Calculation of Thermogravimetric Data by Electronic Digital Computer J. R. SOULEN Research and Development Department, Pennsalt Chemicals Corp., Wyndmoor, Pa.
b A
The amount of computation required to obtain kinetic constants from thermogravimetric data is large, so it is not surprising that relatively few points from the curves have been used. The formidable amount of computation also has discouraged auxiliary studies which would be highly desirable. These include the influence on the kinetic constants obtained of apparatus and procedural variables, such as sample container material and geometry, atmosphere, and heating rate, as well as sample variables, such as initial weight, physical state, and particle size. It would be an improvement, therefore,
method has been developed for automatic computation of temperature, weight, weight loss, and rate of weight loss from raw electromotive force data representing these quantities in thermogravimetry.
I
N AN earlier paper (8) the author discussed several aspects of thermogravimetry which are important in obtaining correct weight and temperature measurements during the course of thermal decompositions, as well as in defining the decompositions precisely by means of atmospheric control. These are particularly important considerations if one wishes to use thermogravimetric data to determine kinetic constants such as the activation energy, order, and frequency factor for reactions. A method for doing this has been published by Freeman and Carroll (4) and used by several other workers ( I , 3, 6, 7'). In examining the linear plots used in this method to obtain the activation energies (from the slopes) and orders of reaction (from the intercepts), it is obvious that only a small fraction of the available data from each thermogravimetric run is utilized. Further, it is doubtful in some cases whether the best lines have been drawn through the relatively few numbers of points used. No uncertainties appear to have been assigned to the values obtained.
to perform the required calculations, using much more of the available data, in considerably less time. The author has developed the program reported here for computation of temperature, weight, and rate of reaction from the values of d.c. millivoltage indicating them during thermogravimetric decompositions. Under development are programs for automatic calculation of the kinetic constants from these values of temperature, weight, and rate of reaction, as well as statistical analysis of the reliability of the results. INPUT DATA
Two sets of input data are required. One is a series of millivoltage values representing temperature, weight, and rate of reaction a t a number of specified times during a run. The first four columns of Table I list a sample of such readings taken a t 1-minute intervals for 1 hour during the decomposition CO, in a nitrogen atCaC03+ CaO mosphere. The decomposition curve is shown in Figure 1, on which the 10minute interval corresponding to the data in Table I is indicated. The second set of input data consists of numbers which relate the observed millivoltages t o the quantities they measure. These are listed in Table 11.
+
Figure 1. Thermogravimetric curve for the decomposition CaCOa + CaO Con
+
Data for Toble I taken at times shown by dots. Temperatures in C.
PROGRAM AND COMPUTATION
Table 1.
Time, min. 49 50 51 52 53 ._ 54 55 56 57 58
Input Millivolts Temp. Weight 7.04 7.13 7.19 7.24 7.30 7.35 7.41 7.51 7.58 7.62
5.02 4.91 4.80 4.68 4.53 4.41 4.30 4.20 4.14 4.12
Input and Output Data"
---
Rateb
Tzmp., K.
output Weight, Loss, mg . mg.
4.49 4.30 4.49 5.28 5.28 4.49 4.10 3.13 1.56 0.39
1051.43 1055.15 1060.70 1065.33 1070.87 1075.48 1081.01 1090.19 1096.61 1100.27
192.20 187.89 183,57 178 86 172.97 168.26 163.94 160,OO 157.63 156.83
+
54.93 59.24 63.57 68,27 74.16 78.87 83.19 87.14 89.51 90.30
Rate, mg./min. 4.49 4.30 4.49 5.28 5.28 4.49 4.10 3.13 1.56 0.39
Total data taken for CaCOa +. CaO GO1 decomposition in N2 included 60 values each of temperature, weight, and rate. Illustrative data selected for this table are for a 10-minute interval, minutes 49-58 of the 60-minute period (see Fig. 1). Can be obtained as millivoltage, but was not for this run. See Discussion.
136
ANALYTICAL CHEMISTRY
The Remington Rand Univac computer has been programmed using the Math-Matic compiling system (9). A 23-sentence, English language program (Table 111) was used to compute 60 values each of temperature, weight, cumulative weight loss, and rate of reaction, and to store these for subsequent computation of the kinetic constants. Calculation of these 240 values (40 of which are shown in the last four columns of Table I) required only a few minutes on a Univac I1 computer. DISCUSSION
In this example, values were read from the strip chart record a t 1-minute
Table II.
Relation
Symbols
Electromotive force temperature E.M.F. = A BT C T 2 ( T = "C.)
