965
Reaction Kinetics and Differential Thermal Analysis
Reaction Kinetics and Differential Thermal Analysis Ralph T. Yang” and Meyer Steinberg Department of Applied Science, Brookhaven National Laboratory, Upton, New York I 1973
(Received October 2, 1975)
Publication costs assisted by Brookhaven National Laboratory
Quantitative relationships between chemical kinetics and DTA curves are studied for the reactions which follow the general rate expression: r = rOe-E/RT(l - 2). Both the frequency factor and the activation energy can be derived from a single DTA curve. From the known kinetic parameters, the DTA curve can also be predicted. The rather simple models developed in this work have been tested by the oxidation reaction of nuclear graphite with C o n and by data in the literature on the thermal dehydration of clays. The results are satisfactory considering the complexities involved in DTA.
Introduction Considerable importance and interest exist in the study of reaction kinetics by differential thermal analysis (DTA). Since the classic work of Murray and White on clay dehydration kinetics,2 DTA has been used to characterize complex chemical reactions, such as the thermal dehydration and decomposition of solid compounds3-7 and homogeneous reaction systems.s The method of Borchardt and Daniels was later extended to heterogeneous r~actions.~-ll I t has not been possible, however, to obtain both the activation energy and frequency factor from a single DTA curve for a heterogeneous gas-solid reaction, and to predict the DTA curve from the known kinetic parameters for the same reaction system. Most of the previous work was concerned with the dehydration and decomposition of minerals with the reaction form: A (solid) B (solid) t C (gas). The order of the reaction is defined as the exponent n in the following rate expression:
Theoretical Considerations
Here r is the rate and is expressed as dx/dt; t is the time; ro the frequency factor; E the activation energy; R the gas constant; T the absolute temperature; and x the fractional completion of the reaction. For heterogeneous gas-solid reactions of the form A (solid) B (gas) C (gas), such as carbon oxidation reactions, the order of the reaction is conventionally related to the dependence of rate on the concentration of the gaseous reactant. The rate of a carbon oxidation reaction ( r ) is normally expressed as -(l/m)(dm/dt), here m is the instantaneous mass of the solid. It can be shown that this expression is identical with (dx/dt)(l - x ) - l by using the relation: 3c = 1 -(m/mo), here mo is the initial mass. When the rate is independent of the burn-off ( x ) , the rate expression is identical with that of the “first order” reaction in the previous case (with the concentration of the gaseous reactant constant). It should be noted here that the quotation mark is used here to differentiate it from the conventional meaning of the order of a gas-solid reaction. The scope of this work is to ( 1 ) calculate both frequency factor and the activation energy from a single DTA curve and (2) predict the DTA curve, or a t least its shape, from known kinetic parameters. The type of reactions considered here is of the general form as in eq 1 with n = 1.
E = ___ (4) k2 RTm2 where k2 is the heating rate. The temperature a t the inflection points (Ti), i.e., at the maximum and minimum slopes, can be obtained by setting d2(AT)/dT2= 0 and solving for Ti. The following relationship is obtained:
-
+
-
In .a DTA curve, the temperature differential (AT) at any given instant can be approximated as being proportional to the instantaneous mass reacting rate:
AT = kly
(2)
Here y is the instantaneous mass rate (=rmo) and K 1 is a constant. The equation is approximate because k l is affected by the heat and mass transport processes which do not remain strictly constant during the reaction (the term reaction is used to include all chemical changes). However, for the case in which only the instantaneous thermal effect is detected and there is no accumulation of the effect, k l is equal to the heat of reaction divided by the total heat capacity of the sample mass. For the “first order” reactions
AT = klrOe-E/RT(l - x)mo
(3)
The peak temperature of a DTA curve (T,) is obtained by simply setting d(AT)/dT = 0 and solving for T,. The following relationship was obtained and tested satisfactorily by Murray and White:2
0e-E/RTm
In eq 5, there exist two solutions for Ti (Til and Tiz) which correspond to the two inflection points. E and ro can be calculated by solving eq 4 and 5 simultaneously with T , and either one of the Ti’s, or by solving eq 5 with Til and Tiz. The calculation is, however, quite tedious. In eq 5, approximation for Til (the first inflection point) can be made by dropping the second term in the left-hand side because it is normally quite small compared to the first term for Til. Thus, one obtains
