53
Anal. Chem. 1988, 60,53-57
Equations for Isothermal Differential Scanning Calorimetric Curves David N. Waters* and John L. Paddy Department of Chemistry, Brunel University, Uxbridge, Middlesen UB8 3PH, England
Equatlons are devebped for differential scanning calorimetric curves In the isothermal case. Kinetlc schemes defined by -d[A],/dt = k[A]:, -d[A],/df = k[A]:[P];", and [A],/IA], = exp(-(kt)"i are examined (A = reactant, P = product). The second of these equations describes autocatalytlc reactions; the third descrlbes Avraml processes. Methods are glven for extracting kinetic parameters dlrectiy from the observed curves, without recourse to graphical integration procedures. Appllcations to data in the dlfferentiai scanning caiorlmetry literature are discussed.
Differential scanning calorimetry (DSC) has been extensively used for the study of physical transformations, e.g. phase changes, and for the investigation of the kinetics of both homogeneous and heterogeneous chemical reactions. The observation of a physical transformation in a material by DSC often requires a controlled variation of the temperature of the sample, and this implies the uge of the temperature-scanning, i.e. "dynamic", mode of operation of the calorimeter. On the other hand, a number of chemical reactions, e.g. polymerizations, decompositions, and oxidations, and some physical changes, e.g. crystallizations, can be observed in both isothermal and dynamic modes of operation. The isothermal mode, where it can be applied, offers the advantage of greater simplicity of interpretation of the data, leading, it may be hoped, to greater confidence in the values of derived kinetic parameters. Even a simple reaction, conforming to eq 3 below, is conventionally characterized by three kinetic parameters: reaction order and any two of activation energy, Arrhenius preexponential factor, or rate constant. The determination of all three parameters, where these are unknown, from dynamic DSC methods is often considered to be unreliable (I). Thus it can be found that several sets of parameter values are equally consistent with the observed data. More complex reactions, e.g. those which are autocatalyzed, are characterised by more parameters, and for these the problem of defining a unique set of values is even greater. It is precisely because the isothermal method introduces fewer experimentalvariables into a single measurement that the scope for ambiguity in the interpretation of the data is reduced. Of course it follows that more than one isothermal measurement (at different temperatures) will be required to establish all the parameters, but, for the investigation of new reactions where the order, or even the reaction mechanism may be unknown, this is a small price to pay for increased certainty. Most general treatments of thermoanalytical kinetic data, e.g. ref 2 and 3, give very condensed accounts of the isothermal DSC method. Therefore, although the principles of the method are well-established, and indeed are straightforward, their systematic development to the cases of classical, or "model", reaction schemes does not appear to have been given. Such a presentation is given here.
KINETIC TREATMENT An isothermal DSC trace is a plot of heat flow against time. We consider the simple reaction 1in which it may be assumed
A-P
(1)
that the instantaneous heat flow (the DSC ordinate, yJ at time t is proportional to the rate of reaction of A at that time, i.e. to -d[A],/dt. Thus yt = q(-d[A],/dt), where q has the dimensions of energy per mole, and is obtainable from total heat evolved = q[A], (2) We shall be concerned with processes that follow the rate laws 3 or 4,i.e. with reactions that may be characterized as "simple" or "autocatalyzed", respectively. In addition, transformations that follow the Avrami rate law (4-6)will be considered. -d[A],/dt = k[A],"
(3)
-d[A],/dt = k[A],"[P],"
(4)
A. Simple Reactions. Writing
1-41, = [Alo(1 - CY)
(5)
where a is the fraction of A reacted at time t , eq 3 becomes -d[A],/dt = [A], d a / d t = k[A]on(l - a)" d a / d t = k[A]O"-'(l - CY)"
(6) (7)
Equation 7 is of the form d a / d t = CY)
(8)
By rearrangement and integration
F(a) = t
(9)
= G(t) This gives on differentiation
(10)
The inverse equation is CY
da/dt = g(t)
(11)
