ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978 (23) A. M. Bond and R. J. O'Halloran, J . Nectroanal. Chem., 68, 257 (1976). (24) D. R. Canterford, J. Necfroanal. Chem., 77, 113 (1977). (25) J. H. Christie, J. A. Turner, and R. A. Osteryoung, Anal. Chem., 49, 1899 (1977). (26) J. A. Turner, J. H. Christie, M. Vukovic, and R. A. Osteryoung, Anal. Chem., 49, 1904 (1977). (27) R. J. Schwall, A. M. Bond, R. J. Lovd, J. G. Larsen. and D. E. Smith, Anal. Chem., 49, 1797 (1977). (28) D. E. Smith, Anal. Chem., 48, 221A (1976). (29) S. C. Creason, J. W. Hayes, and D. E. Smith, J . Electroanal. Chem., 47. 9 (1973). (30) R. J. Schwali, A. M. Bond, and D. E. Smith, Anal. Chem., 49, 1805 (1977). (31) R. J. Schwall, A. M. Bond, and D. E. Smith, J. Electroanal. Chem., 85, 217 (1977). (32) A. M. Bond, R. J. Schwall, and D. E. Smith, J. Elecfroanal. Chem., 85, 231 (1977). (33) Raytheon Data Systems, "Apoilo, The Array Processing System", Brochure No. 44-10044, Raytheon Data Systems, Norwood, Mass., 1976.
1079
(34) Floating Point Systems, Inc., "AP-IPOB Array Transform Processor Handbook", Form No. 7259, Floating Point Systems, Inc., Beaverton, Ore., 1976. (35) CSP, Inc., "CSPI-4000 FFT Array Processor", CSP, Inc., Burlington, Mass., 1976. (36) J. W. Haves, D. E.Glover, D. E. Smith, and M. W. Overton, Anal. Chem., 45, 277(1973). (37) R. de Levie, J. W. Thomas, and K. M. Abbey, J . Electroanal. Chem., 62, 111 (1975). (38) D. E. Smith, in "Electroanalytical Chemistry", A. J. Bard, Ed., M. Dekker, New York, N.Y., 1966, pp 1-155. '
RECEIVED for review February 1, 1978. Accepted April 18, 1978. The authors are indebted to the National Science Foundation (Grant No. CHE77-15462) for support of this work.
Study of Solid State Kinetics by Differential Scanning Calorimetry Kaushal Kishore High-Energy Solids Laboratory, Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore, 560 0 12 India
Dlfferentialscanning calorimetry (DSC) has been used to study the thermal decomposltlon of some secondary explosives. DSC can be used successfully to study the kinetics of solid state reactions. A general procedure, independent of the kinetic law assumptions, for calculating activation energles has been presented which holds equally well for both scannlng and isothermal conditions.
zero adjustment control of the calorimeter was replaced with a ten-turn helipot which allowed the output signal to be easily and precisely adjusted. Thermograms for RDX and HMX decompositions were obtained for scanning operations using scan speeds of 2, 4, 8, 16, and 32 K m i d . Suitable range settings (2,4, 8, 16, and 32 mcal for full scale deflection) and sample mass (0.5 to 1.5 mg) were taken so that a sizable thermogram could be recorded. The sample was sealed into the pan and a small hole was pierced in the lid for the escape of the product gases. At a pre-selected temperature range, thermograms for empty pan, pan containing the sample, and pan containing the product (residue left after the decomposition was over) were obtained for each sample. In isothermal operations, the desired temperature was attained rapidly by manually operating the scanning temperature knob and then holding constant at that temperature. The fraction decomposed ( a )at a particular time was obtained by dividing the area at that time by the total area. The enthalpy changes under the exotherm/endotherm were calculated by comparing them with the peak produced by a known weight of standard indium.