Ti in eq 6 refers to Til only. By knowing T, and Til, eq 4 The Journal of Physical Chemistry, Vol. 80. No. 9, 1976
Ralph T. Yang and Meyer Steinberg
966
and 6 can be easily solved simultaneously for E and rg. Examples will be provided of this method for the kinetics of C COz reaction and dehydration of kaolinite and halloysite. To predict a DTA curve, one has to be able to express ,AT as a function of T. This can be achieved only when the functional form of x(T) is established. We start with
10-1
I
I
I
I
I
I
1200 'C
+
I
or
-dx 1-x
- '0 e-EIRT
dT
kz
1000 O C
(7)
Here t is the time. Upon integration of eq 7 , x(T) can be obtained. However, because normally EIRT >> l, the integration of Je-E'RT d T is the most troublesome task. This integration has been attempted, and the following substitution has been u ~ e d : ~ J ~
Here u = EIRT. The function JI(e-ulu) du has been tabulated for E = 10 - 66 kcallmol and T = 100 - 430 OC.I An approximate solution for the same function was used by Murray and White2 in their calculation of the percentage reaction at the peak DTA temperature. The approximate solution was taken as the first two terms of the Taylor series expansion of e-Ulu. The two terms involve the second and third powers of RTIE. It is nevertheless, more desirable to.obtain an analytical solution or approximation in a simpler form for it not only simplifies the calculation but also characterizes the actual processes. In order to achieve this, we adopt the "method of expanding the exponent" first suggested by Frank-Kamenetskii in developing his theory of combustion.13 The argument of the integrand is approximated as the following:
E, 10-3 W
0
a
0
i --
-
are three factors in this equation: the first one is the Arrhenius factor; the second and third describe the decrease of the reacting material with the linearly increasing temperature. It is also interesting to note that eq 10 can be used to calculate E and ro with two sets of corresponding x and T values, which are obtained with the simultaneous DTAITG apparatus. Applications of the method will be exemplified.
+
IfTo Here 6T = T - To and To can be chosen arbitrarily, provided it is near the reaction range so that 6T/To is small. The approximation is better at higher temperature ranges and is in error by less than a few percent for all practical DTA measurements. Equation 7 can then be integrated conveniently with the boundary condition: T = 0, x = 0, and the solution substituted in eq 3. The following results are obtained:
Here
kq = -RTo2 e-2E/RTo k2 E The lower integration limit of T was chosen to be zero to simplify the solutions. When T is small, the approximation in eq 9 is poor but since EfR >> T , the integrands, both the original and the approximated, are negligibly small. Equation 11 can be used to predict DTA curve. There The Journal of Physical Chemistry, Vol. 80, No. 9, 1976
-
Experimental Section The kinetics and DTA curve of the reaction C C02 2CO were obtained by using a Mettler differential thermoanalyzer Model TA-1. Reaction rates were measured gravimetrically (TG), and are expressed as -(llm)(dmldt) with the units of min-l. Sensitivity of the balance was f3 x 10-6 g. The carbon sample used was pile-grade nuclear graphite. Ash content of the graphite was below 0.01% and its major constituents were less than 0.001% of Si, Ag, and Cu and less than 0.0001% of Mg as detected by emission spectroscopy. Particle size of the sample used in all the experiments was in the range 590-840 ym. The COz and argon gases used were originally bone dry and prepurified grades, respectively, and were supplied by Liquid Carbonic Co. The gases were purified from traces of water and oxygen by passing through a train of silica gel, Mg(C10& (both at room temperature), and copper turnings held at 500 O C . Gas flow rate was maintained constant and was about three orders of magnitude greater than could be consumed at the maximum reaction rate. Same samples and flow conditions were employed in both T G and DTA measurements. In the DTA curve measurement, 20 mg of graphite was mixed with alumina (grade A-10, Alcoa, -50 200 mesh) and diluted to 10%. The reference contained 200 mg of alumina. It is also noteworthy that the thermocouple wells were immersed in the centers of the materials so that any
+
967
Reaction Kinetics and Differential Thermal Analysis
1 \ \,~11 H EA X LPE LO R YIMSEI TNETA , L
-
I0-5
I
t
10-4
6.5
7.5
1.2
1.1
8.5
(I/T)x103,
I / T X 104;~-1
Figure 2. Arrhenius plot
of C 4- Con.
