A new iterative method for determining kinetic parameters from TG data

Thermochimica Acta, 61 (1983) 277-286 Elsevier Scientific Publishing Company,

277 Amsterdam

- Printed

in The Netherlands

A NEW ITERATIVE METHOD FOR DETERMINING PARAMETERS FROM TG DATA

J.E. HOUSE, Department

Jr. * of Chemistry, Illinois State University, Normal, IL 61761 (U.S.A.)

J. DANIEL

HOUSE

Department

of Psychiatry,

(Received

KINETIC

University of Iowa Medical School, Iowa City, IA 52242 (U.S.A.)

22 July 1982)

ABSTRACT An iterative method is described which computes n and E from nonisothermal TG data. The method which requires only three (a,T) data pairs as input, involves successive iterations of E/R and n using data pairs where Aa is first small, then large. A complete algorithm for implementation of the method on a programmable calculator is presented as well as results of analysis of calculated and experimental (a,T) data.

INTRODUCTION

Reich and Stivala have utilized a kinetic analysis of TG data that makes use of an approximate rate equation in a two point form 1 - (1 - a,)‘-”

ln

I 1-

(?)I

(1 -Qz)‘-n

+_+j

(1)

I

These workers have shown that constant E/R values are obtained for pairs of (a,T) data when n has the correct value [ 11. Subsequently, Reich and Stivala described an iterative method that determines the correct n when the values of 1 - (1 -n;)‘-”

ln

I 1 - (1 - a(,+l)‘-n and

* To whom correspondence

- 11 = q.,,

i

T,

2

Y

should be addressed.

0040-603 l/83/0000-0000/$03.00

ma 1983 Elsevier Scientific

Publishing

Company

278

are related by linear regression yielding an intercept closest to zero for the correct value of n [2]. In later work, other approximate rate laws were used that were based on other ways of approximating the temperature integral [3,41. Reich and Stivala have also utilized a technique of minimization of the difference between the left-hand side and right-hand side of eqn. (1) as E and n are varied using various pairs of ((Y,r) data [5]. Finally, these workers have developed a method that makes use of ((Y,T) data obtained at different heating rates [6]. Thus, a number of recent iterative techniques have been described. In the present paper, we present a somewhat different iterative technique for computing n and E. This method requires only three (cy,,T,) data pairs and is adaptable to programmable calculators of intermediate capacity. The implementation of the method is described for the Hewlett-Packard HP-34C machine. METHOD

The method of determining n and E described the two point approximate equation ,n

1 - (1 -cX,)‘-n I 1 - (1 -a;+,)‘-”

[$?)j

=;(*-+)

in this paper makes use of

(4

While other forms are possible when other approximations to the temperature integral are used, they offer no distinct advantages [3,4]. The first step in the analysis is to compute an approximate value of E/R. This is performed using two data pairs, ( (Y,,T,) and (a,,T,), where (Y, and (Y* are small since E/R is nearly constant for any value of n under these conditions. The first value assigned to the reaction order, n,, is zero, and a reasonably good value of E/R results with n, = 0 as long as (Y, and (Yeare small (see rows 1 and 2 of E/R values shown in Table 1). After getting an approximate first iterate value for E/R, two data points (cu,,T,) and (cu,,T,) are considered where CX~ Z+ (Ye. If (YeB (Ye, the maximum variation in the function of (Yoccurs as n’ is iterated. Under these conditions, a recalculation of E/R using these data will give a value close to the test value of E/R only when n has the correct value. Using the first iterate E/R value, the function

is calculated

using T2 and T,. Then, the function

T,

390 400 410 420 430 440 450 460 470 480 490

(K)

a Point numbers

2 3 4 5 6 7 8 9 IO II

I

Point a

computed

400 410 420 430 440 450 460 470 480 490 500

(a.=)

data pair.

0.03177 0.06740 0.13369 0.24365 0.39894 0.57706 0.7385 I 0.8553 I 0.92584 0.96354 0.98235

a2

0.4

11894 11646 11210 10452 9238 7559 5619 375 I 2220 1104 347

n =

using n = S/3, E = 100 kJ mole-’

=, (K)

refer to the left-hand

0.01414 0.03177 0.06740 0.13369 0.24365 0.39894 0.57706 0.7385 I 0.8553 I 0.92584 0.96354

al

Values of (cw.T) and E/R

TABLE 1

0.8

11951 11768 11461 10929 10062 8825 7320 5762 4355 3199 2302

n =

and A/b

12007 11891 11715 11419 1093 1 10216 9311 8321 7371 6539 5857

n = 1.2

12073 12036 12016 12009 12003 12000 11999 11997 11995 II991 11994

n = 5/3

= 3 X 10”’ min-’