The advent of commercial equipment has led to the extensive use of differential scanning calorimetry in the study of chemical kinetics. In spite of the apparent simplicity of the instrument, there are problems involved in the derivation of meaningful exotherms and in the analysis of the same to give the kinetic parameters. While the former has been largely ignored, the latter has aroused considerable interest although little attempt has been made to assess the uncertainities in the results. In the present work, a differential scanning calorimeter has RESULTS AND DISCUSSION been used to study the thermal decomposition of 1,3,5-triT o obtain meaningful kinetic parameters, the precise value nitrocyclo-2,4,6-trimethylene-1,3,5-triamine (RDX) and 1,3,5,7-tetranitrocyclo-2,4,6,8-tetramethylene-l,3,5,7-tetramine of the instantaneous reaction rate and the extent of the reaction at a particular instant are needed. Application of DSC (0-HMX). The RDX and HMX (particle size = 35 to 75 ym) to the study of chemical kinetics depends upon correlating used were more than 99% pure. These compounds have been the output signal with the rate of the kinetic process. I t is studied previously both by conventional (1-9) and scanning also related to the total exothermicity per unit time of all calorimetric techniques (10-12). It is the purpose of the chemical and phase changes of the sample. The output signal present work to consider further the difficulties associated is measured in terms of mcal s-l which is nothing but the with DSC techniques in obtaining kinetic parameters. ordinate deflection of the thermogram. Section A (below) EXPERIMENTAL describes the calibration of the ordinate signal which is the The signal output from a Perkin-Elmer differential scanning measure of the reaction rate. Section B describes the procalorimeter (DSC-1B) was fed to either a chart recorder (Leeds cedure for drawing the correct baseline so that accurate & Northrup Co., Speedomax W) or to a digital voltmeter (Dymeasurements of the reaction rate and the extent of reaction namico Instruments Ltd., DM 2023). Average and differential could be carried out. Section C describes a procedure for temperature settings were calibrated according to the manuevaluating the kinetic parameters which is free from the facturer's instructions (13). Flowing dry nitrogen was used as an assumptions made for the kinetic laws. An attempt has been inert atmosphere inside the calorimetric cell. made to derive relevant kinetics without a priori knowledge The sample and the reference pans were positioned at the center of the holder cells and were covered with aluminum domes. The about the reaction order. This will have a definite advantage 0003-2700/78/0350-1079$01 .OO/O
0 1978 American Chemical Society
1080
ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978
for studying the kinetics of any process without first identifying the topochemistry of the process. Section D describes the relevance of the activation energy of the RDX and HMX decomposition obtained by the above procedure from the bond energy considerations. A. Calibration of the Ordinate Signal. If we assume that there are no temperature lags in the calorimeter then we can describe the scanning program by the following two equations. The term calorimeter is used here to mean the sample holder assembly which comprises the sample and reference pans and lids and domed covers.
=.nL
-"I
-
40
dT C,- = W , + W ( C Y ,T ) - h , ( T - T,) dt dT C,-= W , - h,(T- T,) dt where C is the effective specific heat, W is the electrical power, w is the exothermicity/endothermicity, T is the temperature at time t, h is an effective heat-transfer coefficient which needs no further definition here. In general, h, # h, and h may be a function of T and a both. T, is the ambient temperature. It is assumed that T = T, = T, and dT/dt = dT,/dt = dT,/dt. The subscript s refers to the sample, sample holder, pan, lid, and the domed cover. The subscript r refers to the reference holder, pan, lid, and domed cover. Temperature is the independent variable and is a pre-determined linear function of time. The operating equation of the instrument can be obtained by subtracting Equation 1 from Equation 2.
dT
(C, - C,)= W , - W , - w - ( T - T,)(h, - h,) ( 3 ) dt The output signal s is proportional to (W,-W,). The equation is simplified for the isothermal mode of operation when dT/dt = 0, and for measurement of inert samples when w = 0. The changes in the output signal can be accommodated within the framework of the simple model used here by attributing them to the variations in the heat-transfer coefficients. Some simplification is possible by working with a relative signal in which measurements are made first on an empty pan and then with the same pan containing the sample. Equation 4 now describes the operation of the instrument.
The subscript e refers to the empty pan, lid, and domed cover and the associated holder. The derivation of Equation 4 assumes that no alteration is made to the reference during the measurements. C, - C, is the heat capacity of the sample. The heat-transfer term can be eliminated for inert samples by taking the difference between isothermal and scanning signals recorded a t the same temperature. This difference is proportional to the specific heat term (C, - C,) (dT/dt) and is the basis of the method for the measurement of specific heats. Thus,
dT dt
Y = s, - s, = (C,- C e ) -
(5)
The heat capacity can now be obtained by dividing Y with the corresponding heating rate. We have made use of this procedure for calibrating the output signal. Measurements were made on an empty pan and then with the same pan containing sapphire disks. Disks of different masses were used and measurements were done at 409,509,609, and 709 K a t the scan speed of 8 K min-l. For full scale deflection, three ranges were used: 2,4, and 8 mcal s-'. The heat capacity data
A/
30
-
20
X
I/r
!
