THE CONSEQUENCES FOR DIFFERENTIAL THERMAL ANALYSIS OF ASSUMING A REACTION TO BE FIRST-ORDER

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1 THE CONSEQUENCES FOR DIFFERENTIAL THERMAL ANALYSIS OF ASSUMING A REACTION TO BE FIRST-ORDER By E. C. SEWELL Building Research Station, Watford [Read 5th November, 1954] AKt;TRACT From their isothermal experiments, Murray and White drew the conclusion that the dehydration reaoions of clays are first-order with a rate factor of Arrhenius form. The object of this paper is to discuss whether differential thermal analysis can be used to confirm or refute this hypothesis. The problem, previously considered by Murray and White, of a sample heated at a constant rate throughout is recalled. Particular attention is drawn to the effect of heating rate on the temperature at which the maximum rate of reaction occurs. To apply to differential thermal analysis, the treatment must be extended to take account of thermal gradients; as the mathematics involved is cumbersome, only results are considered. The most important result concerns the effects of sample size and dilution on the position of the turning-point. A disagreement with the results of experiments on kaolinite is pointed out; it is argued that only a small part of this disagreement is due to simplifications made in developing the theory, and it is concluded that the hypothesis that the clay reactions are first-order, with a rate factor of Arrhenius form, is only approximate. INTRODUCTION Studies of the dehydration rates of a number of clays at various fixed temperatures made by Murray and White (1949) and more recently by Vaughan (1955) have led them to conclude that the dedehydration reactions obey a first-order law dcl/dt ~ k(t) (l - c0, (1) and that the temperature variation of the rate factor k(t) is determined by an Arrhenius expression k(t)~g exp( - E/(RT)) (2) In these equations, t---time, T=absolute temperature, m---fractional decomposition, R is the gas constant, A a constant, and E an energy of activation. In this paper are described the results of a mathematical investigation undertaken with the object of finding how far the hypothesis that the clay reactions obey the above kind of law is supported by the evidence provided by differential thermal analysis. The basic problem considered is: supposing that a reaction obeys the law expressed by equations (l), (2), what is the shape of the peak to which it will give rise on a differential thermal curve, and how will the temperature at which this peak occurs be affected by using different heating rates or sizes of sample, or by using samples in which the test material is diluted with inactive material? 233

2 234 E.C. SEWELL The foundations for an investigation of this problem were laid by Murray and White, who calculated, first by a numerical method (Murray and White, 1949), and later analytically (Murray, 1950), the rate at which a reaction obeying (1), (2) proceeds in a sample in which the temperature is the same at all points and rises at a constant rate. The curves of da/dt vs. t which they derived have the shape of differential curves, and they showed that heating rate is likely to have a marked effect on the position of peaks. The model considered by Murray and White is, however, too simplified to represent adequately the conditions which arise in differential thermal analysis. In the latter, the furnace temperature is constrained to rise at a constant rate, and, at least when a heavy metal block is used, this may ensure that the surface temperatures of the test and reference samples rise at a constant rate; but the temperature at a given time is not the same at all points within the samples--the temperature at a point inside a sample lags behind that of the surface by an amount which depends on its position. The amount by which the temperature at a point in the test sample lags behind that of its surface varies during a reaction; this is, in fact, the basis of the technique. The results of an attempt to extend Murray and White's work to take account of the varying thermal gradients in the test sample are described in this paper. As the equations involved are non-linear and their solution gives rise' to cumbersome algebra, enunciation of the problem considered is followed immediately by a statement of results, details of the solution (Sewell, 1953) being omitted. It is necessary, 'however, to begin by defining the temperature TM, say, at which the reaction proceeds most rapidly in the problem considered by Murray and White. Temperature at which a reaction proceeds most rapidly in Murray and White's problem.--if we put T=flt, whcre fl is a constant, we have, by (1), (2), de A - E/(RT) dt --/3 e (1 - a). (3) By logarithmic differentiation, d2a I da _ E 1 da [ dt RT a "dt E A - E/(RT) --RT 2 /3 e, by (3). Thus the temperature at which the reaction proceeds most rapidly is given by exp (E/(RTM)) -~ A RTM 2/(/3 E). (4) Figure 1 shows how, according to (4), TM varies with heating rate/3 when A and E are given the values which Murray and White found for Supreme kaolin; the relation between TM and log/3 is approximately linear. In the remainder of this paper, (4) is regarded as defiining a temperature TM as a function of A, E and/3 ; the purpose of introducing

