306 21.902 21.516 21.143 20,784 20.437 20.104 19.784 19.474 19.178 18.885 18.605 18.334 18.073 17.821 17.573 17.334 17.104 16,878 16.656 16,449 16.241 16.038
29.342 28.646 27.986 27,356 26.757 26.190 25.643 25.120 24 624 24.145 23.688 23.249 22.827 22.424 22.034 21.661 21.298 20.952 20.615 20.291 19.976 19.675 19.383 19.100 18.825 18.559 18.303 18.055 17.812 17.578 17.348 17.127 16.910 16.702
30,552 29.826 29.135 28.480 27.855 27.257 26.690 26.145 25.625 25.129 24.651 24.194 23.751 23.330 22.922 22.533 22.156 21.793 21.443 21.107 20.779 20.465 20.159 19.862 19.575 19.301 19.031 18.770 18.517 18.271 18,037 17.803 17.578 17.362
31.761 31.004 30.283 29.602 28.951 28.327 27.738 27.161 26.627 26.109 25.612 25.134 24.678 24.235 23.815 23.407 23.013 22.632 22.270 21.916 21.580 21.248 20.929 20.624 20.323 20.035 19.757 19.488 19.223 18.967 18.720 18.481 18.247 18.018
32.968 32.164 31.430 30.722 30.041 29.396 28.780 28.192 27.627 27.086 26.573 26.073 25.598 25.138 24.701 24.276 23.869 23.475 23.095 22.728 22.374 22.034 21.702 21.384 21.075 20.774 20.483 20.200 19.926 19.661 19.416 19.155 18.912 18.674
34.175 33.353 32.574 31.837 31.124 30.462 29.822 29.211 28.626 28.066 27.527 27.014 26.519 26.041 25,585 25.143 24.723 24.312 23.919 23.538 23.172 22.814 22.474 22.143 21.820 21.507 21.207 20.915 20.629 20.355 20.085 19.825 19.575 19.328
35.375 34.528 33.724 32.954 32.226 31.528 30.865 30.229 29.625 29.041 28.484 27.950 27.437 26.942 26.469 26.013 25.571 25.147 24.741 24.348 23.964 23.597 23.244 22.900 22.564 22.242 21.929 21.625 21.330 21.047 20.770 20.501 20,237 19.980
36.579 35.701 34.867 34.073 33.313 32.592 31.904 31.246 30.619 30.014 29.440 28.888 28.354 27.846 27.351 26.879 26.424 25.986 25.562 25.152 24.759 24.380 24.009 23.652 23.307 22.972 22.651 22.338 22.034 21.734 21,448 21.170 20,897 20.638
37.782 36.869 36.008 35.183 34.403 33.657 32.945 32.267 31.613 30.991 30.395 29.822 29.270 28.744 28.237 27.747 27.275 26.820 26.383 25.959 25.553 25.156 24.777 24.407 24.050 23.706 23.371 23.045 22.732 22.424 22.131 21.838 21,561 21.289
38.984 38.040 37.150 36.298 35.491 34,719 33.983 33.282 32.606 31.962 31.345 30.753 30.189 29.642 29.117 28.610 28.125 27.657 27.203 26.766 26.343 25.936 25.544 25.161 24.795 24.434 24.090 23.756 23.430 23.115 22.805 22.510 22.220 21.938
40.184 39.211 38.290 37.413 36.579 35,781 35.023 34.296 33.599 32.941 32.298 31.689 31.103 30.538 29.996 29.476 28,973 28.489 28.022 27.572 27.135 26.717 26.307 25.914 25,535 25.165 24.809 24.461 24.126 23.801 23.484 23.176 22.877 22.587
41.382 40.380 39.429 38.524 37.663 36.841 36.057 35.309 34,592 33.907 33.250 32.619 32.015 31.434 30.875 30.337 29.819 29.320 28.794 28.374 27.926 27.492 27.073 26.668 26.275 25.894 25.525 25.269 24.823 24.487 24.160 23.842 23.534 23.235
42.581 41.548 40.568 39.635 38.747 37.904 37.094 36.322 35.583 34.877 34.201 33.552 32.927 32.328 31.753 31.199 30.665 30.151 29.656 29.177 28.714 28.268 27.837 27.419 27.014 26.623 26.243 25.875 25.518 25.172 24.836 24.511 24.196 23,882
I
perimental data. If the minimum value, is small enough, we can assume the validity of a kinetic equation of the same type as eq 3, with t'he corresponding f(a) and the found E , value as the apparent activation energy of the process. The obtained value can be used for the calculation of the apparent frequency factor 2, according to the equation log Z =
B
alytical forms of g(a) and of B for the case of validity of equations of this type, calculated by means of eq 8 and 13, respectively, are listed below for different 6 values in the equation f (a)= (1 - a) b
=
0
- log P(X)
(17)
+ log [l - v1 - a ] - log p(z)
(19)
go(a)
=
a
Bo = log a
+ log Rp - log E ,
(15) derived from eq 13. An apparent activation entropy can be calculated too, using the equation S'
Zh
= 2.303 log -
kT
From these calculations for T can be taken the temperature at which the weight loss is half of the total weight loss during the considered step. Applying the above method, the first problem is to calculate g(a) from the experimental data, presuming different analytical forms for f(a). The simplest cases are equations of the same type as ( 5 ) . An-
log 2 b
=
'//a
g*/,(a) = 3[1 log 3
-v
z ]
B,,, =
+ log [l - G] - log p(z)
(20)
Volume 72,Number 7 July 1968
J. ZSAK;)
2410 b = l gl(a!)