I
I
I
1
20
30
40
50
€
C p ( m l l l l c o i . 4 ) Exptl
Figure 1. Dependence of C, (experimental)on C, (known) at various temperatures
-1
;
'400
500
600
Temperature
700
(OK)
Figure 2. Dependence of C,(experimental)/C,(known) on temperature
for sapphire disks were obtained a t various temperatures by changing the range settings. The experimental heat capacity values were then plotted vs. the literature values. The solid line in Figure 1 shows the expected curve whereas the experimental points are distributed around it. In Figure 2, the calibration factor expressed as C,(known)/ C,(exptl) has been plotted vs. the temperature. The measurements show that the proportionality constant for the sample holder is approximately unity. The possible change in the calibration with temperature was ignored when deriving the relative reaction rates because the selected temperature ranges involved were small. B. Estimation of the Extent of Reaction and Reaction Rate. For kinetic studies, it is important to have accuracy in the measurement of the fraction decomposed ( a ) . This accuracy depends upon the accurate measurement of the area under the thermogram, which in turn depends upon the accuracy of the drawing of the baseline. In general the baseline is drawn by joining two extreme points on the inert linear part of the signal on either side of an exothermic/endothermic process. The baseline drawn in this way is only approximate, particularly when the substance undergoes decomposition. As a consequence the heat capacity and the heat-transfer properties of the sample change considerably during the reaction. The approach of drawing a correct baseline in a scanning thermogram has been given by Brennan e t al. ( 1 4 ) and is now being set down in an explicit form for the scanning mode of operation. Suppose that we have the data for four experiments conducted over the same temperature range on an empty pan, pan and reaction mixture, pan + reaction product, and finally the fourth, a hypothetical experiment, on the pan and inert reaction mixture (Le., the extrapolation of the inert part of
ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978
n I
I Programme temperature
-
I
Figure 3. Baseline correction in a scanning thermogram. AB = Empty
+
pan thermogram. AEIFD = Pan reaction mixture thermogram. CFD = Pan 4- product thermogram. EG = Extrapolated inert reaction mixture thermogram. EF = Rough baseline. EHH'F (curved) = Corrected baseline
the signal from pan and reaction mixture run). The relative ordinates can be calculated by using the following equations:
Ym
=
w - ( T - Ta)(hm
-
he)
-
dT -Cm dt
(6)
1081
using Equation 10 is carried out by iterative procedure. Generally, it has been found in the case of RDX and HMX thermograms that two to three operations yield the correct baseline. Typical baseline correction for an RDX decomposition thermogram is represented in Figure 3. It may be seen that the baseline curves after the correction is applied. The baseline corrections not only affect the total area but also the shape of a-time curves, the ordinate at a particular instant, and the a values. Brennan et al. (14) have pointed out that in some cases the total area can change by as much as 10% when the corrections are applied. It may be emphasized that only after the possible errors in ordinates and areas are considered, will the derived kinetic parameters have any real significance. C. Calculation of t h e Kinetic Parameters. Equation 16 can be used to calculate the reaction rate and the activation energy for a particular decomposition reaction. If AH is the total exothermicity (cal g-') under the curve and dm/dt (g s?) is the rate of decomposition (Le., weight loss rate), then Equation 16 can be written as follows:
dm Z=----AH dt
o r -drn =dt
Z
AH
(17)
We have assumed that the kinetics of decomposition can be expressed as a separable function of the extent of decomposition and temperature as follows:
where
Equation 18 can be written in a general form
= s m - s r9
m '
Sreact -
= Sprod - s r 9
'prod
and
da =
Yreact =
(9)
Sr
( Y , is the signal for the reaction; Yprod, for the inert product; and Yreact,for the inert reaction mixture). Subtraction of the weighted amounts of the signals due to inert reactants and products from Y , gives Equation 10. a is obtained by drawing a rough baseline and then determining the ratio (area/total area).