3 D.T.A. OF FIRST ORDER REACTIONS 235 it in terms of Murray and White's problem was to suggest why it is likely to be related to the turning-point temperature of a differential thermal curve. It is worth emphasising that when a temperature gradient is present T~ no longer has the simple physical meaning which can be ascribed to it in Murray and White's problem. Its importance lies in the fact that it is possible to derive a relationship between the quantity TM, the peak height and the measured peak temperature, the latter depending on the dilution and size of sample as well as on the heating rate. A relationship of this type then enables one to express the measured peak temperature in terms of a quantity TM, which, for a given substance (A and E constant), lagma = 7./c Eg / d; O r,, ('c ) C, O S50 L% C o!s ~.'o ;5 io FIG. 1--Relation between turning-point temperature TM and heating rate in Murray and White's problem, using values of A and E found by Murray and White for Supreme Kaolin. Values of TM calculated from Speil's experiments are indicated by X. depends on the heating rate alone. The use of the temperature TM rather than the peak temperature should therefore facilitate comparison of the results of different observers. ENUNCIATION OF DIFFERENTIAL THERMAL ANALYSIS PROBLEM. Because of the difficulty of the problem, a simplified model was considered: it was supposed that: (i) the temperature is the same at all points on the surfaces of the test"and reference samples, and rises at a constant rate; (ii) the samples are spherical; (iii) the junctions of the differential thermocouple are at the centres of the samples; (iv) conduction of heat along the thermocouple wires may be neglected.

236 E.C. SEWELL The first of these suppositions is satisfied almost exactly when a heavy metal block is used, but is only approximately satisfied when a ceramic block is used.

4 236 E.C. SEWELL The first of these suppositions is satisfied almost exactly when a heavy metal block is used, but is only approximately satisfied when a ceramic block is used. Although, in practice, samples are not spherical, it is easy to allow for this fact, and the limitation is not a serious one. The most serious limitation is the neglect of conduction along the wires: in many apparatus this plays an important role (Stegmiiller, 1953, Sewell, 1955). RESULTS A solution of the problem just enunciated, has been used to calculate the shape which the differential thermal curves of kaolinite and mixtures containing it should have, if the hypothesis that the dehydration of kaolinite is governed by equations (1), (2) is correct. Figure 2 shows curves for a sample of kaolinite and for samples containing 50% and 25% of kaolinite heated at 10~ Figure 3 shows curves for the same samples with the differential temperature scale multiplied by factors of two and four for the dilute samples. Figure 4 shows curves for undiluted samples heated at different rates. It may be concluded that, when the curves of reactions obeying (1), (2) are replotted with suitable scales for the differential temperature, their shape is almost independent of the amount of active material, and is only slightly affected by changes in heating rate. Among the more important characteristics of a differential curve, are the temperatures in the test and reference samples and in the block which correspond to the turning-point of the curve. The following simple formulae have been derived for these temperatures: centre of test sample, TM - O. 3 (peak helght) + 0.4f~(O)/(1 + m) ; (5) centre of reference sample, TM +0.7 (peak height)+0"4)fs(0)/(l +~bm)-(a0)o; (6) block, TM +0"7 (peak height) +fs(o)~ /(1 +)~bm) ; (7) where TM is defined by (4); ~(o) --surface minus centre temperature in the test sample before the reaction; (do)o :-: differential temperature before the reaction; r =(peak height)/oo, where 0o-RTM2/E--40~ for the dehydration reactions in clays. TM is, by definition, a function of heating rate only; the effects of thermal gradients are represented by the remaining terms in each of the above expressions. The coefficients 0.7, - 0-3, are exact only for small peaks, but the deviation from these values is so small that it may be heglected. The terms involving fs(0) are not very important: e.g., in the experiments of Robertson, Brindley and Mackenzie discussed below, fs(0) is likely to be- 3~ Comparison of the coefficients rmaltiplying the peak height in (5)-(7) shows that variations in amount of active material should

5 40o I -5 <.! t2_ - IO I-- z 13 D.T.A. OF FIRST ORDER REACTIONS DEGREES CENTIGRADE / 6501 ~ / 237.CUR,CA CE (BI.. OCt~j "BMPERA TU/~ ES. ~. -/5 Cl ~162 \ Ioo~/ "4/ I FIG. 2--Calculated d.t.a, curves for samples of mass ca. "45 g containing 25 %, 50% and 100% kaolinite heated at 10 ~ C/min; plotted on same differential temperature scale. 40oi 450t 50o DEGREES CENTIGRADE E 550, 600 6SO, 7001 SU,~FACs (~LOC,~) T~MP~RA TUf( ES. Fie. 3--Curves of Fig. 2 replotted with differential temperature scales for samples containing 25 % and 50 ~ kaolinite multiplied by 4 and 2 respectively.

