J. Woodfin, Anal. Chem., 48, 1395 (1976). NIOSH Manual of Analytical Methods, HEW (NIOSH)75-121 (1974). (4) K. D. Reiszner and P. W. West, Environ. Sci. Technol.,7 , 526 (1973). (5) A. E. O'Keeffe and G. C. Ortman, Anal. Chem., 38, 760 (1966). (6) F. P. Scaringelli. S. A. Frey, and 8.E. Saltzman, Anal. Chem., 42, 871 (1970). (7) B. E. Saltzman, W. R . Burgh, and G. K. Romaswamy, Environ. Sci. Technol., 5, 1121 (1971). (8) A. P. Altshuller and I. R. Cohen, Anal. Chem., 32, 802 (1960). (9) D. R. Bell, K. D. Reiszner, and P. W. West, Anal. Chlm Acta, 77, 245
(1975).
(3)
RECEIVED for review December 30,1976. Accepted March 14, 1977. We wish to acknowledge support for this research from the National Science Foundation, RANN Grant AEN7418932-A01and the National Institute of Occupational Safety and Health, US€" Grant R10H-00666A.
Differential Thermal Analysis and Reaction Kinetics for n th-Order Reaction Ralph T. Yang" and Meyer Stelnberg Department of Applied Science, Brookhaven National Laboratory, Upton, New York 11973
Relatlonshlps are derived for determinlng the three klnetlc parameters from a single DTA curve for a general nth-order reaction, and for predlctlng the DTA curves from the known kinetic parameters. Gorbachev's approximation for the Integral of the Arrhenius exponentlal function Is used. The relationshlps have been applied to a solldjlas reaction and decomposltlon reactions In solid-state and In aqueous solutlon. Fair agreement between the rather simple theories and the experimental data was obtained.
The subject of determining kinetic parameters from differential thermal analysis (DTA) data has attracted considerable attention and interest. Comprehensive reviews on this subject and discussions on related problems have been made by Barrall (I),Blazek (2), Wendlandt (3), etc. In a previous study (4),quantitative relationships between the kinetic parameters and the characteristic temperatures of a DTA curve were derived for the first-order reactions. However, it is obviously more desirable to evaluate the kinetic parameters for a general nth-order reaction without a priori knowledge of the value of the order. In this work, relationships are derived for evaluating the three kinetic parameters, i.e., reaction order, activation energy, and frequency factor from a single DTA curve and, conversely, with known kinetic parameters, the DTA curve will be predicted. The approximation proposed recently by Gorbachev (5) for the integral of the Arrhenius exponential function has been used in the derivations and it was shown to be superior to that by Frank-Kamenetskii (6). This approximation has also been shown by Gorbachev (5) to be more accurate than that by Coats and Redfern (7). The application of the method outlined in this work is demonstrated for a gas-carbon reaction and the decomposition reactions in the solid state and in aqueous solution. All the reactions are assumed to follow the simple rate expression shown in Equation 1. It is also noted that deduction of the mechanism of the reaction is not attempted in this study.
