2863
Anal. Chem. 1984, 56, 2863-2867
Determination of Arrhenius Kinetic Constants by Differential Scanning Calorimetry Jan C. M. Torfs,* Leen Deij, Anthonie J. Dorrepaal, and Josef C. Heijens
Dow Chemical (Nederland) E. V., Terneuzen, The Netherlands
A method to obtain Arrhenius kinetic constants of chemlcai reactlons by differential scannlng calorimetry (DSC) has been developed. The results of a temperature-programmed DSC scan and an isothermal test are combined to yield the desired kinetlc constants with a relative standard devlation of 4 %, Peroxides and an epoxide have been tested by this new method and were compared wlth a method based on variable heating rates. The latter procedure was found to be more time-consuming and generally less accurate than the disclosed method. The presently developed very rapid method yielded kinetic constants whlch were in excellent agreement wlth data obtained from various other tests.
It has been stated by various workers (1,2)that differential scanning calorimetry (DSC) offers a great potential for the determination of the kinetic constants of chemical reactions. Well-established methods to study the kinetics of reactions are based on the determination of the amount of reactant decomposed after various intervals of time. These experiments must be carried out isothermally at a range of temperatures and are therefore rather time-consuming. The DSC technique has the advantage of being very rapid and versatile. A single, temperature programmed DSC experiment can be used to compute the Arrhenius preexponential factor, activation energy, and order of the reaction within a few hours (3,4). Other thermal methods, such as isothermal and adiabatic storage tests (5, 6) and accelerating rate calorimetry (ARC), which employs a nearly adiabatic calorimeter (7), also yield kinetic data, but they are more time-consuming than DSC. A major problem with DSC is, however, that there is much confusion in literature about the accuracy of the obtained kinetic constants (8). Large discrepancies were found between the results of various kinetic methods based on DSC (9). Some authors (5, 6, 10-12) preferred the heat evolution method, which is based on the shape of a single DSC curve, whereas others (1,13) found an approach using a series of DSC experiments a t various heating rates much more satisfactory. The reason for these discrepancies seems to be the approximative character of the various calculation procedures and the presence, in some cases, of side reactions and other thermal phenomena such as melting, evaporation, etc. occurring simultaneously with the investigated reaction. In our opinion the solution to these problems can be found in the use of exact computer calculations and in testing a substance under various experimental conditions, which generally reveals the presence of interfering phenomena. The objective of the present study was to compare the kinetic constants obtained by using DSC with the results of well-established techniques, in order to assess the reliability of kinetic data obtained by DSC. EXPERIMENTAL SECTION DSC experiments were carried out with the Du Pont DSC controlled by the Du Pont 1090 thermal analyzer. Data are transferred to a HP 9835 A desk-top computer via the RS-232C serial interface of the 1090. In most cases large volume capsules (LVC) supplied by Perkin-Elmer were used. Loading about 5 mg
of sample in these crucibles was generally done in a nitrogen chamber, since it was found that the presence of oxygen may affect the degradation pattern. LVC sample pans can withstand pressures up to 24 bar and temperatures up to 300 "C. Setaram (France) supplies DSC crucibles which can be used up to 500 "C and 100 bars. In a number of experiments cold-welded aluminum crucibles supplied by Du Pont were employed. Calibration of the temperature and heat flow measurements with the DSC was carried out by using high-purity metals indium and gallium. The isothermal tests were usually carried out by aging test materials in the LVC Capsules which could afterwards be run in the DSC. Only glycidol, which boils at 57 "C, was aged in a steel bomb having a larger volume. The air was removed by applying vacuum previous to heating the bomb to the temperature of aging. PROCEDURES TO DERIVE KINETIC INFORMATION FROM A DSC EXPERIMENT Heat Evolution Method. In the DSC the heat flowing from a sample to an inert reference is measured. Figure 1 is a schematic representation of the DSC trace of an unstable substance. Baxter (14) has shown that the sensitivity of the Du Pont DSC is independent of sample size and thus also independent of heat flow. Moreover the high resolving power of this device was illustrated in his work. Therefote, if samples (below 20 mg) not too large and heating rates not greater than 20 OC/min are employed, the heat flow signal is as a good approximation directly proportional to the instantaneously evolved heat. Accordingly, the instantaneous reaction rate can be derived from the measured heat flow. Under these conditions the reaction rate constant, k , at any temperature, TI, may be calculated from eq 1 (10) which is similar to that employed by Borchardt and Daniels (1, 5). b=-