12120 12140 12233 12441 12807 13378 14150 15039 15916 16668 17258

n = 2.0

12177 12265 12498 12973 13813 15142 16966 19076 21120 22804 24047

,, = 2.4

280

is computed with no = 0 and the result is compared with the value of F,. It is easily shown that if F, > F,, then the iterated value of the order, n’, is smaller than the “correct” n (n’ < n when F, > F,).The process continues by incrementing n’ by 0.100001 (so that n’ is never exactly one) and repeating the calculations. At the point where F, > F2, n’ is greater than n by an amount less than 0.1 for the first iterate value of E/R. This fixes an approximate upper limit of n’, usually within 0.1 of the “correct’ n. At this point, the value of n’ calculated from (a,,T,) and (a,,T,) using the first iterate E/R value is reduced by 0.1 (n’ --) n’ - 0.1) and An’ is reduced from 0.1 to 0.01. The resulting n’, the first iterate to n, is used in recalculating a second iterate E/R value using the first two data pairs. This E/R value is very nearly correct because the value of n’ used is correct to within approximately 0.1. Having a very nearly exact second iterate to E/R, the function F, is calculated using points (cy,,T,) and ( a3,T3). Processing continues by computing the function F2 iterating with An = 0.01. When F2 < F, occurs, computation ends and n’ has been determined with an upper limit within 0.01 of the “correct” value. If desired, the entire process can be repeated to provide a third iterate to E from which n’ can be obtained using iterations of 0.001. The flow chart shown in Fig. 1 illustrates the logic of processing and the complete program for the HP-34C calculator is shown in the Appendix.

TESTING

THE METHOD

Using the program Implementing the program shown in the Appendix on the HP-34C programmable calculator uses the entire capacity of the machine with partitioning for 105 program lines and 15 registers. The computation can easily be adapted to other advanced calculators and microcomputers. As written, the program utilizes the following locations (given as register number, use): R, and R,,, ‘Y,; R,, q; R,, ‘Ye; R, and R,,, T,; R,, T,; R,, 7;; R,, n’; R,, R,, R, and R,,, computed results; R,, E/R; R,,, An. The (Y,, q, n,, and An values are entered and processing is initiated by pressing label A. The first stop displays the first iterate n’. Pressing R/S restarts processing which ends with the second iterate n’, etc. The value of E/R after each iteration is stored in R,. Testing the method with calculated (CX,T) data The iterative method was tested using the calculated values of Table 1 for which n = 1.666.. . and E = 100 kJ mole-‘. Since involves using three values of (Y, it was necessary to determine produced by the nature of that selection. In order to show the

(Yshown in the method the effects trends and

281

effects that changing (Yhas on E/R, values for the latter were calculated and they are also shown in Table 1. One of the first factors to be studied was the effects produced by choosing

Compute F, using E/R and last two temperatures P Compute F, using ax, OL), and n’

n’=n’+An

No

HALT

Rotate

2 aI -P a2,

0~2 + as. T, + T, T, + TX 4 ~

n’=n’-An An = 0.1 An

Fig. 1. Flowchart

for the three-point

iterative

method.

different points for use in the calculations. For example, E/R is approximately constant for small values of (Y,and cyz. It would appear, then, that the first iterate n’ should depend on which points are chosen. Table 2 shows the results obtained when various combinations of three points were chosen. In each case, the (a,,T’) point was the same (point 10) although when other points for which (Ye> 0.7 are used similar results are obtained. From the data shown in Table 2, it is immediately obvious that as the second point represents successively larger values of (Y,the first iterate n’ and

282 TABLE 2 Kinetic parameters obtained and A//3 = 3 x 10” min- ’ Points used ’

1, 1, I, I, I, 3, 4,

2, 3, 4, 5. 6. 4, 5.

10 IO 10 IO IO IO 10

a Numbers

for (ol,T) data calculated

First iteration

using

n =

1.66667,E = 100 kJ mole- ‘.

Second iteration

n’

E (kJ mole-‘)

n

E (kJ mole- ’ )

1.7 1.6 1.6 1.6 1.5 1.5 I.4

98.430 97.159 95.256 92.425 88.417 91.157 83.060

1.69 1.67 1.67 I .66 1.64 I .64 I .60

100.300 99.918 99.628 99.268 98.043 98.468 95.982

refer to points

in Table I.