+ +
Weight correction temperature Correction = (G X T ) H ( T = "K.)
A B C
+
G H
Weight sensitivity, mv. deflection per mg. of weight change
S
Rate sensitivity, mv. deflection per mg. per minute rate of weight loss
Q
Table 111.
Auxiliary Data Input Numerical Value for This TGA Run
- 294.40
8.20345 0.00165055 0.0041 0.33
39.08
Remarks Depends on thcrmocoupla used, which should be calibrated. A . B, C can be obtained from calibration curve. Depends on crucible support, crucible and sample, buoyancy of atmosphere used, convection currents, apparatus design. See (6, Figure 1) for graphical illustrations. Linear relation is usually satisfactory above 100" C. Determined for each run by calibration with known weights. Changes slightly from day to day. Higher degree equation can be used if sensitivity is not constant throughout entire weight range.
1.ooo
To be determined by calibration with known rates of millivolt change. See (?, Figure 8) for illustration of calibration curve.
Program for Calculation of TGA Data
.
READ-ARRAY D(60,3) SERVQ 7 TYPE-IN A B C G H E S Q . SET TQ 0 M . CQKTAIN TEMP(60) . CQNTAIN WGT(6O) CQNTAIN RATE (60) CQNTAIN LOSS(60) . CQNTAIN K(60) . CQNTAIN WEIGHT(6O) . VARY I 1 (1) 60 SENTENCES 11 THRU 15 . TEMP(1) = (SQR (B POW 2-4*C*(A-1000*D(I,1)))-B)/(2*C)+273.15 . WGT(I ) = D( 1,2)*S- (G*TEMP( I ) -H) LQSS(1) WGT(1)-WGT(1) . RATE(1) = Q*D(I,3) IGNQRE . PRIKT-QUT M . I = K(l) . WEIGHT(1) = LQSS(60)-LOSS(1) WRITE-ARRAY CQNVERT TO 2 DECIMALS TEMP(E0) SERVQ 2 LABEL TEMPERATURE WRITE-ARRAY COXVERT TO 2 DECIMALS WGT(60) SERVQ 3 LABEL ITEIGHT. WRITE-ilRRAY CONVERT TO 2 DECIMALS R.4TE(60) SERVQ 4 LABEL RATE . WRITE-ilRRAY CQNVERT TQ 2 DECIMALS LQSS(60) SERT'Q 5 LABEL LOSS . STQP .
.
.
.
.
.
.
intervals to obtain temperature and weight millivoltages, and rate of reaction values are from measurement of the slope of the weight curve a t each point. This is a poor method to obtain data for input to the computer, but it sufficed to demonstrate the feasibility of the automatic computation. Addition of a differentiating circuit ( 2 ) so that an electromotive force representing rate of reaction can be obtained from the weight curve is in progress. To obtain full benefit from this program, a data logger is being designed t o make simultaneous measurements of temperature, weight, and rate of reaction a t specified intervals, recording this information on tape suitable for direct input to the computer. The use of an English language program is generally inefficient. A more efficient program could no doubt be developed using machine language. For one without previous computer experience, however, an English language program is attractive. It can be understood and used after relatively little study, and alterations and refinements are easy to make. ACKNOWLEDGMENT
Thanks are due E. T. Domboski for the thermobalance runs, Arthur Anicetti and John Hunter of Remington Rand for assisting in programming and carrying out the Univac calculations, and the Computing Center of the Franklin Institute for its course in MathMacic programming which was taken by the author. LITERATURE CITED
(1) Anderson, D. A., Freeman, E. S., J . A p p l . Polymer Sci. 1,192 (1959). (2) Campbell, C., Gordon, S., Smith, C. L., ANAL.CHEM.31,1188 (1959). (3) Deadmore, D. L.,Ph.D. thesis, Univ. of -.Illinnis. ___ I 1960. -----
(4) Freeman, E. S., Carroll, B., J. Phya. Chem. 62, 394 (1958). (5) Lumme, P., Suomen Kemistilehti 32B, 198 (1959). (6) Newkirk, A. E., ANAL. CHRM.32. --, 1559 (1960). (7) Padmonbhan, V. M., J. Inorg. Nuclear Chem. 12, 356 (1960). ( 8 ) Soulen,. J. R., Mockrin, I., ANAL. CHEY.,33, 1909 (1961). (9) Sperry Rand Corp., "Math-Matic Remington Ran!, Automatic Programming System, 1960. RECEIVEDfor review July 14, 1961. Accepted October 23, 1961. Work s u p ported in part by the Office of Naval Research.
VOL 34, NO. 1, JANUARY 1962
0
137