dH
1
In eq 1 dH/dt is the heat flow (more exactly the heat flow deviation from the base line), AHbt is the total heat of reaction evolved, which can be derived from the integral area of the reaction exotherm in Figure 1, AHrestis the reaction heat evolved above a temperature T, (see Figure l),and n is the order of the reaction. The reaction rate constant k is computed by using eq 1for n = 0, 1,2, and 3. This yields a number of Arrhenius plots of which the one corresponding to the true reaction order should be linear (7,15). Figure 2 is an example of such a plot of log k vs. the reciprocal temperature for the degradation of tert-butylperpivalate in the DSC. The most perfectly linear curve is selected. However, since none of the Arrhenius plots are really straight, the selection of a best linear curve can be complicated in some cases. The determination of the preexponential factor, activation energy, and order of reaction from a single DSC run can be very difficult because all these constants are interrelated (8): several triplets of constants can be found which describe the DSC curve reasonably well. In our opinion, accurate determination of reaction order is best done from a series of DSC experiments at various concentrations. The dependence of reaction rate, at a certain temperature, on concentration yields reaction order, as is described in standard textbooks on kinetics (16).
0003-2700/84/0356-2863$01.50/00 1984 American Chemical Society
2864
0
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
~~~~
TI
~~
Temperature T
Flgure 1. Schematic plot of a DSC curve of an unstable material. The area between the curve and the broken base ling represents the heat of reaction, AHtot, and the fraction of material, which has not yet reacted at temperature T , , is related to AHrest.
Indeed, some authors employed more sophisticated base lines than the straight line used here (4, 17). Variable Heating Rate Method. The ASTM-E698 procedure was followed. This DSC method which is based on Ozawa's work (11) measures exothermic peak maximum temperature variations with changes in linear programmed DSC heating rates, between 1 and 20 OC/min. The temperature shift of the peak maximum is used to calculate the kinetic constants. Isothermal Check. In order to conduct a quick check on the obtained constants, the isothermal test described in ASTM-E-698 was used. This test amounts to aging a sample in an oven for a predicted half-life time and then measuring in the DSC whether the remaining heat of reaction, AH,is, as expected, 50% of the original value, A",. The test result can also be employed to compute the reaction rate constant, k , at the temperature of aging. The equation for a first-order reaction is
k = [In (m~/AfOlt
(3)
where t is the duration of aging. RESULTS OF KINETIC MEASUREMENTS tert -Butyl Perpivalate (TBPP). Heat Evolution Method. TBPP is selected because it has been thoroughly characterized by well-established tests (5, 6). Using the heat
Figure 2. Arrhenius plot of 30 % TBPP in mineral oil, calculated from DSC data according to eq 1, for various reaction orders, indicated in the figure. Broken curve is the best linear fit to the data assuming an order of 1.
In many cases, however, the order of the reaction is known from other sources. Most diluted peroxides, as used in this work, decompose following a first-order mechanism. After the order is assessed, the straight part of the selected curve in Figure 2 is obtained. This is done by eliminating data points a t the high- and the low-temperature side, while the correlation coefficient of a linear least-squares fit is followed. This coefficient increases at first because nonlinear parts are discarded, until the noise on the heat flow data causes a decrease of the correlation again. The relative importance of this noise increases with the decreasing number of data points. This procedure has the advantage that it is reproducible and independent of the operator's skill. The kinetic parameters of the substance, Le., the preexponential factor, A , and the activation energy, E, which are related to the rate constant, k , by the Arrhenius equation
k = A exp(-E/RT)
(2)
can be derived from this plot. In eq 2 R is the gas constant. Figure 2 shows also the best linear fit. In the range from 90 to 120 "C the agreement is good. This portion of the curve is employed to calculate kinetic constants. The nonlinearity outside this range may be due to poor base line selection.