THEORETICAL CONSIDERATIONS The rate expression for a general nth-order reaction is:
where x is the fractional completion of the reaction; t, the time; ro, the frequency factor; n, the order; E , the activation energy; R, the gas constant; and T, the absolute temperature. It is assumed that the temperature differential (AT) in a DTA curve can be approximated as being proportional to the mass reaction rate and is expressed as the following: -EIRT
A T = h l r o m o ( l- x)"e
here mo is the initial mass of the reactant and kl is the proportionality constant which can be approximated as the heat of reaction divided by the total heat capacity of the sample mass. The validity of Equation 2 has caused considerable controversy in the past. On the one hand, Equation 2 has been assumed in DTA involving some gas-solid and solid decomposition reactions, as in the classic work by Murray and White (81, by Kissinger (91, and by others. On the other hand, in DTA involving phase transitions, the reactions were terminated at the peak temperatures and Equation 2 is not applicable (10-12). However, by using simultaneous DTADTG techniques, for the types of reactions that will be discussed in this work, it has been clearly shown that the DTA and the DTG correspond quite well (13). Detailed analyses of the various reaction systems in connection with the validity of Equation 2 will be published elsewhere. It is noted here Equation 2 is indeed a good approximation to the reaction systems herein studied. For a DTA curve with a linear heating rate kz, dt = dT/kz. Equation 1 can be rewritten with the above substitution and integrated as the following:
(3) In integrating the Arrhenius exponential function, the approximation by Frank-Kamenetskii has been adopted previously (4). It has been found since then that the approximation for the integral proposed recently by Gorbachev (5) is more suitable for the kinetic analyses of the DTA data because of its simpler form and perhaps higher accuracy under normal conditions. By using Gorbachev's approximation: nm2
(4) the following solution is obtained for Equation 3: 998
ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
1 k2 E
+ 2RT
e-E / RT
(5)
E+2RTe E + 2RTiI
here n # 1. Equation 5 can be rearranged and substituted into Equation 2 and A T is expressed as a function of T
AT
Knowing the values of the kinetic parameters, Equation 6 can be used to predict a DTA curve. In this equation, for n < 1,value of the square bracket is less than 1and the exponent is positive; the exponent increases from 0 and asymptotically approaches a as n increases from 0 to 1. Therefore, AT converges to zero more rapidly a t a greater n as T increases. For n > 1, value of the bracket is greater than 1 and the exponent is negative; the exponent increases from -a asymptotically to -1 as n increases from 1. Consequently, AT also converges to zero as T increases, but at a slower rate with a greater n. This indicates that the portion of the DTA curve beyond the peak temperature is less steep at higher reaction orders. This fact has also been indicated by Kissinger in his analyses of inflection points of a DTA curve. Now we proceed to derive the relationships for calculating the three kinetic parameters from a single DTA curve. For n > 1, the end temperature ( T J ,or the temperature where the DTA curve returns to the baseline, is obtained by setting AT = 0 in Equation 6 and it follows that k2 ( n - 1)ro e-E/RTe + -0 E + 2RTe RT,~
(7)
For the peak temperature T,, Equation 6 is differentiated with respect to T and by setting d(AT)/dt = 0,
For the two inflection temperatures (Ti, and T,,), second derivative of Equation 6 is taken and by setting d ( A T ) / d p
s
T e ) 23-a
TiI
- E + 2RT, ne (n- l ) E
= klromoe-E/RT
-(
:(&-+)
( ~ ; C Y ) ~
:(&
..
P
,
Equations 13 and 14 can be approximated by cancelling all the terms containing E in the numerators and the denominators, but leaving E in the exponents, and are subsequently solved simultaneously. The following equation containing n is thus obtained:
(TeITil ) 2
Tm (Te - Ti1 ) + LL] n-13-Lu = Ti1 (Ye - Tm )
By solving Equations 8, 9, and 10 in a similar fashion and making similar approximations, it can be shown that
log[",
Ti,
3 + a ( n - 1 ) ( 3- a ) - np 3 - a ( n - 1 ) ( 3+ a ) - no
1
From a DTA curve, the four characteristic temperatures, i.e., Tll,T,, Tlz,and T, are obtained. For the reactions with n < 1, Equation 15 can be conveniently solved by trialand-error methods for the value of n. Equations 13 and 7 can be used subsequently to calculate E and ro. However, Equation 15 does not converge or yield a solution for n when n is equal to or greater than one. In this case, Equation 16, which is valid for all values of n except n = 1, can be used although it is more complicated. Also in this case, Equations 8 and 9 can be used to calculate E and ro. For the case that n = 1, neither equation converges and the method outlined previously ( 4 ) or the following method which is probably superior can also be used. For n = 1, Equation 3 can be integrated and the results substituted into Equation 2 to yield the following equation for predicting the DTA curve from E and Ro:
-E/RT -
r,RT2
,-E/RT
k , ( E -2RT)
AT
=
k,romoe
(17)
To calculate the peak temperature, we set d(AT)/dt = 0 and obtain:
E
p
=
For the inflection temperatures, the following is derived by setting d2(AT)/dp = 0:
2(2 - );
From Equations 7 and 8,
:(k- &)
(E + 2nRTm)(E+ 2RTe) e E(E + 2 R T m ) ( 1- n )
=(e) 2
From Equations 7 and 9,
El
2
--ro e-ZE/RTi ro -E/RTj - - (19) RT2 k22 Equations 18 and 19 can now be solved simultaneously without much difficulty and can be used to calculate E and ro from the characteristic temperatures of a DTA curve. It is appropriate here to discuss the assumptions made in the above derivations. Equation 2 is valid only if the thermal effect is detected instantaneously and is then dissipated ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
999
0
I!