hprod
+ (1- & ) h e a c t - h m
(11)
and
+ (1- Q)Creact - C m Equation 10 can be written as follows:
Q=
Cprod
Z= w
+P(T-
Ta)
+
dT Qdt
(12)
(13)
Q from Equation 12 vanishes because c m = Cprod
+ (l-a)Creact
(14)
The thermal lag coefficients in Equation 11may be taken to be of equivalent magnitude since no alteration is made in the calorimetric assembly. We would then expect P to be independent of a and to be small and hopefully negligible because, hreact
E
hproduct
hm
(15)
Thus Equation 13, on combining with Equations 11, 12, 14, and 15, becomes
z=w
k(l
- a)"
where n = order of reaction. If the temperature dependence can be expressed as an Arrhenius type relationship
k = A exp
(-&)
where E is the activation-energy for decomposition in cal mol-', then on combining Equations 17, 19, and 20, we get
--Z dt AH
d mda _ --
By substituting the following
P=
dt
(16)
Z is the correct ordinate representing the instantaneous rate alone. It may be noted that the correct baseline obtained by
dt
If a and n are maintained constant in Equation 21, then
AZH = B exp(-
&)
where B = A ( l - a)". Equation 22 is equally valid for both scanning as well as isothermal thermograms provided the necessary conditions are satisfied. The plot of log Z/AHvs. 1/T will directly yield the value of E. The essential conditions in the validity of this procedure, as pointed out above, lies in the fact that a and n should be maintained constant for different scanning operations. The conditions that n should be constant require that the kinetic behavior at different heating rates (different temperatures in isothermal runs) should be the same. This condition is tested from the so-called "Reduced-time plots" where a vs. time plots at various heating rates are superimposed on each other (15). A typical reduced-time plot for RDX decomposition is shown in Figure 4. Figure 4 clearly shows that E estimations can be carried out without any difficulty. The conditions to achieve a to be a constant were obtained by selecting a particular a value for calculating log Z / A H . For example E for RDX was calcd. for a = 0.3, 0.5, and 0.7. The E plot was made according to Equation 22. The derived E
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ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978
Table I. Literature Data on Activation Energy ( E ) for RDX and HMX Decomposition Authors E, kcal mol-' Technique A. RDX Decomposition Robertson 47.5 Pressure measurements Adams 45.5 Pressure measurements Batten and Murdie 67.0 Pressure measurements Rogers and Morris 67.5 DSC Hall 45.2 DSC B. HMX Decomposition Robertson 52.7 Pressure measurements Maycock and Pay Verneker (i) 59-62 TGA (ii) 65 DTA Sinclair and Hondee ( i ) 44.2 DTA (ii) 63.2 DTA (iii) 52.6 DTA Rogers and Morris 228 t 24 DSC Rogers 51.3 DSC Hall 180-210 DSC
Temp range, "C
Ref.
213-299
1
203.5-261
4
179-200
5
... ...
10 12
271-314
1
245-280
2 2
220-2 33 233-241 241
3 3 3 10 11 12
... ...
271-285
...
I
RDX (Scannlnql o 2 ' ~min'
A 8*C m l d 16% mln"
Time
-
Flgure 5. Baseline correction in an isothermal thermogram. A'A = Uncorrected baseline. BA = Corrected baseline. BC = Extrapolation of the initial thermogram
Reduced time(&,)
-
Figure 4. a vs. reduced-time plot for RDX scanning thermograms
for RDX were 43.0 f 3, 42.0 f 1, 41.0 f 2 kcal mol-I. The average E value for a = 0.3 to 0.7 came out to be 42 f 3 kcal mol-'. The uncertainties were calculated from the error in the slope of the line assuming that the error resides in the rate. It is felt that the uncertainty in the heat capacity of the liquid may be responsible for the uncertainty in the E values. The uncertainty in the measurement of T may be small because the instrumental lag time was small. They will be greater in some cases such as the decomposition of liquid HMX where there is no experimental heat capacity data and are likely to be smaller where the mass change is smaller. In HMX, uncertainty becomes greater because HMX decomposes significantly before melting and this makes the derivation of a proper baseline impossible for the liquid-phase decomposition. That is why E for HMX decomposition (scanning runs) was found to have a value of 44 10 kcal mol-l. The decomposition of RDX and HMX has also been studied using the DSC in the isothermal mode. The advantage of this mode is that the heat capacity correction is zero and the changes in the heat leakage modulus are considerably reduced. The drawing of the baseline for an isothermal signal
*