evolution method we calculated an activation energy of 123 kJ/mol and a frequency factor, In A, of 37.6 (min-l). This agrees extremely well with the results of isothermal and adiabatic tests: 123 kJ/mol and 37.7 (min-l). Also the ARC results agree within 3%, as is shown in Table I. The kinetic constants obtained with the heat evolution method are practically independent of the heating rate used in the DSC, as shown in Table 11. Variable Beating Rate Method. Table I11 presents the peak temperatures obtained a t various heating rates, for the degradation of T B P P measured in cold-welded aluminum crucibles. The repeatability of the peak temperature measurement is good: the standard deviation was found to be about 0.5 "C. The plot of log (heating rate) as a function of the reciprocal peak maximum temperature, as required in the ASTM procedure, was found to be in line with the anticipated linearity, in the range from 0.5 to 20 OC/min. An activation energy of 114 kJ/mol and a frequency factor, In A (min-l), of 34.9 were obtained. From the point of view of the DSC technique, these results are not entirely satisfactory, because it was found that part of the sample evaporated during the DSC experiment. Weighing crucibles including the sample before and after the DSC experiment indicated a weight loss of about 13%,for a solution of 24.3% T B P P in mineral oil. Therefore, the experiments were repeated using pressure-tight crucibles. The peak temperatures found for the various crucibles are presented in Table I11 together with some data regarding the crucibles. It is clear that the peak temperatures depend on the type of crucibles employed. This dependency leads to different values for the activation energy E, which amounted to 114,103, and 80 kJ/mol for the various sample containers. The reason for these variations is not completely clear but may well be associated with the variation in the masses of crucibles. Their weights range from 55 to 1050 mg. when the weight is increased the activation energy decreases and simultaheously the calibration factor for the measurement of heat flow increases, indicating that heat losses increase from about 0% to 20%. These heat losses might cause the deviation of the results from the expected activation energy value. Nevertheless, experiments with the relatively heavy LV crucibles led to correct results when using the heat evolution principle, and the results did not depend on the heating rate
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
2865
Table I. Kinetic Coostants of the Arrhenius Equation (eq 2) far the Decomposition of tert-Butyl Perpivalate (TBPP), tert -Butyl Perbenzoate (qBPB), Di-tert -butyl Peroxide (DTBP), and 2,3-Epoxy-l-propanol (Glgcidol) as Obtained by a Number of Techniques and Methods"
DSC, heating, rate, %/min
E, kJ/mol
In AI, min-'
30% 25%
in mineral oil in mineral oil; variable heating rates solution in mineral oil pure
10 2, 5, 10, 20
123 103 124.1 123
37.6 31.5 38.1 37.7
DSC DSC ARC is0 and adiab. tests (12)
6.4% in trimethylbenzene 10% in mineral oil 10% in trimethylbenzene
2 10
152.0 171.2 149.2 144.7
41.9 48.1 40.9 39.7
DTBP DTBP DTBP DTBP DTBP DTBP DTBP
DSC DSCb DSCb ARC ARC ARC' chem anald
30% 15%
in mineral oil in mineral oil variable heating rates, 15% in mineral oil 20% in mineral oil 20% in toluene 10-100% in toluene (results are averaged) various
10
154.6 153.8 129.1 166.0 158.3 158.9 158.3
38.9 39.0 32.4 42.9 41.2 41.2 40.5
glycidol glycidol glycidol glycidol g1y cido1
DSC DSC DSC DSC ARC
pure pure pure variable heating rates; pure pure
5
85.8 77.1 77.5 65.0 84.0
19.5 17.6 17.6 14.1 18.8
material
technique
conditions
TBPP TBPP TBPP TBPP
DSC DSC ARC is0 and adiab. tests (3)
TBPB TBPB TBPB TBPB
pure 10 2, 5, 10, 20
10 20 2, 5, 10
E = activation energy. In A = the natural logarithm of the frequency factor. In the DSC work carried out in our lab stainless steel LV capsules were used. bReference9; Perkin-Elmer DSC-2 apparatus was used. 'Reference 7. Reference 20; an average of a large number of measurements was made.