6 238 e.c. S~WELL, 0 50~ 6 O0 7O0 5~ rain -I S i -.~0 - i FIG. 4--Thermal curves for undiluted samples heated at different rates. Measurements are in C ~.

D.T.A. OF FIRST ORDER REACTIONS 239 not affect peak temperatures corrected to the centre of the test sample as much as the peak temperatures measured in the block or the reference sample.

7 D.T.A. OF FIRST ORDER REACTIONS 239 not affect peak temperatures corrected to the centre of the test sample as much as the peak temperatures measured in the block or the reference sample. Support is thus provided for the suggestion of Grimshaw, Heaton and Roberts 0945) that corrected temperatures should be used in the correlation of experimental results. Another conclusion which may be drawn from (5) is that the larger the peak, the lower is the turning-point temperature at the centre of the sample; in particular, this turning-point temperature should be higher for dilute samples than for undiluted ones. COMPARISONS WITH EXPERIMENT. Spell (1945) performed a series of experiments in which he heated similar samples of Pioneer kaolin at different rates between 5 ~ and 20~ The formula (7) has been used to derive values of T M from the turning-point temperatures which Spell measured in his nickel block; these values of TM are plotted in Figure I. In these TABLE 1.--Results of experiments on dilution (Robertson, Brindley and Mackenzie. 1954). Clay Cornish kaolinite Iowa hatloysite Active material per cent Peak temp. in sample ~ Peak height ~ 12"7 7"3 3"6 16"0 7"8 4"0 calculations, the terms involving fi(o) were obtained by estimating the diffusivity of the sample; peak heights were estimated semiempirically, from the measured peak area. [To convert areas from mm z to ~ sec. a factor 1-55 was used instead of the factor 2-36 given by Spell because it appears (p. 9) that in calculating this factor Spell erronously used the e.m.f./temperature difference relation for Pt - Pt 10% Rh thermocouples at room temperature instead of at the reaction temperature]. The effect of heating rate on the values of TM as derived from Spell's experiments is in fairly good agreement with that predicted by (4) using Murray and White's values of A and E. No such agreement is found when the predictions of (5)--(7) are compared with the experimental results obtained with dilute samples. In predicting that the turning-point temperature at the centre of the test sample should increase as the fraction of active material in the sample is decreased, (5) contradicts what is accepted to be a general experimental phenomenon, which is typified by the results of Robertson, Brindley and Mackenzie (1954), shown in Table 1 (the peak temperatures were measured in the test sample). These observations are in accord with relations--turning-point temperatures (peak height) ~ for Cornish kaolinite and (peak height) ~ for Iowa halloysite; i.e., instead of the slight increase in peak

8 240 E.C. SEWELL temperature on dilution predicted by (5), there is a large decrease. We are led to ask: does this discrepancy arise because the formula (5) is derived for too simple a model, or must it be concluded that the law expressed by equations (1), (2) does not fully describe the dehydration of kaolinite? Possible causes oj" discrepanc),.--robertson, Brindley and Mackenzie used a ceramic block, whereas the model used in deriving (5) is more analogous to a metal block; but an approximate calculation indicates that for a model representing a ceramic block the coefficient of the peak height in (5) would still be negative. Neglect of conduction along the thermocouple wires is a more serious limitation of the model. Approximate calculations, (Sewell, 1955) supported by Stegmtiller's (1953) experiments, show that in some apparatus the heat so conducted may reduce the peak area to as little as a third of the value it would have if wires of negligible thickness could be used; formulae (5)--(7) cannot be expected to hold very accurately for such an apparatus. The apparatus used by Robertson, Brindley and Mackenzie was of this type, and so a considerable difference between the predictions of (5) and their results was only to be expected. But it is felt that, although taking account of conduction along the wires might reduce the discrepancy, it would not account for the greater part of it; this view is supported by approximate calculations still in progress. It thus seems probable that at a given temperature and degree of decomposition, the dehydration of kaolinite or halloysite really does proceed more rapidly in a dilute sample than in a sample of the pure mineral. This might be explained by the hypothesis that the water vapour pressure in the atmosphere surrounding the sample affects the rate of dehydration. Evidence that water vapour pressure is an important factor at temperatures rather lower than those of differential thermal peaks is provided by the isothermal studies of van Nieuwenberg and Pieters (1929) and Stone's (1952) differential thermal analysis work in controlled atmospheres. Mackenzie's (1954) observation that the temperature of the endothermic peak of kaolinite is higher for nickel blocks than for ceramic blocks, and higher for nickel blocks with lids than for those without, also sup, ports the hypothesis. If the conclusion that the water vapour pressure in the furnace influences peak temperatures is correct, the reason that Spell's experiments were nevertheless found to be in fair agreement with (7) is probably that a water vapour pressure of one atmosphere was built up during each of his experiments. CONCLUSIONS When the consequences of the hypothesis that the dehydration of kaolinite obeys a first-order law (1) with a rate factor of Arrhenius type (2) are developed mathematically for a simplified model of a differential thermal analysis apparatus, the results agree with those of one set of experiments designed to reveal the effects of heating rate, but not with those of experiments showing the effects of dilution.