wL
0.5
= t
c
a
N -
0
1.5
4
1 1000
I100
1200
T, Figure 1.
1300
1400
- /
OG
Calculated DTA curves for the reaction C
1
+ C02
rapidly. Equation 3 assumes that dT/dt = k 2 and it is not a function of Tor t. This is not true since only the “reference” temperature is controlled at a constant heating rate while the “sample” temperature deviates from linearity in the range where the reaction takes place. Under the normal DTA conditions, the deviations are small but they can amount to significant errors. To demonstrate the applicabilities of the above equations, data were obtained and treated on the decomposition of a calcitic limestone. Furthermore, published data on the C + COZ reaction ( 4 ) and on the decomposition of benzenediazonium chloride in aqueous solution (14) have also been employed and treated with this method. EXPERIMENTAL The apparatus and experimental procedures for measuring the rates and the DTA curves are essentially the same as described previously (4,15). Briefly, the rates of decomposition of the calcitic limestone were obtained gravimetrically in a constant flow of nitrogen at three temperatures. The rates are expressed as -(l/m)(dm/dt), m being the instantaneous mass, with the units of m i d and were measured as a function of x . The rate expressed in this fashion, although it has the same units as dxldt, is equal to (dx/dt)(l - x)-’. The DTA curves were measured at three heating rates in nitrogen flow with the same flow rates as in the rate measurements. The sample used was of the grade of high purity and low porosity, obtained originally from Guernsey, Wyo. Chemical analyses showed that it contained 95% CaC03,4% Si02,and 1% MgC03. The particle size of the sample was in the range of 590 to 840 ym. The N2 was of prepurified grade (99.996% purity) supplied by the Liquid Carbonic Company. The gas flow rate was about 1 cm/s passing the sample for both rate and DTA measurements. The amount of sample used in both the rate and DTA measurements was 20-30 mg. In calculating rates from the gravimetric results, SiOnwas excluded. Limestone sample rather than a reagent-grade pelletized sample was employed because of its practical importance and interest (16), and it does suitably serve the purpose of delineating the quantitative relationships outlined above. COMPARISONS OF EXPERIMENTAL DATA WITH THEORIES Experimental results for three reactions are used to test the validity and applicability of the theories derived. First, the kinetic data of the C C 0 2 reaction ( 4 ) are substituted in Equation 6 to synthesize the DTA curve. Second, the results on the decomposition of limestone obtained in this study are used to demonstrate the quantitative relationships between the kinetic parameters and the DTA curve. The well-known results on the decomposition of benzenediazonium chloride in aqueous solution (14) are then used to show the applicability
+
1000
-
ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
L
1
0
20
40
60
COMPLETION OF REACTION,
Figure 2.