Table 11. Influence of Heating Rate in the RSC on the Kinetic Constants of tert-Butyl Perpivalate (15% Solution in Shell Sol1 J',' Supplied by Shell), Using the Heat Evolution Method
heating rate, K/min
activation energy E, kJ/mol
preexponential factor In A , min-'
2 10 20
128.0 122.6 122.5
39.27 37.31 37.31
____
?SC
Ildrldble
-.earin9 P d r e I
CSC ' H e d t evoluilonl
=
A D I A B A T I C 4ND ! S O T I E R H A L TESTS
I
ARC 0
DSC I Iiothermd' check I
Table 111. Peak Temperatures Found for the Degradation of 24.3 wt % of TBPP in Mineral Oil, Using Various Sample Crucibles"
heating rate, OC/min 0.5
1 2 5 10 20 50
activation energy E , kJ/mol wt of sample crucible, mg DSC calibration factor (at 10 OC/min)
stainless aluminum steel stainless cold-welded LVC steel (Du Pont), (Perkin- (Setaram), OC Elmer), "C OC 88 94 101 110 117 125 138 114 55 1.0
104 114 121 130
90 98 112 123 128
103
80
320 1.15
1050 1.27
"Also some physical data of the crucibles are given. Peak temperatures are corrected for thermal lag and heat rate as described in ASTM-E-698-79. (Table 11). The most clear comparison of all data results from the inspection of the Arrhenius plots of Figure 3 which are calculated (eq 2) by using the kinetic constants derived from the various methods. The data agree well over a broad temperature range except for those obtained by the variable heating rate method (LV crucibles). Isothermal tests carried
2.4
2.8
3.2 i(10-3K-l
I
1
Figure 3. Arrhenius plot of TBPP showing the results of various techniques (calculated by using eq 2).
out in our laboratory confirmed the heat evolution method in DSC and agreed with ARC and literature values (see Figure
3) * Since the method of variable heating rate led to erroneous kinetic constants, one of its basic premises, viz., the assumption that the peak maximum represents a point of constant conversion independent of heating rate, was investigated. By partial integration it was found that the conversion at the peak maximum was ranging from 56% to 61% €or six experiments in the range from 5 to 20 K/min. No systematic variation was found. These variations were found to account for at most a 1-K variation in peak temperature and cannot completely explain the too low activation energy found with the variable heating rate method. tert-Butyl Perbenzoate (TBPB). The kinetic constants for the decomposition of TBPB, obtained by DSC, ARC, and a combination of isothermal and adiabatic tests (5,6,18,19), are given in Table I. Most results for the activation energy are scattered around 150 kJ/mol (A5 kJ/mol). A DSC run obtained at 10 OC/min
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
2866
Table IV. Repeatability of Kinetic Constants, and of Reaction Rates Calculated from These Constants, for Di-tert-butyl Peroxide (30% in Mineral Oil) Obtained by DSC Using the Heat Evolution Method"
no.
E , kJ/mol
std dev re1 std dev, %
157.69 166.85 147.90 154.21 156.61 156.49 151.96 152.73 154.6 6.0 4
1 2 3 4 5 6 7 8 mean
kinetic constants In A , min-' 39.73 42.12 37.23 38.86 39.49 39.46 38.27 38.46 38.9 1.6 4
k200, min-'
J/g
1234 1165 1309 1308 1306 1221 1205 1283 1261 56 4
rate at various temp k100, min k20, lo-" m i d
0.69 0.73 0.68 0.70 0.72 0.71 0.69 0.68 0.70 0.02 2.6
1.4 0.8 2.8 1.9 1.6 1.6 2.1 2.0 1.78 0.58 33
1.4 0.4 6.3 2.5 1.7 1.8 3.4 3.0 2.7 1.9 68
"The obtained activation energy, E, and preexponential factor, A , are used in the Arrhenius equation (eq 2) to calculate the rate constant, AH is the heat of reaction, based on pure peroxide.
12, at various temperatures.
n
s
n 0
1
L
___ 2
O
-- -
- _ ... -. . .