9 D.T.A. OF FIRST ORDER REACTIONS. 241 Although it is difficult fully to assess the part played by conduction along the thermocouple wires, it seems doubtful if the discrepancy can be attributed to the simplified nature of the mathematical model. It is more likely that equations (1), (2) cannot be regarded as more than a first approximation to the laws describing the dehydration of kaolinite. It is possible that considerable light would be thrown on the causes of the discrepancy pointed out in this paper if more differential thermal curves were obtained for diluted and undiluted samples in an atmosphere the water vapour pressure of which was carefully controlled. Acknowledgements.--The mathematical investigations of which the results are described in this paper were carried out as part of the programme of the Building Research Board of the Department of Scientific and Industrial Research and this paper is published by permission of the Director of Building Research. The author is indebted to Mr D. B. Honeyborne for his comments and suggestions. REFERENCES Grimshaw, R. W., Heaton, E. and Roberts, A. L Trans. Brit. Ceram. Soc., 44, Mackenzie, R. C. I954 Nature, 174, Murray, P. and White, J Trans. Brit. Ceram. Soc., 48, Murray, P~ 1950 Thesis, Sheffield. van Nieuwenberg, C. J. and Pieters, H. A Rec. Tray. Chint. Pays-Bas, 48, Robertson, R. H. S., Brindley, G. W. and Mackenzie, R. C Amer. Min., 39, Sewell, E. C Research Note, Building Research Station, D.S.I.R. Sewell, E. C Research Note, Bui!ding Research Station, D.S.I.R. Speil, S U.S. Bur. of Mines tech, pap. 664, Washington. Stegmiiller, L Spreechsaal, 86, 1-8. Stone, R. L J. Amer. Ceram. Soc., 35, Vaughan, F Clay Min. Bull., 13. DISCUSSION Mr D. B. Honeyborne said that from conversations he had heard during the tea interval he gathered that some misunderstanding had been caused by the fact that in the figures peak temperatures decreased on dilution, whereas Mr Sewell had said they should increase. Mr Sewell stated that in the figures shown, temperature was supposed to be measured in the block; the appropriate formula for peak temperatures was (7), which did predict a decrease on dilution. The subsequent discussion of the effects of dilution concerned peak temperatures at the centre of the sample, for which (5) which predicted an increase on dilution, was the appropriate formula. Mr F. Youell wished to point out that weight loss data could give rise to curves resembling first order reactions, when such was not actually the true state of affairs. Dr P. Murray emphasised the need for more experimental work in this difficult field with especial need for relation between kinetic, d.t.a. and weight loss data with varying particle size and packing densities. Mr R. HI. Nurse also wished to point out that particle size was a very important variable in evaluations in this field of research. Dr J. White stated that because a reaction was first order this did not of necessity mean that such a reaction was unimolecular.

THE ROLE OF WATER VAPOUR IN THE DEHYDROXYLATION OF CLAY MINERALS

THE ROLE OF WATER VAPOUR IN THE DEHYDROXYLATION OF CLAY MINERALS By G. W. BRI~qDLE~C and M. NAtOa~RA Department of Ceramic Technology, The Pennsylvania State University, University Park, Pa., U.S.A. [MS.