ACKNOWLEDGMENT
The author is grateful to J. P. Williams for advice and encouragement in this research and to J. A. Marley who prepared the tin oxide samples. LITERATURE CITED
(1) Beard, H. C., Natl. Acad. Sci.-,Vatl. Research Council, Nuclear Sci. Ser. NAS-NS-3002 (1960). (2jtBowen, H. J. hl., Gibbons, D.,
Radioactivation Analysis,” Oxford University Press, London, 1963.
(3) Cali, J. P., “Trace Analysis of Semi-
conductor Materials,’’ Pergamon Press, New York, 1964. (4) Dyer, F. F., Leddicotte, G. W., Xatl.
Acad. Sci.-Natl. Research Council, Nuclear Sci. Ser. NAS-NS-3027 (1961). (5) Leddicotte, G. W., Mullins, W. T., Bate, L. C., Emery, J. F., Druschel, R. E., Brooksbank, W. A., Jr., Proc. U . N . Ztern. Conf. Peaceful Uses At. Energy Znd., Geneva, 28,478 (1958). ( 6 ) Lewis, J. E., Natl. Acad. Sci.-Natl. Research Council, Nuclear Sci. Ser. NAS-NS-3032 (1961). (7) hlarley, J. A., MacAvoy, T. C., J. Appl. Phys. 32, 2504 (1961).
(8) Mullins, W. T., Leddicotte, G. W., Natl. Acad. Sci.-Natl. Research Council, A-uclear Sci. Ser. NAS-NS-3055 (1962). (9) Nervik, W. E., Natl. Acad. Sci.-Natl. Research Council, Nuclear Sci. Ser. NAS-NS-3023 (1960). (10) Sullivan, W. H., “Trilinear Chart of
-
Nuclides.” Oak Ridge National Laboratory, 1957. H. E. RAUSCHER Research & Development Laboratories Corning Glass Works Corning, N. Y. 1 6 PITTSBURGH ~ ~ Conference on Analytical Chemistry and Applied Spectroscopy, hlarch 1-5, 1965, Pittsburgh, Pa.
Use of the Computer to Calculate Thermodynamic Constants from Thermogravimetric Curves SIR: A digital computer program has been written for the determination of activation energy and the pre-exponential factor of the Arrhenius equation for the raw data obtained from the thermogravimetric curves. The program was designed to accept sample weight, (w), and sample temperature, (2‘) , values as a function of time, ( t ) . The program was written for first order reactions only; however, with a very slight modification it would be applicable for any order of reaction. Copies of the programs may be obtained from the authors. The computer program makes use of a least squares polynomial fit of the timesample weight values to the following equation :
where n is the desired order polynomial, C the coefficient of the polynomial, and t the time. The POLY 2 Fortran program that currently appears in the library file of The Computation Center, The Pennsylvania State University, was employed to fit the T G h curve with a n nth order polynomial. From the weight-time curve thereby generated, an additional Fortran subroutine, FREEB, calculates the reaction rate constant for any point chosen on the T G h curve, from the following equation
(8: k = n-I
where n is the desired order polynomial. Data manipulation within the computer then prepares values of log K a t corresponding values of lOOO/T. h simple least squares analysis of the values of log k vs. 1000/T can be obtained for the following first order polynomial,
520
ANALYTICAL CHEMISTRY
Log k
(1) of this type of equation gives the following relationships for the slope and the intercept on the ordinate:
energy as indicated by the above equations. This revised program is shown by the flow diagram as indicated by Figure 1. An alternate approach to determining the pre-esponential factor and activation energy is to enter the time-weight-temperature data with the revised POLY 2 program and calculate the two thermodynamic values for a series of polynomial order fits to the time-weight values. A simple inspection of the curve fit output data, as in the
where N , is the number of the tinietemperature data points chosen. Ea is therefore easily obtained from the above equation. I n addition, the preexponential factor A can be easily calculated from the following relationship, (2)
first method of solution, determines the polynomial order along with the preesponential factor and activation energy in a single computer pass. Convenience and conservation of computer time dictates which manner of solution should be used.