_ . . . ...._-. --
~
-
-
-4~ 100
-~
-
200
300 Temperature ('C
Figure 4. Arrhenius plot of TBPB showing the results of various measurement techniques (calculated by using eq 2).
-___
JSC
r r d r l a t , e "ea1 r g "$:el
I S C ,Heel B V O . ~ L 3
yields a 10% higher value of 171 kJ/mol. This might be , found that at high related to the remarks of Duswalt ( l )who heating rates runaway conditions may be met, leading to erroneous results. The DSC obtained at 2 OC/min resulted in a value of 152 kJ/mol. Figure 4 reveals that the reaction rates obtained by ARC and DSC (at low heating rate) correspond well with results from extensive testing by isothermal and adiabatic methods (5,6,18,19). An isothermal check by DSC also confirms the results (see Figure 4). Di-tert-butyl Peroxide (DTBP). DTBP has also been tested because it is used as a model peroxide, and it has been the subject of many kinetic studies, reviewed by Shaw and Pritchard (20). Figure 5 is a DSC plot of the decomposition of DTBP. The Arrhenius constants obtained by the various methods are presented in Table I. DSC of a solution in mineral oil, using the heat evolution method, yields an activation energy of 154.6 kJ/mol. A nearly eqilal value of 153.8 has been found elsewhere (9). In line with the results for TBPP, the method of variable heating rates yields a much lower value of 129.1 k J / mol. Activation energies obtained by ARC are in the range 155-166 and confirm DSC results obtained by the heat evolution method. Based on a wide range of measurements (20) a value of 158 was found, in good agreement with the DSC and ARC data. This indicates that the anomaly at 150 "C (Figure 5) in the decomposition of DTBP does not lead to a significant inaccuracy in the kinetic data. Figure 6 shows the Arrhenius plots corresponding to some of the results. Only the slope of the curve corresponding to the DSC method of
I
Figure 5. DSC of a 30 wt % solution of DTBP in mineral oil (heating rate, 10 OC/min). ~ 1
A O l t B A T ! C hN3 ISCTHLRHAL TESTS ARC
~ _ _ _ _ _ 2.0
2.5
3.0 1(!0-31-11
Flgure 6. Arrhenius plot of DTBP showing the results of various measurement techniques (calculated by using eq 2). variable heating rates deviates significantly from the majority of the data. This is evidently due to the low activation energy found with this method (see Table I). Table IV illustrates the good repeatability of the DSC results. Relative standard deviations on activation energy, E , preexponential factor, In A , and reaction enthalpy, AH,are all 4% for DTBP. For the degradation of T B P P a value of 3% was found. The rate constant of DTBP at 200 "C, as calculated from the Arrhenius constants, displays a relative standard deviation of 2.6%. Upon extrapolation to low temperature, Le., outside the temperature range of the DSC measurement, the uncertainty naturally becomes greater. The relative standard deviation amounts to 33% and 68% at 100
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
activation energy of glycidol is very close to 80 kJ/mol. Again the activation energy obtained by the variable heating rate method, 65 kJ/mol, seems to be too low. The results of this work suggest strongly that the variable heating rate method should be used with considerable caution since it led in all investigated cases to values of the activation energy which were 10-25% too low. This conclusion is supported by the work of Barendregt and Verhoeff (5,6,18,19) and by that of Grentzer et al. (21). The reason may well be the strong dependence of the activation energy value on small variations in measured peak temperature values. Moreover, the method is rather time-consuming, since it requires a range of DSC experiments of which some use a low heating rate.
ARC 0
DSC ! !50Lhermdl Check 1
-3
2.40
2867
ACKNOWLEDGMENT
3-20
i,10
-3K
-'
I
Figure 7. Arrhenius plot of glycidol, allowing a comparison of various measurement techniques (calculated by using eq 2).