E/R become less accurate. This is expected since the first value of E/R is established using n 0 = 0 and assuming that E/R is constant for all n’ values. Obviously, this approximation is worse for larger values of (Y. The second iteration largely removes this error since E/R is established using the first iterate n’. If the results obtained using points (1, 5, lo), where (Ye= 0.24365, are considered, the second iterate n’ is 1.66 and E = 99.268 kJ mole-‘. Clearly, even though (Y*is hardly “small”, the results are as accurate as the five-decimal (Y values will allow. However, when points (1, 6, 10) are used, the second iterate values n’ = 1.64 and E = 98.043 are probably outside the accepted tolerance for these parameters. However, in the case of the computation using points (1, 6, lo), the third iterate values are n’ = 1.667 and E = 99.740 kJ mole-‘. In the case of the computation using points (4, 5, lo), the third iterate values are n’ = 1.653 and E = 99.030 kJ mole- ‘. Certainly in this case neither (Y, nor (Y*fit the criterion of “small” (Y.The results of three iterations appear to exhibit sufficient accuracy in the derived n and E values as long as both LY,and (Yevalues are less than about 0.25 with better accuracy resulting when (Y, < 0.1. Because An = 0.001 during the third iteration, performing three iterations greatly increases computing time. Numerical solutions of rate equations are available for various values of n from 0 to 2 [7] and they have been analyzed by the method of Reich and Stivala [2]. These data have also been used as a test of the present method with the results shown in Table 3. These results show that as long as (Y, and (Yefit the criterion of being small, two iterations produce n’ and E values that are as accurate as can be expected. This is especially true when it is recalled that the value for n’ represents the value which first produces F2 -c F, so that the actual n is smaller than the value shown by less than 0.01.

283

TABLE 3 A comparison of results calculated ((Y,T) data

obtained

using the Reich and Stivala

E (kJ mole-‘)

a

Present

Reich-Stivala

h

0.01 0.34 0.51 0.67 I.01 I .34 1.67 2.01

100.23 I 100.848 101.526 100.912 101.065 99.62 1 99.323 100.707

n

Actual

Reich-Stivala

0

0.02 0.34 0.50 0.67 I .oo I .34 1.67 2.00

l/3 l/2 2/3 I 4/3 5/3 2

h

and present

methods

with

Present ’ 99.999 99.842 100.069 99.713 100.065 99.956 99.628 100.071

’ Actual E = 100 kJ mole-’ used in numerical solution of rate equation. h Using the Reich and Stivala method [2]. ’ Using two iteration cycles. These data were calculated using (Y,< 0.05. (Ye< 0.15. and a3 > 0.85 (usually (Ye> 0.95).

Testing the method with TG data In order to test the described method using actual TG data, the data for decomposition of NH,HCO, were used [9]. In each case, (Y, was in the range 0.04-0.10, cx2 in the range 0.15-0.20, and CQ was > 0.67. Table 4 shows the results of the analysis. Again, it is clear that the present method gives satisfactory agreement with established methods. TABLE 4 A comparison of n and E values iterative methods Run

for the decomposition

Reich and Stivala a

Present

n

n

E

(kJ mole-‘) I 2 3 4 5 Ave (I ’ Computed

1.20 1.32 1.33 I .06 I .66 I.31 0.22

99.97 90.68 99.1 I 80.62 94.57 92.99 7.86

[8] using the method

of NH,HCO,

obtained

method E

(kJ mole- ‘) I.16 I.61 1.58 0.93 1.52 1.36 0.30

of Reich and Stivala (21.

98.70 95.71 103.20 75.80 89.67 92.96 IO.61

by two

284

It must be cautioned that in TG studies (Yis determined by comparison of an observed mass loss with that corresponding to the complete reaction. If a given reaction is accompanied by only a few per cent mass loss, CYis generally not determined as accurately as when a larger mass loss is involved. The present method, and those based on other computations, will not generally give kinetic parameters of high accuracy for such cases.

REFERENCES 1 2 3 4 5 6 7 8 9

L. Reich and S.S. Stivala, Thermochim. Acta, 24 (1978) L. Reich and S.S. Stivala, Thermochim. Acta, 36 (1980) L. Reich and S.S. Stivala, Thermochim. Acta, 41 (1980) L. Reich and S.S. Stivala, Thermochim. Acta, 53 (1982) L. Reich and S.S. Stivala, Thermochim. Acta, 52 (1982) L. Reich and S.S. Stivala, Thermochim. Acta, 55 (1982) J.E. House, Jr., Thermochim. Acta, 55 (1982) 241. J.E. House, Jr., Thermochim. Acta, 47 (1981) 379. J.E. House, Jr., Thermochim. Acta, 40 (1980) 225.

9. 103. 391. 121. 337. 385.

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