is entirely different from that described in the scanning mode. In this case, the zero knob position is changed to bring the thermogram on the chart and the difference in the isothermal signals before and after the reaction is taken into account in drawing the baseline. A typical isothermal thermogram with a baseline correction is shown in Figure 5 . The difficulty associated with the isothermal run is that the start of the reaction, especially if it is very fast, is very difficult to locate. The only way to do this is to roughly extrapolate the curve back to the starting point. The extrapolation is shown in Figure 5. The isothermal experiments were carried out at temperatures of 477,481.9,487,493.9,and 498 K for RDX and 524.1, 528.1,532.1, and 536.1, for HMX. E determinations were done by using Equation 22. Signal 2 at different temperatures was estimated from the thermograms for a fixed value of a ( a = 0.5). The condition for similar kinetic behavior a t different temperatures was tested from reduced time plots. Typical 01 vs. reduced time plots for isothermal decomposition of RDX are shown in Figure 6. E values for RDX and HMX were estimated to be 41 f 1 and 42 f 2 kcal mol-', respectively. These values compare well with the scanning E values giving credence to the present procedure for E estimations. It may now be said that it is possible to derive E , both for dynamic as well as isothermal runs, without recourse to the assumptions, which are often erroneous, about the "Order of the reaction". Moreover, the identification of the exact kinetic law, particularly in a solid-state reaction, is difficult unless precise topochemical information is available. In Table I the literature E values of RDX and HMX decomposition are listed; most of them involve assumptions concerning the order of the reaction. However, some of the values are in agreement
ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978
A
x 0
A 0
1083
decomposition. However, in their calculations, they considered the high frequency dielectric constant for cy and y polymorphs and not for the 6 form. This dielectric constant varies from 4.6 to 3.9 and therefore there is a possibility of getting a value of around 40 kcal mol-l for the band to band transition if the dielectric constant of 6-HMX is less than 4. The same authors have also pointed out that the E value corresponds to the N-C bond dissociation energy which is around 72 kcal suggesting that the rate controlling step is the dissociation of the N-C bond. It may be noted that the bond dissociation energy of the N-N bond is of the order of 39 kcal and this corresponds to the observed E value in the present work for (3-HMX decomposition. RDX also yields the E around 40 kcal mol-l, showing thereby that the rate controlling step in the pyrolysis of RDX may also involve N-N bond dissociation. This may not be surprising since both compounds possess a ring structure of the same basic unit of
4770OK 481,9*K 4836.K 487.8.K 493.9'K 49BO'K
H2
NO2 I
,C--N, ACKNOWLEDGMENT Thanks are due to P. G. Laye, Department of Physical Chemistry, University of Leeds, U.K, for his keen association with the work and his helpful suggestions at various stages. The author is also thankful to P. Gray F.R.S., Chairman, Department of Physical Chemistry, University of Leeds, U.K., for his encouragement.
IP *O
-0
2.0 Reduced t i m e
30
(=I--t
Figure 8. a vs. reduced-time plot for RDX isothermal thermograms
with our values within an uncertainty of 3-5 kcal mol-'. D. Significance of t h e E V a l u e s for RDX a n d HMX Decomposition. Maycock et al. (16) from their mass spectrometric studies of the decomposition products of 0HMX by UV light suggest that the following intermediate unit is formed during the decomposition n-
Suryanarayana et al. also suggested the formation of the same intermediate from their I5Ntracer study of the decomposition products. Maycock and Pai Verneker ( 2 ) have further suggested that E for the thermal decomposition of 0-HMX (60-65 kcal mol-') corresponds to the energy required for the band to band electronic transition in the solid during the
LITERATURE C I T E D (1) A. J. B. Robertson, Trans. Faraday Soc., 45, 85 (1949). (2) J. N. Maycock and V. R. Pai Verneker, Exploslvstoffe, 1, 5 (1969). (3) J. E. Sinclair and W. Hondee, Inst. Chem. Treib-Exploslvstoffee, Berghausen, Germany, 5771 (1971); Chem. Abstr., 81, 108070J (1974). (4) C. R. H. Bawn, "chemistry of the Solid-State", Butterworths, London, 1955, p 261. (5) J. J. Batten and D. C. Murdie, Aust. J Chem., 23, 749 (1970). (6) J. J. Batten, Aust. J . Chem., 24, 2025 (1971). (7) J. J. Batten, Aust. J. Chem., 24, 945 (1974). (8) F. C. Rauch and A. J. Fanelli, J . Phys. Chem., 73, 1604 (1969). (9) J. D. Cosgrove and A. J. Owen, Combust. Flame, 22, 19 (1974). (10) R. N. Rogers and E. D. Morris, Jr., Anal. Chem., 38, 412 (1966). (11) R. N. Rogers, Thermochlm. Acta, 3, 437 (1972). (12) P. G. Hall, Trans. Faraday Soc., 67, 556 (1971). (13) Instructions Differential Scanning Calorimeter No. 990-9556, Perkin-Elmer Limited, Norwalk, Conn., Nov. 1966. (14) W. P. Brennan, B. Miller, and J. C. Whitwell, Ind. Eng. Chem., Fundam., 8, 314 (1969). (15) M. Selvartnam and P. D. Garn, J . Am. Ceram. Soc., 59, 376 (1976). (16) J. N. Maycock, V. R. Pai Verneker, and W. Lochte, Phys. Status Solidi, 35, 849 (1969).
RECEIVEDfor review January 27,1977. Resubmitted February 1, 1978. Accepted March 27, 1978.
