Reaction Kinetics in Differential Thermal Analysis

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When a reaction occurs in differential thermal analysis (DTA), the change in heat content and in the thermal properties of the sample is indicated by a deflection, or peak. If the reaction proceeds at a rate varying with temperature i.e., possesses an activation energy the position of the peak varies with the heating rate if other experimental conditions are maintained fixed. In a previous paper (10), it was demonstrated that this variation in peak temperature could be used to determine the energy of activation for first order reactions. The present paper extends the method to reactions of any order, and proposes a method for determining the order of reaction from the shape of the differential thermal analysis peak.

DIFFERENTIAL TEMPERATURE AND REACTION RATE

It was assumed that the temperature of maximum deflection in differential thermal analysis is also the temperature at which the reaction rate is a maximum. Because the proposed method for determining kinetic constants depends on the accuracy of this assumption, a more detailed discussion of its validity is given. The temperature distribution in the differential thermal analysis specimen holders obeys the general heat flow equation.

where T is the temperature, t the time. k the thermal conductivity, p the density, c the specific heat, and dq/dt the rate of heat generation due to a chemical reaction per unit volume of sample. No heat effects occur in the reference sample, so the temperature distribution in the reference is given by:

If the sample is assumed to be a cylinder of radius a and of infinite length, with the temperature of the outside given by T = To + ϕt, where is a constant rate of temperature rise and To the initial temperature, the temperature at Tr at the center of the reference sample is, by integration of
Equation 2 with proper limits,


Equation 2 gives the reference temperature Tr recorded when no reaction occurs. When the corresponding boundary condition is applied to the sample holder, the analogous expression for the sample temperature Ts contains an extra term proportional to the rate of heat generation dq/dt. The differential signal actually recorded by the instrument, ΔT = Ts − Tr, is therefore governed by the rate of heat evolution or absorption rather than by the total heat released, and it is this rate, not the area under the differential thermal analysis curve, that passes through a maximum when the reaction itself is proceeding at its fastest. The assumption used throughout this paper, that the temperature of maximum deflection coincides with the temperature of maximum reaction rate, follows directly from this result and holds closely whenever the thermal lag within the sample holder is small in comparison with the width of the reaction peak.

For a reaction of general order n, the rate of conversion can be written dα/dt = A exp(−E/RT)(1 − α)n, where α is the fraction of material already reacted, A the preexponential factor, E the activation energy, R the gas constant, and T the absolute temperature. Locating the peak by setting the second derivative of α with respect to time equal to zero, and introducing the heating rate β = dT/dt, gives after rearrangement ln(β/Tp2) = −E/(RTp) + ln(AR/E) − ln[n(1 − αp)n−1], in which Tp is the peak temperature and αp the extent of reaction already attained at the peak. Because the bracketed term depends only weakly on the order of reaction and on αp, this expression is closely approximated by ln(β/Tp2) = −E/(RTp) + constant for a reaction of any order. A series of runs carried out at different heating rates should therefore give a straight line when ln(β/Tp2) is plotted against 1/Tp, and the slope of this line, −E/R, provides the activation energy independently of the order of the reaction.

Since the activation energy obtained in this way does not depend on the order of reaction, a separate criterion is needed to establish the order itself, and the shape of the differential thermal analysis peak supplies it. A reaction that follows first order kinetics gives a peak that is very nearly symmetrical about Tp, while reactions of higher order give peaks that are increasingly skewed toward the low temperature side. Comparing the two portions of the peak on either side of Tp, measured between the points at which the curve leaves and rejoins the baseline, gives a shape index that increases in a regular way with the order of reaction, so that an approximate order can be assigned once the activation energy has already been obtained from the slope of the ln(β/Tp2) against 1/Tp plot.

The chief advantage of the method described here is economy of effort. A conventional kinetic analysis requires that the extent of reaction be followed as a continuous function of time at each of several fixed temperatures, while the present method requires only the single peak temperature recorded at each of several heating rates. This advantage is the reason the peak shift method, now generally known as the Kissinger method, has remained in wide use for estimating activation energies from differential thermal analysis and differential scanning calorimetry traces, and continues to be applied to the study of dehydration, decomposition, crystallization, and other solid state reactions.

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