=
-In (1
amine and to the formation of the nonvolatile nonelectrolyte Co(DH)zAmX as a quite stable intermediate product, accordingly to the equation
- a!)
[Co(DH)zAmz]X= Co(DH)2AmX b = 2 a!
gda) = -
1-a
a
Bz = log -- log p(z) 1-a
(22)
If the thermogravimetric step corresponds to a unitary process, the fraction of the initial compound reacted can be given as the ratio of the actual weight loss to the total weight loss during the considered step, i.e. a!=
wo - w wo - Wt
where W , Wo,and Wt are the actual, initial, and final weights of the sample, respectively. Equation 23 enables us to calculate g(a) for different presumed kinetic equations. To facilitate the obtaining of the activation energy, we have tabulated the integrals p(z) for temperatures between 100 and 430" and for activation energies between 10 and 66 kcal/mol. The -log p(x) values, given in Table I, have been calculated by using Doyle's tabulated values and, in the case z > 50, by using the approximate formula
+ Am
The corresponding thermogravimetric steps were studied by means of the Horowitz and Metzger method. We give now two examples for the application of the above-explained method, using experimental data published in our earlier paper.15 The two complexes we take here are [Co(dimethylglyoxime)z(p-eth~laniline)~ ]NCS (I) [Co(dimethylgly~xime)~( y-pic~line)~]r\'CS(11) Using the Horowitz and Metzger method, we have found the following kinetic data for these compounds, where b is the apparent reaction order, corresponding to eq 5 b
=
2;
b = 1;
E, = 42 kcal/mol;
E*
= -22 eu
E,
8"
=
= 33 kcal/mol;
(I)
(11)
+l eu
For applying the above-suggested method, we have calculated go(a) for these compounds, presuming zero-, first-, and second-order reactions. The weight of the sample, W , at different temperatures and the corresponding gb(a!)values are given in Table 11. Table 11: Sample Weight and Log g(a) Data for Compounds I and I1 at Different Temperatures
For intermediate E, values, the corresponding -log p(z) can be found by means of linear interpolation. The suggested trial-and-error method consists in calculating Bi for different f(a) functions and different E, values with the help of Table I and formulas 17-22 or by means of the corresponding g(a) function, obtained according to eq 8 from the presumed f(a), for all the temperatures at which the weight of the sample was measured. By means of eq 14, the standard deviation, 6, will be calculated, and the minimum of this will indicate the best f(a) and the corresponding E, value. The apparent frequency factor can be calculated by means of eq 15, and an apparent activation entropy can be also obtained by means of eq 16. Applications In our earlier papers14-17 the kinetics of the thermal decomposition of complexes of the type [Co(DH)zAmz]X have been studied: (DH)zis dimethylglyoxime or cyclohexanedione dioxime; Am is different aromatic amines; and X is C1, Br, I, or NCS. The first step of the decomposition of the above complexes is the substitution of an amine molecule by the external-sphere anion, which leads to the volatilization of the aromatic The Journal of Physical Chemistry
Corn-
Temp,
W,
Log
pound
C
mg
0
160 170 180 190 200 210 220 140 150 160 170 180 190 200 210 220
99.3 97.9 95.9 92.4 87.9 82.1 81.1 101.4 101.2 100.9 100.2 98.2 95.5 91.2 86.0 84.3
-1.469 -0.992 -0.701 -0.433 -0.231 -0.061 -0.037 -1.948 -1.648 - 1.406 -1.104 -0.719 -0.465 -0.233 -0.057 -0.012
Ia
116
Wo = 100.0 mg; Wt = 79.4 mg. 83.8 mg.