Figure 3. and 4).
1.3 ( O K
)-I
Rates of dehydration of kaolinite and halloysite (from ref 3
TABLE I: Kinetic Parameters for Some Reactions I
Predicted from, eq 4 and 6 Reaction C + C02
2co
-
Dehydration of Georgia kaolinite Dehydration of Eureka halloysite
E,a kcal/ mol 72
ro, l/s
3.25 X lo7
I
I
I
I
0
-" 0 0
Expt value E , kcal/ mol 71.16
r
w I 0.5-
ro, l/s Source
35
1.23 X lo6
34.97
3.32 X This 107 work 1.29 X 3,4 106
36
2.05 X lo7
35.85
1.66 X 107
-
LL
+ 0
-
EXPERIMENTAL
-
D Z
- 1.0w
-
7 1.5
3,4
Two significant figures are derived by considering the error limits of T, and Ti of h0.5 "C. a
thermal effect could be detected promptly. Platinum and platinum-10% rhodium thermocouples were used. In a typical rate measurement, between 10 and 20 mg of graphite was used. The sample particles were spread into a layer of one to two particles thick on a flat disk sample holder made of alumina or platinum. The thermocouple joint was, in this case, underneath the disk and was in contact with its surface. The sample was heated to the desired reaction temperature in a stream of argon a t a rate of 10 OC/min. The purified C02 was then introduced and reaction rates were calculated from the DTG curve (time derivative of the weight). Reaction rates of nuclear graphite + C02 were measured a t six temperatures ranging from 870 to 1200 "C. No difference was observed between the rates obtained with the
I
1000
Figure 4. DTA
rate).
I 1100
I
1200 T, "C
I 1300
I 1 1400
curves of nuclear graphite in COP (10 OC/min hbating
platinum and the alumina holders. The rate was plotted against burn-off of carbon and shown in Figure 1. As seen in this figure, the rate was quite constant throughout the reaction. One is then justified in assuming that the reaction is approximately "first order". An Arrhenius-type plot of the rates a t 20% burn-off (Figure 2) shows that the reaction was within the chemical controlled region and diffusional effects were negligible.14 Also from Figure 2, the following values were calculated for the kinetic parameters: PO = 3.32 X lo7 s-l and E = 71.16 kcal/mol. The DTA curve of nuclear graphite in COz was measured at a heating rate of 10 OC/min and is shown in Figure 4. The temperature scale was enlarged by using fast chart speed so that more accurate readings of Ti's and T, could be obtained. The temperatures were 1221, 1293, and 1343 O C for Til, T,, and Ti2, respectively. The Journal of Physical Chemistry, Vol. 80, No. 9, 1976
Ralph T. Yang
988
Comparisons of Experimental Results with Theories Equations 4 and 6 (or eq 4 and 5) are used to calculate the activation energy and frequency factor from a single DTA curve, based on our data on the C COz reaction and the well-known dehydration data of kaolinite and halloysite obtained from the literature. The examples were chosen because they are "first order" and they differ in a wide range of activation energies. Table I gives the comparison of the values of the two kinetic parameters calculated from eq 4 and 6 and the experimental data for the above reactions. The peak and inflection temperatures of the reactions are: Til = 1221 OC and T , = 1293 "C for C COQreaction; Til = 555 "C and T , = 600 "C for kaolinite dehydration; Til = 496 "C and T , = 533 "C for halloysite dehydration. The heating rate was 6"C/min for the two latter cases. It should be noted that the values calculated from eq 4 and 6 are approximate. The cumbersome eq 4 and 5 would give exact solutions. A comparison was made in this respect for the C COz reaction. By using eq 4 and 5 the following values were obtained: E = 66 kcal/mol and ro = 4.11 X lo6 s-l. The predicted rates calculated from the predicted E's and ro's are also shown in Figure 2. These predicted values are lower than the experimental data determined isothermally. I t is because the temperatures measured (Tiand T,) were the reference temperatures and during the reaction, the sample temperature was lower than the reference and the heating rate was also lower before the peak temperature. Consequently, the