and 111, we find that the average of their four values is 44 (for the powder and pellet experiments). Here the order is assumed to be 1/2. Using a value for the order of 2/3 their results yield E = 46 kcal/mole. It would be interesting to see the final results for Method I when the smoothing procedure is used and when the points are weighted in accordance with their probable error. There is one other example in the literature where Method I was compared with an integral (modified Doyle) method. This is the recent work of Zsak6 (8) on the thermal decomposition of some cobalt complexes. Here again the author found “good agreement” between both methods. Apparently, this solid state reaction also showed a constancy in parameters during the course of the reaction.
BENJAMIN CARROLL Department of Chemistry Rutgers, The State University Newark, N.J. 07102 EMANUEL P. MANCHE Division of Natural Sciences and Mathematics York College of the City University of New York Flushing, N. Y. 11 365 RECEIVED for review February 13, 1970. Accepted May 14, 1970.
(8) J. Zsakb, J. Phys. Chem., 72,2406 (1968).
Kinetic Parameters from Thermogravimetric Data-A THEPRECEDING CORRESPONDENCE by Carroll and Manche ( I ) comments on the recent conclusions of Sharp and Wentworth ( 2 ) that the method of obtaining kinetic parameters from a T G curve developed by Freeman and Carroll (3) (method I) is less satisfactory than the methods of Coats and Redfern ( 4 ) (method 11) and Achar et a/. (5) (method 111). Although the earliest significant procedure developed to obtain kinetic parameters from T G is probably that of van Kreleven et a/. (6), method I is important because it was the first method to be used extensively. It may be that under favorable circumstances and when modified by the incorporation of a smoothing procedure (1) method I provides satisfactory kinetic parameters. However, since 1958, many alternative methods have been developed [as reviewed by Flynn and Wall (7)] and we suggest that some of these are superior to method I, especially to the unmodified procedure which is, of course, that used most frequently in the past. Although in principle the order of reaction can be determined from method I, the scatter of data is often so great that a reliable value cannot be obtained. This was so in our investigation of the decomposition of calcium carbonate (2) ; the line drawn in Figure 2 of that paper is a theoretical line based on the kinetic data obtained from methods I1 and 111. Without this further information, lines with other intercepts (and hence, orders of reaction) could have been drawn equally well. Indeed the kinetic parameters changed markedly according to whether 8, 14, or an intermediate number of points were used. Many solid state reactions do not follow an order of reaction, but some other kinetic equation. Thus the dehydroxylation of kaolinite was reported to follow first order kinetics by Jacobs (8) using method I, but later workers under iso(1) B. Carroll and E. P. Manche, ANAL.CHEM.,42, 1296 (1970). (2) J. H. Sharp and S. A. Wentworth, ibid., 41, 2060 (1969). (3) E. S. Freeman and B. Carroll, J. Phys. Chem., 62, 394 (1958). (4) A. W. Coats and J. P. Redfern, Nature, 201,68 (1964). (5) B. N. N. Achar, G. W. Brindley, and J. H. Sharp, Proc. Znr. Clay Conf., Jerusalem, 1,67 (1966). (6) D. W. van Kreleven, C. van Heerden, and F. J. Huntjens, Fuel, 30, 253 (1951). (7) G. H. Flynn and L. A. Wall, J. Res. Nar. Bur. Sfand., A , 70, 487 (1960). (8) T. Jacobs, Nature, 182, 1086 (1958).
Reply
thermal conditions (9, 10) and from T G using method I11 (5) have shown that the reaction does not obey the order of reaction equation. We accept that constancy of kinetic parameters does not establish the mechanism of a solid state reaction, but this applies to all three procedures. In methods I1 and I11 a change in the order of reaction should be detected by a marked deviation from linearity. Carroll and Manche ( I ) point to the inconvenience of trial and error methods, but with modern computational methods these may be less tedious and certainly more justified than a procedure of “smoothing out the random errors of the observed slopes.” We have found method I1 more convenient to program than either method I or 111, since slope measurements are not involved. The necessity for the smoothing procedure underlines the weakness of method I. This is clearly stated by Carroll and Manche ( I ) themselves, “Method I is based on the differences in the slopes of the thermogravimetric trace and in the initial and final regions of a reaction these differences are usually extremely small, thus magnifying enormously the errors inherent in slope measurements.” We conclude that method I should be regarded as of historical importance in the development of methods for obtaining kinetic parameters from T G curves, but should not be recommended as a modern method. Other methods including methods I1 and 111, lead to more satisfactory kinetic analyses of TG curves. Department of Ceramics with Refractories Technology
The University Sheffield, England
SALLYA. WENTWORTH~ J. H. SHARP
RECEIVED for review April 9,1970. Accepted June 29,1970. 1 Present address, Department of Soil Science, Ontario Agricultural College, University of Guelph, Ontario, Canada.
(9) G. W. Brindley, J. H. Sharp, J. H. Patterson, and B. N. N. Achar, Amer. Mineral., 52, 201 (1967). (10) H. B. Johnson and F. Kessler, J. Amer. Ceram. Soc., 52, 199 (1969).
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