DIFFERENTIAL THERMAL ANALYSIS AND REACTION KINETICS SIR: In their recent paper, Reed, Weber, and Gottfried (4) observed that for benzenediazonium chloride the ratio Ellog A was remarkably constant, with a value of 1.56 ( A was measured in min.-’ and E in kilocalories per mole), and that the coefficient of variation of the ratio was far smaller than those of either E or log A . I t appears to the present author that this ratio is simply related to the temperature a t which the rate constant, k , is unity, which happens to occur within the range of the measured values of log k. The integrated Arrhenius equation is:
k
= Ae-E/RT
or
I n k = In A
E RT
--
This confusion was wrought when Equation 1 was applied twice to the same data and was intensified when, after E and log A had been obtained for each of 10 DTA curves, the values of log C / K X kp,& (k,& calculated from Equation 1) were plotted against the appropriate temperature. From this graph E is redetermined (the final value) and A is obtained from Ellog A . I t is difficult to see any justification for this manipulation of the results. A simple statistical treatment of the individual values of E and log A , which was given but apparently discarded, would have been sufficient. Also in Reed et al.’s paper it is stated that Kissinger (2, 3 ) incorrectly assumed that the maximum reaction rate occurs a t the peak of the DTA curve. This is a fair statement; however, under certain experimental conditions Kissinger’s assumption can be closely approached. This can be appreciated by considering the dimensionless relation expressed by Equation 23 of (4):
Hence the slope of the corresponding Arrhenius plot is:
E
-
R
d(1n k)
In A
- -
d(l/T)
(1/T)hl
k=o
which in its dimensional form becomes:
or
E
- = 2.303 RT,,, log A
k=o
where, as in Reed et al.’s paper, E is the thermal activation energy, A is a frequency factor, k is the rate constant, R is the gas constant, and T is the absolute temperature. Ellog A is, therefore, proportional to the temperature a t which the Arrhenius plot intersects the log k = 0 abscissa. I t is not surprising that the coefficient of variation is small, since Ellog A is merely redetvrmining a temperature in the experimental range. This also explains why the results obtained by Kissinger’s (2, 3 ) analysis, although independently in very poor agreement with the results of the Borchardt and Daniels (7) analysis, lead to the same value of Ellog ,4. The fallacy of attributing any significance to the ratio Ellog A can be seen if the time scale is altered. The value of E is independent of time, while A is modified by a suitable factor, so that log A is changed by a constant amount. If a time scale is chosen so that log k = 0 no longer lies near the experimentally determined values of log k, scatter will be more apparent. In the case of Reed el ul.’s results, if the time scale had been days instead of minutes, so that the log k scale was displaced by an amount - 3.2, according to the Borchardt and Daniels analysis: Ellog A
=
1.34 f 2% (approximately)
while using Kissinger’s analysis gives
where subscript m refers to the maximum reaction rate. This equation is never valid at the peak of the DTA curve, since the left-hand side is always a maximum and the right-hand side equals zero a t the peak. With decreasing values of C / K (= o( in Reed et al.’s paper) the point on the DTA curve where Equation 3 is satisfied will approach the peak. I t can never reach the peak, but would be acceptably close if, for example, it lay in a temperature range similar to the error in determining the peak temperature, or some other criterion. The ratio C / K has dimensions of time and may be regarded as a “response” time of the system. In work a t these laboratories where the specimens are solid aluminum cylinders separated from a silver heating block by a 20-micron anodic film, the C / K ratio is estimated to be a few seconds, so that the maximum reaction rate occurs within 1’ C. of the DTA peak. Such was not the case in Kissinger’s or Reed et al.’s work.
literature Cited
(1) Borchardt, H. J., Daniels, F . , J.A m . Chem. SOC.79, 41 (1957). ( 2 ) Kissinger, H. E., Anal. Chem. 29,1702 (1957). ( 3 ) Kissinger, H. E., J . Res. h’utl. Bur. Std. 57, 217 (1956). ( 4 ) Reed, R. L., Weber, L., Gottfried, B. S., IND. ENG.CHEM. FUNDAMENTALS 4, 38 (1965).
D. S. Thompson
Ellog A = 1.19 these values are significantly different.
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I&EC FUNDAMENTALS
Reynolds M e t a l s Co. Richmond, Vu.