=
Log A
- Ea/2,303R (1000/T)
This equation can be treated as the equation, y=a+bx
h least squares analysis
N%
The input data to the computer program is explained by comments cards a t the beginning of the program. I t is necessary to determine the order of the polynomial that gives the best fit to the time-weight values by using a succession of trials for the order n in the POLY 2 library program of The Computation Center. The equation order is conveniently determined in a single computer pass with orders of up to 40 being possible. After the order for the polynomial giving the best fit to the TGA curve is obtained, this polynomial order along with the weight-time, and temperature-time data are submitted to the POLY 2 program that had been revised to include the subroutine FREEB. The revised POLY 2 calculates the coefficients of the polynomial for the prescribed polynomial and the subroutine F R E E B then calculates the pre-exponential factor and activation
1
Kinetic values were obtained from the dynamic thermograms as previously described using this computer program. This technique afforded several advantages over the manual calculations. First, it eliminated the problem of mechanically obtaining slopes from the thermogram. This was accomplished by using a high order polynomial to fit the TGA\ curve ( n = 10 to 15). The accuracy of the computer fit of the TGA curve was 0.2 mg. and the limit of accuracy for reading a weight value was 0.1 mg. Once an equation was obtained for the curve, the computer then differentiates the curve and finds the slope using the data submitted. Second, after the computer calculates the values for the hrrhenius plot of log k us. 1000/T it does a least squares fit on these points and calculates the activation energy for the reaction. These machine operations eliminate much of
Computer Flow Diagram
POLY 2
Number of Data Points Time-Weight Data Points I
IPOLY
2 Program1
Subroutine F R E E B . I
ITransfer Polvnomial Coefficients1 Points Sample Weight at time euual to infinitv
and Correct slope and reject bad data points I
Convert data to reaction rate log values and 1000/T for analvsis Calculate activation energy and Pre-exponential factor by least squares analysis \Data :utputl /End of Program1 Figure 1 the human error involved in the calculations. Table I compares the weight values generated by the equation for the TGA curve developed by the computer program to actual values from the thermogram for the decomposition of calcium oxalate. Table I also shows the input data supplied to the computer. The results obtained by taking the slope manually compared favorably with those obtained from the computer program as is shown in Table JI. The computer, however, gives a better straight line plot whereas the manual values are more scattered. The use of a computer program to obtain the equation of an experimental curve can be of considerable value. The use of this approach has been demonstrated above for use with TGX
curves. However, this program could be adapted to other experimental curves. Once a fit has been obtained any mathematical manipulation can be accomplished with ease and accuracy. For curves which are difficult to fit it is possible to break the curve into sections which will enable a stepwise solution. This program also has the added advantage that a number of curves can be run simultaneously in a single computer operation. I n order to add subroutines for other calculations simply replace the statement “1395 CALL FREEBERG (K, COEFX),” in our revised POLY 2 program. This statement can be changed t o call any subroutine desired to do any calculations with the POLY 2 curve generated in the main program simply by writing a new sub-
Table 1.
Computer Fit of TGA Curve for CO Loss
ExperiInput data Furnace mental Time toemp., weight, C. min. mg.
Predicated weight, mg.
78.40 78.10 78.00 77.80 77.60 77.10 76.80 76.20 75.50 74.80 73.70 72.30 71.00 69 .OO 66.70 64.10 61,50 58,80 56.20 54.00 52.10 50.20 48.90 47.80 47.50
78.38 78.20 77.95 77.75 77.50 77.18 76.78 76.26 75.61 74.77 73.72 72.40 70.79 68.87 66.66 64.21 61.58 58.90 56.29 53.90 51.85 50.23 49.05 47.76 47.50
0.00 2.50 5.00 6.25 7.50 8.75 10.00 11.25 12.50 13.75 15.00 16.25 17.50 18.75 20.00 21.25 22.50 23.75 25.00 26.25 27.50 28.75 30.00 32.50 35.00
416.0 423.0 430.0 437.0 444.0 461 .O 458.0 465.0 472.0
Table 11. Arrhenius Plot Comparison for Computer and Manual Results
Log k J
computer lOOo/!P 1.451 -2.122 min.-’ 1.437 -1,906 1.422 -1.678 1.408 -1.443 i.394 -1.218 -1,010 1.382 1.368 -0.818 1.355 -0.640 -0.469 1.342
Log k , manual -2.001 min.-‘
-1.910 -1.857 -1.494 -1.331 -1.009 -0.845 -0.638 -0.542 Activation Energy Manual 69.0 Kca1.l
mole. Activation Energy Computer 70.7 K cal./mole.
routine and specifying a new namee.$. “1395 CALL ROYER (K, COEFN).” X set of data cards and a complete program deck can be obtained from D. J. Royer. LITERATURE CITED
( I ) Bennett, C. A., Franklin, N. L.,
“Statistical Analysis in Chemistry and the Chemical Industry,” p. 222, Wiley, New York. 1954. (2) Ibid., p. 224. (3) Newkirk, A. E., ANAL. CHEM. 32, 1658 (1960).
J. M. SCHEMPF F. E. FREEBERQ D. J. ROYER F. M. ANGELONI
The Pennsylvania State University The Department of Chemistry University Park, Pa.
VOL. 30, NO. 3, MARCH 1966
521