80
Yo
Rate of decomposition of limestone
of the theories in solution kinetics. The calculated DTA curve for the C C02reaction using Equation 6 is shown in Figure 1. This DTA curve is compared with the previously calculated curve using the approximation method of Frank-Kamenetskii (4). Better agreement with the experimental data is shown by using Equation 6 over the previous one. Another advantage in using Gorbachev’s approximation method in DTA is that it is simpler in form, and the theoretical treatment based on it yields simpler results. In this connection, it should be noted that other approximations have also been proposed but it seems that in DTA applications, Gorbachev’s approximation is superior in the combined benefits of simplicity and accuracy ( 7 , 1 7 ) . For decomposition of limestone, the isothermal rate data are presented in Figure 2. The rates are plotted against the percentage completion for three temperatures. The order of the reaction, the activation energy and the frequency factor were determined from the rate data shown in Figure 2 and are 0.55,44 Kcal/mol, and 1.6 X lo6 s-’, respectively. These results are in fair agreement with literature data (7,9,18-21) although such agreement is not particularly meaningful owing to the complexities and the many variables involved in the decomposition process. DTA curves a t three heating rates are shown in Figure 3. In the DTA measurements, a high chart scanning speed was used to enable accurate readings of the characteristic temperatures. The temperatures are, in O C : (a) for k2 = 6 OC/min: Til = 762, T,,, = 798, T , = 840; (b) for k 2 = 10 OC/min: TL1 = 777, T,,, = 823, T,= 870; (c) for k 2 = 15 OC/min: T,,= 801, T , = 840, T, = 888. In Figure 3, the three predicted DTA curves are also shown. The three kinetic parameters obtained from Figure 2 are used along with Equation 6 in calculating the curves. The agreement between the calculated and the experimental curves is satisfactory. The fact that the magnitude of the experimental AT is higher than the calculated values indicates that the thermal effect is partially accumulated; while, in the theoretical treatment, the thermal effect is assumed to be released instantaneously. The DTA curves are used to calculate the kinetic parameters by using Equations 7, 13, and 15. Thus, using the above characteristic temperatures, n is calculated from Equation 15 and E and ro are calculated from Equations 13 and 7 subsequently. Three kinetic parameters are obtained
+
Table I. Comparison of Kinetic Parameters by V a r i o u s Methods Reaction Source of method h , , C/min Isothermal TG Decomposition of limestone Calculated from 15 DTA, this method 10 6
Isothermal ( 2 2 )
Decomposition of benzenediazonium chloride in solution
Calcillated from DTA ( 1 4 ) This method, from DTA
n 0.55
E , Kcal/mol 44
0.56 0.57 0.55 1
44 43 44 27.2
1
0.9
To,
l/s
1.6 X
lo6
1.9 x 1.2 x 1.9 x
lo6 lo6 lo6
1015.2
28.3
1016.9
30
10’7.4
Daniels is better in calculating E and ro. This may be partially attributed to the fact that the heating rate in their experiments was not strictly constant and that their method in calculating E and PO does not require a constant heating rate (14). However, this method has the advantage of being applicable to a general nth-order reaction wherein the value of n can also be calculated. In summary, the comparisons between the experimental results and the theoretical values are in fair agreement. Equation 6 can be used to predict the DTA curve from known kinetic parameters. Equations 7, 13, and 15 or Equations 8, 9, and 16 can be used to calculate the three kinetic parameters from a single DTA curve for a nth-order reaction (n # 1). The method outlined in this study can be applied to both solid state and solution kinetics. For n = 1,the method outlined earlier (4) or Equations 17 to 19 can also be used with perhaps higher accuracies in the normal ranges of DTA measurements.