kinetic parameters calculated are lower than the values determined isothermally. Correction in the calculations cannot be made at this point because the heating rate of the sample during the reaction is not a constant, being smaller than the reference heating rate before the peak temperature and greater after the peak. However, by reducing the peak height experimentally, closer values can be obtained. Figure 3 gives the comparisons between the predicted rates from DTA and the experimental rates determined isothermally for kaolinite and halloysite. The results are surprisingly satisfactory considering the complexities involved in DTA. The lower rates predicted for halloysite can be again attributed to the lower temperature in the sample during the endothermic reaction. To predict a DTA curve from the kinetic parameters, the reaction C COz is used as an example. From the isothermal TG data we obtained: E = 71.16 kcal/mol and ro = 3.32 X lo7 s-l. The value of To is chosen, with the knowledge of E and ro, to be 1500 K where the rate is appreciable. As discussed previously, the approximation in eq 9 is good as long as GT/To is small. The initial mass of carbon mo was 20 mg. Substituting the above values into eq 11,we obtain
+
+
+
+
The Journal of Physical Chemistry, Vol. 80, No. 9, 1976
AT = 6.64 X 105k1exp
and Meyer Steinberg
[ -35 8T12*8
- 2.2893 X 10-11e0.01592T + 2.2893 X
1
1
Here k1 can be approximated as the heat of reaction divided by the total heat capacity of the sample; both change slightly with temperature. Now AT can be calculated as a function of T and is plotted in Figure 4. The DTA curve predicted in this fashion gives lower values of the AT than the experimental curve. This discrepancy indicates that heat transport is not fast enough and the thermal effect is partially accumulated. In summary, from a single DTA curve, eq 4 and 6 can be used to estimate the two important parameters, namely, activation energy and frequency factor. Knowing the two kinetic parameters, eq 11 can be used to predict the DTA curve. Some examples are given in this work of such applications. Comparisons of predictions by the simple models with the experimental data are considered satisfactory in view of the complexities involved in DTA. In general, the rates predicted from DTA are lower for the endothermic reactions and are higher for the exothermic reactions. This is because both the temperature and the heating rate are measured and controlled for the reference in all the commercial instruments. The DTA curve predicted by eq 11 gives somewhat smaller temperature differential than the experimental value because the heat transfer rate is not rapid enough and the thermal effects are partially accumulated. Acknowledgments. Discussions with Drs. G. Adler and C. R. Krishna of Brookhaven National Laboratory were most helpful. Mr. R. Smol is acknowledged for his able technical assistance. References a n d Notes
(1)This work was performed under the auspices of the Office of Molecular Science, Division of Physical Research, U S . Energy Research and Development Administratlon, Washington, D.C. (2)P. Murray and J. White, Trans. Br. Ceram. Soc.,48, 187 (1949);54,
137, 189,204 (1955). (3) H. E. Kissinger, J. Res. Natl. Bur. Stand., 57,217 (1956). (4) H. E. Kissinger, Anal. Chem., 29, 1702 (1957). (5)E. S.Freeman and B. Carroll, J. Phys. Chem., 62,394 (1958). (6)G. 0.Piloyan, I. D. Ryabchikov, and 0. S. Novikova, Nature (London), 212, 1229 (1966). (7)J. Zsako, J. Phys. Chem., 72,2406 (1968). (8)H. J. Borchardt and F. Daniels, J. Am. Chem. Soc., 79,41 (1957). (9)A. A. Blumberg, J. Phys. Chem., 63, 1129 (1959). (IO)H. J. Borchardt, J. Am. Chem. SOC.,81, 1529 (1959). (11) H. J. Borchardt, J. lnorg. Nucl. Chem., 12, 133 (1959);12, 252 (1960). (12)C. D. Doyle, J. Appi. Poly?; Sci., 15,285 (1961). (13)D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics", 2nd ed, Plenum Press, New York, N.Y., 1969,Chapter VI. (14)P. L. Walker, Jr., F. Rusinko, Jr., L. G. Austin, Adv. Catal., 11, 133 (1959).
