DIFFERENTIAL T H E R M A L ANALYSIS AND REACTION KINETICS SIR: Although Thompson’s observation that Ellog k = 2.303RTl,, 0 is interesting, the attendant inferences are not convincing. H e has evidently transformed the equation Ellog A = 1.56 + 0.9% [Equation 37 ( 2 ) ]to the new time scale using (E)*”and (log A),” directly and thereby obtained a n increase in data scatier. This procedure is incorrect. T h e proper procedure is to une all data in Table I (2) and recalculate Ellog A for each run, using the appropriate value of E and the value of log A adjusted to the new unit of time. \Yhen this is done, one obtains Ellog A = 1.34 =k 0.8%, so that data scatter is about the same as before. This is more easily seen by observing that In ( A / k ) == EIRT. Hence, each pair of values ( A , E ) determines a straight line passing through the origin of the appropriate graph. If one now transforms the time scale, the coirection for A just cancels that for k and the data fall on precisely the same lines a s prior to the transformation. Since the values of E ,and log A determined by the method of Kissinger are incorrect, we are prepared to accept the interpretation that the behavior of the ratio may not be meaningful. Analysis of the reproducibility of individual runs showed that data scatter exhibited in Table I could not be attributed solely to experimental errors and errors in data reduction. This meant that some factor was not properly accounted for in the theory or, perhaps, in the application of the theory. Therefore, the “simple statistical treatment” suggested by Thompson was not appropriate. T h e graphical averaging procedure finally used employed all of the information a t our disposal in the least prejudiced manner, a technique we believe to be in the best statistical tradition.
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mum reaction rate occurs near the D T A peak. This, in itself, is of no consequence. The question of significance is: Precisely how close does it have to be to ensure that the error incurred in, say E, using the subject approximation, shall be acceptably small? Thompson gives an example where T , - T , = 1’ C . and implies that this is acceptable. We have determined T, for benzenediazonium chloride using machine calculation. A typical result is T , - T , = 0.91’ C., contrary to Thompson’s expectations. Nevertheless, the corresponding error in E resulting from use of Kissinger’s approximation is 42%. Thus, we have demonstrated a case where the maximum reaction rate occurs a t a temperature of 0 . 9 9 6 T P and yet this is clearly not sufficiently close. I t happens that 0.91 ’C. is not within experimental error of the peak temperature. However, even if it were, there is no justification for assuming that the corresponding error in E would be acceptably small. Finally, it appears that Thompson is using the method of Kissinger for the analysis of DTA data obtained from solid aluminum cylinders. This, of course, cannot be done without theoretical justification. Thompson’s argument using the magnitude of C / K is not valid, since the system is essentially three-dimensional and unstirred. I t was precisely this kind of consideration that led us to study the stirred system before proceeding to the more complex stagnant case. literature Cited
(1) Kissinger, H. E., Anal. Chem. 29, 1702 (1957). (2) Reed, R. L., Weber, L., Gottfried, B. S., IND.ENG. CHEM. FUNDAMENTALS 4, 38 (1965).
Ronald L. Reed
Method of Kissinger
O u r statement was: ‘ Kissinger argued (incorrectly) that the maximum reaction rate occurs a t the peak of the D T A curve, . . .” We intended to convey that Kissinger’s attempt ( 7 ) to give a theoretical proof of this point is incorrect. I t is obvious that experimental conditions may exist under which the maxi-
Drexel Institute of Technology Philadelphia, Pa. Leon Weber Byron S. Gottfried Carnegie Institute of Technology Pittsburgh, Pa.
SOLUTION OF TlHE EQUATION OF T H E T H E R M A L DIFFUSION COLUMN SIR: I n a recent paper Ruppel and Coull (2) have presented a solution of the equation of the thermal diffusion column, valid over the full range of concentration for batchoperated columns closed a t both ends, using the linearization transformation used by Majumdar. T h e solution is particularly useful for short experimental times, but the authors have put the more readily applicable equations in a form requiring an assumption not explicitly stated, so that their use may be misleading in certain cases. In fact, coefficient K in the transport equation is composed of two terms
K = K,
and solution constituent variables. Now, K can be written
or
and Equations 8, 36, 37, and 39 of the authors’ paper should read, respectively,
+ Ka
where
K , = (2~)7g*pB(AT)~@~/(9 !v2D) represents the effect of convection, and
X,j = (2w)pDB is the contribution of ordinary diffusion. Though the authors follow generally the nomenclature of Jones and Furry. they systematically ignore the Kd term, except i n their Equation 6 , when they substitute K for their column VOL. 5
NO. 2
MAY 1966
287