c
a
1
I
520
- EXPERIMENTAL - - - - CALCULATED, E q . ( 6 )
580
I
I
640
700 T,
760
820
880
940
O C
Figure 3. DTA curves for the decomposition of limestone
from each curve and the results are summarized in Table I. The agreement between the calculated and experimental results are satisfactory considering the complexities involved in DTA. The kinetics of decomposition of benzenediazoniumchloride in aqueous solution was studied isothermally by Crossley et al. (22) and the DTA curve was measured and analyzed by Borchardt and Daniels (14). It should be appropriate, at this point, to clear the notations so that their data can be translated and used in this work. In solution or homogeneous kinetics, rates are expressed generally by the following equation:
where C is the concentration of a reactant species and zo is the frequency factor. It can be shown that by substituting C = Co(l - x ) into this equation, it is identical with Equation 1 and where ro = Z ~ C O ~ In - ’ . the case where n = 1, zo = ro, which is the case for the decomposition of benzenediazonium chloride in aqueous solution (22). In this study, the DTA curve by Borchardt and Daniels is used in conjunction with Equations 7, 13, and 15 to calculate the kinetic parameters. The results are included in Table I, along with the experimental data and the values for E and ro calculated by Borchardt and Daniels using their method (14). From the comparison, it is seen that the method of Borchardt and
ACKNOWLEDGMENT We gratefully acknowledge R. Smol for performing the experiments and R. T. Liu for carrying out part of the calculations and for their valuable discussions. G. W. DePuy of the US.Bureau of Reclamation kindly supplied the limestone sample. LITERATURE CITED (1) E. M. Barrall 11, in “Guide to Modern Methods of Instrumental Analysis“, T. H. Gouw, Ed., Wiley-Interscience, New York, 1972, Chap. 12. (2) A. Blazek, “Thermal Analysis”, Van Nostrand, London, 1973. (3) W. W. Wendlandt, “Thermal Methods of Analysis”, 2nd ed., WileyInterscience, New York, 1974. (4) R. T. Yang and M. Steinberg, J. Phys. Chem., 80, 965 (1976). (5) V. M. Gorbachev, J. Therm. Anal., 8, 349 (1975). (6) D. A. Frank-Kamenetskii, “Diffusion and Heat Transfer In Chemical Kinetics”, 2nd ed., Plenum Press, New York-London, 1969. (7) A. W. Coats and J. P. Redfern, Nature (London), 201, 68 (1964). (8) P. Murray and J. White, Trans. Br. Ceram. SOC.,48, 187 (1949); 54, 137, 189, 204 (1955). (9) H. E. Kissinger, Anal. Chem., 29, 1702 (1957). (10) V. P. Equnov and Y. V. Afanasjev, Therm. Anal., Proc. Inf. Conf., 4 t h 1, 303 (1974). (11) D. A. Vassallo and J. C. Harden, Anal. Chem., 34, 132 (1962). (12) C. B. Murphy, Anal. Chem., 46, 451R (1974). (13) R. T. Yang and M. Steinberg, unpublished results. (14) H. J. Borchardt and F. Daniels, J. Am. Chem. Soc., 79, 41 (1957). (15) R. T. Yang, M. Steinberg, and R. Smol, Anal. Chem., 48, 1696 (1976). (16) R. S.Boynton, “Chemistry and Technology of Lime and Limestone”, Wiley, New York, 1966. (17) J. Zsako, J . Therm. Anal., 8, 593 (1975). (18) H. T. S.Britton, S.J. Gregg, and G. W. Windsor, Trans. Faraday SOC., 48, 63 (1952). (19) E. S. Freeman and B. Carroll, J. Phys. Chem., 62, 394 (1958). (20) D. M. Speros and R. L. Woodhouse, J. Phys. Chem., 72, 2846 (1968). (21) P. K. Gallager and D. W. Johnson, Jr., Thermochlm. Acta, 6, 67 (1973). (22) M. L. Crossley, R. H. Klenle, and C. H. Benbrook, J. Am. Chem. SOC., 62, 1400 (1940).
RECEIVED for review January 7, 1977. Accepted March 16, 1977. This work was performed under the auspices of the Office of Molecular Sciences, Division of Physical Research, U.S. Energy Research and Development Administration, Washington, D.C. ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
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