and 20 "C, respectively. When it is realized that extrapolation to 20 "C involves a reduction of reaction rate over a factor of 10" the obtained uncertainty seems well acceptable. Prolonged isothermal testing at lower temperatures can be used to decrease the uncertainty, which is merely limited by the time available to the analyst. Glycidol(2,3-Epoxy-l-propanol). The kinetic constants for the decomposition of glycidol are given in Table I. ARC results are in line with DSC recorded a t 5 "C/min. DSC recorded at 10 and 20 "C/min yields a lower activation energy value. DSC using the method of variable heating rates leads to an activation energy which is completely out of the range found with ARC and DSC. Figure 7 shows the corresponding Arrhenius plots. All methods agree well in the temperature range of 140-260 "C, where the measurements have been carried out. Accordingly the reaction rates are most accurate in this range, and deviations come about upon extrapolation to higher or lower temperatures. If accurate kinetic constants for this substance are required, some additional tests should be carried out, preferably in a wide temperature range. A substance can be made to decompose in the DSC a t low or high temperatures by using slow or fast scan speed, respectively. As a first attempt to improve accuracy an isothermal test was carried out at 105 "C, Le., in the extrapolated low temperature range. Aging for 21 h a t this temperature led to a reduction of the heat of reaction to 47%. According to eq 3 this reduction corresponds to a rate constant of 6.1 X min-'. This result, indicated in Figure 7, is reasonably in line with DSC and ARC data. This allows us to conclude that the
H. Loonen and F. Pijper are acknowledged for their experimental assistance. R. Stehl, W. Graf, T. Klingler, J. Ravenstijn, E. Davies, P. Shepherd, A. Oomens, T. van Sint Fiet are acknowledged for critical reading and discussion of the manuscript. Registry No. TBPP, 927-07-1; TBPB, 614-45-9; DTBP, 110-05-4; glycidol, 556-52-5. LITERATURE CITED Duswalt, A. A. Thermochim. Acta 1974, 9 , 57. Ozawa, T. Netsu Sokutei 1977, 4, 45. Kah, A. F.; Koehler, M. E.; Grentzer, T. H.; Niemann, T. F.; Provder, T. ACS Symp. Ser. 1982, No. 197, 297. Schlichenmaier, V.; WMmann, G. Thermochim. Acta 1977, 21, 39. Barendregt, R. B.; Verhoeff, J.; Van den Berg, P. J. "Thermal Analysis"; Birkhauser Verlag: Basel, Boston, 1980; Voi. 1, p 105. Barendregt, R. B., Ph.D. Thesis, Prim Maurits Laboratory of TNO, RijswiJk, The Netherlands, 1981, Tou, J. C.; Whitlng, L. F. "Thermal Analysis"; Birkhauser Verlag: Basei, Boston, 1980; Vol. 1, p 177. Arnold, M.; Veress, G. E.; Paullk, J.; Paulik, F. "Thermal Analysis"; Birkhauser Verlag: Basel, Boston, 1980; Vol. 1, pp 89-73. McDonald, R. A.; Steiner, E. C., The Dow Chemical Co., Midland, MI, unpublished results, 1982. Carroll, 8. Thermochim. Acta 1972, 3 , 449. Freeman, E. S.; Carroll, B. J. Phys. Chem. 1958, 62,396. Borchardt, H. J.; Daniels, F. J. Am. Chem. Soc. 1957, 79, 41. Orawa, T. J. Therm. Anal. 1975, 7 , 601. Baxter, R. A. "Thermal Analysis"; Academic Press: New York, 1969. Operating manual of: "Accelerating Rate Calorimeter"; Columbia Scientlflc Industries Corp: Austin, TX, 1979; Vol. 1, pp 65-84. Glasstone, S.; Lewis, D. "Elements of Physical Chemistry"; MacMillan: London, 1976. Heuvel, L. M.; Lind, K. C. J. H. Anal. Chem. 1970, 42, 1044. Verhoeff, J., Ph.D. Thesis, Prins Maurlts Laboratory of TNO, Rijswijk, The Netherlands, 1983. Verhoeff, J., personal communication, 1983. Shaw, D. H.; Pritchard, H. A. Can. J. Chem. 1988, 46, 2721. Grentzer, T. H.;Hoisworth, R. M.; Provder, T.; Kline, S. Org. Coat. Piast. Chem. 1981, 44, 673.
RECEIVED for review January 17, 1984. Resubmitted and accepted July 30, 1984. Thanks are given to The Dow Chemical Co. for permission to publish this work.
