reactions have been proposed for other systems (3-8). Such reactions could alter the dipole moment of the salt without (3) S. Bruckenstein and A. Saito, J. Amer. Chem. Soc., 87, 698 IlOhz;)
(4) J. Steigman and P. Lorenz, ibid., 88, 2083 (1966). (5) I. M. Kolthoff and S. Bruckenstein, ibid., 78, 1 (1956). (6) G. Barrow and E. Yerger, ibid., 76, 5211, 5248 (1954). (7) E. Yerger and G. Barrow, ibid., 77,4474, 6206 (1955). (8) E. K. Ralph and W. R. Gilkerson, ibid., 86,4783 (1964).
greatly affecting the UV-visible spectrum. This could qualitatively explain the apparent anomaly in the presence of a large excess of pyridine. It is to be noted that dielectrometric measurements are capable of revealing the presence of species which could not be detected by simple spectrophotometric measurements.
RECEIVED for review August 27, 1969. Accepted June 25, 1970.
CORRESPONDENCE Kinetic Parameters from Thermogravimetric Data SIR: In a recent article in this Journal (I), parameters derived from a thermogravimetric trace for the decomposition of calcium carbonate were compared using three different methods of kinetic analysis. These methods were labelled Method I, the difference-differential method of Freeman and Carroll ( 2 ) ; Method 11, the integral method of Coats and Redfern (3); and Method 111, the differential method of Achar, Brindley, and Sharp ( 4 ) . As a consequence of their findings Sharp and Wentworth ( I ) concluded that they could not recommend Method I because of an apparent poor precision for this method. Because Method I appears to be one of the more widely used thermogravimetric method for obtaining kinetic data (5) for solid-state chemical reactions, we would like to call attention to several points that may help to clarify this situation. One important advantage that attracts the chemists to Method I is that the order of a simple chemical reaction may be determined analytically, whereas in the case of Methods I1 and I11 an order has to be assumed. A trial and error procedure for the latter methods is then used, the criterion of acceptance of the numerical value for the order being that value which leads to the best fit for thermal trace over the entire range of the chemical reaction. Aside from the inconvenience of trial and error methods, the question arises whether the criterion of constancy of kinetic parameters is appropriate even for the case of a simple chemical reaction since a solid state reaction is a very complex process. In our present state of knowledge, constancy of kinetic parameters in general, for reactions in the solid state, is an unwarranted assumption (6, 7). Apparently in the case of the thermal decomposition of calcium carbonate there does not appear to be significant changes in the kinetic parameters during the course of the reaction. In this case all three methods should yield about (1) J. H. Sharp and S. A. Wentworth, ANAL.CHEM., 41, 2060
(1969). (2) E. S . Freeman and B. Carroll, J. Phys. Chem., 62, 394 (1958). (3) A. W. Coats and J. P. Redfern, Nature, 201,68 (1964). (4) B. N. N. Achar, G. W. Brindley, and J. H. Sharp, Proc. Znt. Clay C o n f , Jerusalem, 1, 67 (1966). (5) J. H. Flynn and L. A. Wall, J. Res. Nat. Bur. Stand., 70A,
487 (1966). (6) B. Carroll and E. P. Manche, J. Appl. Polym. Sci., 9, 1895 (1965). (7) I. A. Schneider, Makromol. Chem., 125, 201 (1969). 1296
the same results, provided a good guess is made for the order of the reaction for Methods I1 and 111. The results of Sharp and Wentworth demonstrate that this is the case when the initial portion of the reaction for Method I is omitted. Similarly in a previous publication (6),it was shown that both the integral method of Doyle and Method I yielded about the same results for the volatilization of a liquid. In this particular instance both order and activation energy were known, the order being obviously zero and the activation energy being that for the latent heat of vaporization. Again, whereas the order was assumed for the integral method, the value was determined analytically via Method I. As for the activation energy, Method I yielded a value that checked the thermodynamic value within a tenth of a kilocalorie. It should be noted that the latter results were obtained with the inclusion of the initial part of the process provided that the random errors in the slope measurements (weight us. temperature) were smoothed out, particularly for the initial and final stages of the process. The necessity for this is clear. 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. Precision can be enhanced considerably by the simple expedience of plotting the experimental slopes cs. the reciprocal of the temperature and smoothing out the random errors of the observed slopes. Further, the resulting curve may be subdivided into equal intervals of A(l/7'), thus simplifying the subsequent plot for yielding both order and activation energy (6).
The original thermogravimetric trace for the Sharp and Wentworth paper was not published, making it impossible to illustrate the effect of the above procedure for Method I. We are rather surprised at the valid results they obtained for Method I for the range of 25 to 8 4 x conversion of calcium carbonate, this being 0.55 for the order and 43 kcal/mole for the activation energy. Because the thermal decomposition of calcium carbonate is an endothermic reaction its activation energy should be equal to or greater than the thermodynamic value for the reaction. The thermodynamic value is 43.0 kcal/mole at the temperatures of the decomposition. The activation energy of the reverse reaction is known to be small and is probably in the neighborhood of one kilocalorie or so. Examining the Sharp and Wentworth values for Methods I1
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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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