which, after multiplying by q[A],, yields the required DSC ordinate. This three-stage process-integration, inversion, and differentiation-will be applied to rate equations of the form of eq 3, with n = 0, 1, 3/2, and 2. For each resulting equation simple procedures are given for extracting the rate constant k. We may note that analyses of DSC data have usually been based on equations of the form of eq 10. That is, the observed curve in the form of eq 11is integrated, using some method for area determination, to give extent of reaction as a function of time. With the use of eq 11directly this step becomes unnecessary, and errors introduced by the processes of area determination are avoided. (Flynn (7,€9, however, has based the analysis of isothermal DSC rate data on equations of the form of eq 11). The Case n = 0. In this case -d[A],/dt = k, and the equation of the DSC curve is obtained immediately as
y = q[A]o d a / d t = g k
(12)
The DSC curve for this case is shown in Figure 1,curve a. The rate of heat evolution remains constant from the start of the reaction and falls sharply to zero when all the reactant is
0003-2700/88/0360-0053$0 1.50/0 0 1987 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 60,
NO. 1, JANUARY 1, 1988 The foregoing procedure leads to eq 24 as the DSC curve:
Y '
y = q[A], d a / d t = 8qk[A]03/2/(k[A]o'/2t+ 2)3
(24)
Properties of this curve are shown in Figure 1, curve d. A t t = 0 y takes the value qk[A]:/'. As with the previous case, it is useful also to consider the value of t corresponding to one half of the peak ordinate. Denoting this by tIj2, we find Qlj2
= 1 - 2-2/3 = 0.3700
(25)
k[A]o'/zt1/2 = 16'13 - 2 = 0.5198
(26)
The Case n = 2. The rate equation is
da/dt = k[A]O(l - a)'
(27)
The derived DSC curve, Figure 1, curve e, is y = q[A]oda/dt Flgure 1. Differential scanning calorimetric curves for reactions conforming to -d[A],ldt = k[A]:, for (a) n = 0, (b) n = ' I 2 ,(c) n = 1, (d) n = 3/2, and (e) n = 2.
consumed (t = t f ) . Equating the product qkt, to the total energy given by eq 2, we obtain ktf = CAI0
(13)
The Case n = ' I 2 .We have
da/dt = k[A]o-'/2(l - a)'/'
(14)
This gives, after introducing the requirement that a = 0 at t=O 2(1 - (1 - CY)'/^] = k[A]o-'/2t CY
= 1 - (k[A]o-'/2t - 2)'/4
y = q[A]o da/dt = qk([A]01/2 - kt/2)
(15) (16) (17)
Equation 17 is plotted in Figure 1,curve b. At t = 0, y has the value There is a h e a r fall in y until the reaction reaches completion at t = tf, where
k t , = 2[A]01/2 The Case n = 1. The rate equation is da/dt = k(1 - CY)
(18) (19)
and the DSC curve is obtained by the above procedure as
y = q[A], da/dt = qk[A], exp(-kt)
(20)
The curve is plotted in Figure 1,curve c. To test whether an experimental curve conforms to eq 20 it is best to plot In y against t. The slope of the resulting straight line gives k . If the first-order law has already been established, it may be simpler to consider the time tl/z, corresponding to half the maximum ordinate (not half the total extent of reaction!) The time tllzis easily measured on a given isothermal curve. The ratio Qllz of the area lying between the ordinates t = 0 and t = t l j 2to the area under the whole curve is easily obtained as Qij2
= 0.5
(21)
Measurement of this area ratio provides a convenient check on the assignment of reaction order. We also have
-ktlj2 = In
y2 = -0.6931
+
(28)
At t = 0 we find y = &[AIo', and at time tllz, corresponding to half the initial ordinate, we obtain
Ql/z = 1 - 2-'/' = 0.2929
(29)
k[A]Otl/z = 2'1' - 1 = 0.4142
(30)
B. Autocatalyzed Reactions. We are concerned here with reactions obeying eq 4,with m > 0. If [PIo = 0 initially, such a reaction can never get started. We therefore postulate the presence of a "seed" of the product P, or alternatively the occurrence of a simultaneous parallel reaction of a type conforming to eq 3, in which k is small. The effect is to get the reaction "off the ground". Autocatalyzed reactions often appear to show an "induction" period during which no apparent reaction occurs: this can be interpreted as a manifestation of the parallel or seeding reaction. The "duration" of the induction period thus depends critically on the concentration of the seed, or the value of k of the parallel reaction. For this reason there is no special significance in the duration of the induction period, at least in our present context. A consequence is that the zero or origin of the reaction time scale can be chosen arbitrarily, and it may be convenient to choose this origin to be different from the actual start of the experiment. In the following we shall choose the time origin in a manner which gives greatest simplicity to the defining equations for the curves. Since only time differencesare to be used for extracting kinetic information, an alternative choice of time origin makes no operational difference. The procedure is similar to the above, eq 8-11. A complete treatment would ideally include all cases (0 6 n 6 2, 'Iz
