## Theory of DTA

The main applications of DTA are (i) to describe the thermal decomposition and transitions occurring on heating a material through a programmed temperature range, (ii) to measure the heat of reaction, and (iii) to determine the kinetic parameters. The first is reasonably clear, the second requires an explanation, and the third, once the energy terms have been related to the weight changes, is similar to the treatment of kinetic data on the thermobalance.

In dealing with the measurement of the heat of reaction, the theories of DTA can be placed in two categories: (i) those which deal with heat transfer alone and (ii) those which deal with the reaction equation, that is, take into account the chemical nature of the reaction.

A simplified heat transfer theory is based on the method developed by Vold (1949).

The equation of heat balance for the cell containing the reaction is

and that for the cell containing the inert material is

where C is the heat capacity of each cell; T1, T2, and T3 are the temperatures of the reactant, reference material, and block, respectively; K is the heat transfer coefficient between the block and the cell; and dH is the heat evolved by the reaction in time dt.

In writing these equations two assumptions are made: (i) the heat capacities of the two cells are the same and do not change during the reaction, and (ii) there is a uniform temperature throughout the sample at any instant. Otherwise a single value (T1 and T2) could not be written for the temperature of the material in the two cells.

To determine the total heat of reaction it is necessary to integrate from t = 0 to t = x:

AH = C(AT - AT0) + Kl AT dt f Jo when t = 0, T = 0, and when t = x, T = 0:

The influence of physical properties on the baseline can be considered more realistically by assigning different values of C and K to each cell and considering the simple case where there is no reaction, that is, dH = 0; then

C2 dT2 = K2(T3 - T2) dt. C1 and K1 refer to the reactant cell, and C2 and K2 to the reference cell. Rearrangement gives and

Ci dTi Ki dt

C2 dT2

K2 dt dTi /dt and dT2/dt represent heating rates and should be identical, that is.

dTi dT2 dT dt dt dt

Then dT (KiC2 - K2Ci) AT = Ti - T2 = -——-— (5)

We now have three cases. C2 Ci

this is demonstrated by the zero value of AT (Fig. 7). C2 Ci

K2 Ki ), which gives a positive constant value of AT (Fig. 7). C2 Ci

K2 Ki -TJK J, which gives a negative constant value of AT (Fig. 7). If

C2 Ci

FIGURE 7 DTA baseline behavior according to Vold. Note that in the top plot CK is not a function of temperature; and in the bottom plot (C2/K1).

or if

then the curved plot in Fig. 7 results. If, however, C2/K2 > C1/K1, the slopes are in the opposite direction.

For a more comprehensive treatment the publications of Wilburn and his coauthors should be consulted.

The practical tests of the use of DTA equipment in this way are to check whether the peak area is proportional to the quantity of material under examination and also to check the area under the peak for materials of a known heat of reaction. If the equipment responds properly to calibration tests of this kind, then it would seem, within the limits of the calibration range, to be proper to use it as a scanning calorimeter.

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