Log
[In 1/(1 -
LOP a)J
-1.464 -0.969 -0.654 -0.337 -0.053 0.308 0.395 -1.948 -1.643 -1.397 -1.087 -0.674 -0.377 -0.057 0.320 0.553
a/(l
-
a)
-1.454 -0.945 -0.605 -0.233 0.153 0.821 1.046 -1.945 -1.648 -1.388 -1,069 -0.627 -0.283 0.148 0.851 1.539
W O= 101.6 mg; Wt =
(14) J. Zsak6, Cs. Vlrhelyi, and E. KBkedy, Studia Univ. BabesBolyai, Chem., 2, 7 (1965). (15) J. Zsak6, Cs. Vhrhelyi, and E. KBkedy, Analele Univ. Bucuresti, Ser. Stiint. Nat., 14, No. 15 (1965). (16) J. Zsak6, Cs. Vhrhelyi, and E. KQkedy, J. Inorg. Nucl. Chem., 28, 2637 (1966). (17) J. Zsak6, Cs. Vfirhelyi, and E. KBkedy, Acta Chint. Acad. Sci. Hung., 51,53 (1967).
KINETICANALYSISOF THERMOGRAVIMETRIC DATA In order to determine which of the three tested b values is the most consistent with experimental data, we had to calculate for different presumed E, and those b values that we wanted to test. On the basis of eq 17, 21, and 22, and by means of tabulated -log p(x) values in Table I we have calculated Bo, BI, and Bz for all the temperatures. I n each case the arithmetical mean of B was calculated and 6 was obtained accordingly to eq 14. I n the case of complex I, we obtained the following intermediate results = I---.
-b
= 0--7
b-
koal/ mol
6
Ea, koal/ mol
20 22 24
0.1149 0.1086 0.1192
26 28 30
Ear
6
-b Ea, kcal/ mol
0.0900 0.0714 0.0755
38 40 42
= 2-
6
0.0917 0.0900 0.1044
These data are sufficient to see that standard deviations are less if we presume a first-order reaction. Thus we can consider that from the tested b values b = 1 is the best. Using interpolated -log p(z) values for the case b = 1, we have 28.8
E,, kcal/mol--28.9
29.0
0.0694
0.0693
0.0698
r---
6
Le., 6 has a minimum value for E,
= 28.9 kcal/mol. This corresponds to BI = 16.291. By means of eq 15, we obtain log 2 = 11.712. Apparent activation entropy can be estimated using relation 16, and we obtain S* = -5.84 eu. We can see that the considered method gives quite different results from that obtained by means of the Horowitz and Metzger method. Even the found reaction order is 1 and not 2. In the case of complex I1 the minimum values of standard deviation are Smin
b = O
b - 1
b = 2
0.0767
0.0610
0,0743
i.e., this reaction seems to be of first order, too. The corresponding activation energy is E, = 28.3 kcal/mo14 the frequency factor log Z = 11.446, and the activation entropy S* = -7.06 eu. These values are also flatly different from those obtained by means of Horowitz and Metzger’s method. Apparent reaction order and activation energy have
2411 been calculated also by means of the Freeman and Carroll method,ls using the same experimental data. In Table I11 are compared the b and E, values, obtained by means of the three methods. Table I11 : Reaction Order b and Activation Energy E, in the First Thermogravimetric Step of the Complexes I and I1 -Complex
II-
b
I-E.¶, kcrtl/ mol
b
kcal/ mol
1 1.16 2
28.9 29.2 42
1 0.82 1
28.3 28.7 33
--Complex
Method
Modified Doyle Freeman-Carroll Horowitz-Metzger
Ea,
We can see that the data given by the Freeman and Carroll method are in good agreement with those found by means of our method, but the Horowitz and Riletzger method gives quite different results. The Freeman and Carroll method can be used successfully, though it is based on graphical determination of thermogravimetric-curve slopes. It gives nearly the same results as our more accurate method. We cannot say the same about the Horowitz and Metzger method, which is affected by considerable errors.
Conclusions The suggested method, using the real values of the integral p(x), is free from the errors introduced by the approximative Horowitz and Metzger formulas. Meanwhile, the statistical way of working up experimental data and the minimization of standard deviation ensures an accuracy in estimating apparent activation energy which is consistent with the presumed kinetic equation. This method is more expeditive than the initial Doyle method. Calculations can be carried out relatively simply using Table I and can be easily programmed for electronic computers. Since the standard deviation, 6, is a quantitative measure of the consistency between experimental data and the presumed kinetic equation, the suggested method can be very useful in testing the validity of different kinetic equations and thus can be applied in the study of solid-state reaction mechanisms. (18) E. S. Freeman and B. Carroll, J . Phys. Chem., 62, 394 (1968).
Volume 78, Number 7 July 1968
