Isothermal
ecornposition of
xplosives
41. A . COOK AND 31. TAl*-LBR ABEGG Cnicersity of Utah, S a l t Lake City, Utah
AKY explosive compounds decompose isothermally over at least a part of their decoiiipoPition range, by the first-order law
(See Nomenclature list for definition of si-mbols.) hutocat,alysis frequently complicates t'he kinetics of decomposition. These effects are associated with the accumulation of products of decomposition in the sample. I\f oreover, self-heating effects are difficult to avoid in the thermal decomposition of explosives owing to the highly exothermic nature of such reactions. Many investigators have studied the kinetics of high temperature decompositions of explosives under adiabatic or nearly adiabatic conditions because of the difficulty of maintaining isothermal conditions (2, 3, 6, 8, IO). The heat balance equation
-KA2T
+
pc
dT j j = p&k'(T)f
may be integrated only under one or the other condition, hence kinetic data obtained under nonisothermal, nonadiabatic conditions cannot be treated theoretically. I n the above equation the first term on the left accounts for external heat loss, the second for the heat accumulated in the explosive. and that on the right for the heat generated by chemical renction. The adiabatic method ( K A T = 0 ) yields time-lag T data nhich may be related to the kinetics of decomposition through the equation (3)
I n the final analysis however, the adiabatic method is a t least as difficult t o control as the isothermal method (bT/bt = 0). Isothermal conditions should be maintained a t temperatures as low as possible consonant with measurable decomposition rates, since high temperatures increase the t,endency tonard selfheating. However, under carefully controlled conditions and a t temperatures sufficiently low that' self-heating is avoided, the isothermal method readily lends itself to an accurate determination of decomposition kinetics. Isot,hermal studies have been carried out by techniques similar t o those used by Robertson ( 1 2 ) and Poffe ( 2 0 ) which measure the rate of decomposition by the rate of pressure increase of the decomposition products. Measurementx have usually been made a t temperatures sufficient'ly high to ensure completion of the reaction in a period of a few seconds to several minutes. This method has confirmed the occurrence of first-order decomposition for some explosives in certain regions of the decomposition curve. However, it has been applied in many instances a t t'ernperatures generally too high t o observe and/or evaluate accurately the true first-order decomposition without encountering autocatalytic and self-heating efYects. These effects contribute, no doubt, to the general lack of agreement in data reported in the literature. It has been shown hy work in this laboratory that accurate, uncatalyxed thermal decomposition data for detonating explosives may be used t o predict the region of "nonideal" detonation -i.e., the diadeter range over which the measured detonation velocity, D, is lower than the "ideal" or hydrodynamic velocity, D* ( 4 , 6). Gnfortunately most thermal decomposition data have been obtained under conditions of autocatalysis or/and
undei conditions nheie temperatures and thus true activation energy data could not be established. By carrying out measurements a t sufficiently low temperatures it is possible in many cases t o separate unambiguously the autocatalyzed portions from the uncatalyzed or true pure explosive thermal decomposition region of the decomposition-time curve. This article describes the development of a neF7 isothermal method applicable in quantitative studies considerably lox er in 7' than those used in previous studies. I t permits evaluation oi the true rate constants in many instances before autocatalytic, auto-inhibitorj- 0 1 self-heating effects become noticeable. APPARATUS AND EXPERIMESTAL .METHOD
The method presented here employs the direct measurement' of the weight loss by the use of a sensitive quart,z spring balance. It applies strictly to explosives for nhich the total weight loss is due to decomposition alone-i.e., where vapor pressure is sufficient,ly low that no weight, loss from vaporization occurs. Samples of explosive (0.0500 t,o 0.2500 gram) were placed in a small, very thin-walled borosilicate glass bucket and were attached t,o a chain through a port ( A ) (Figure 1). This was lowered into position inside the furnace by lowering a rod ( B ) until the bucket was positioned correct'ly inside the furnace. This position was then secured by a locking clamp (C). Figure 2 shows a detailed sketch of t,he interior of t'he furnace. The bucket was positioned centrally inside a copper tube (D).The initial position of the bucket, was measured by a cathetometer ( E )(Figure 1)from a marker on the chain through a window ( F ) . The furnace !vas made from a 3-inch cylinder of aluminum 12 inches long with a 3/a-inch hole drilled axially through its cent>er (Figure 2). This was wrapped with a layer of asbeet>osupon which a Sichrome heating element was uniformly viound, spaced a t 1 / 4 inch per turn. Four washer-shaped aluminum baffles were placed above the mouth of the copper tube ( D ) to minimize convection currents. This copper tube which surrounded the sample of explosive was connected to a threaded plug ( H ) by means of a l/s-inch diameter steel rod. A 3is-inch well was drilled 8 inches deep in the furnace wall to accommodate a thermopile ( J ) consisting of ten iron-constantan thermocouples connected in series (Figures 1 and 2). The tvell cont'ained silicone oil t o provide good thermal contact between the furnace walls and the t,hermocouple junc,tions. The cold junctions \\-ere placed in a small glass tube containing kerosine R-hich mas submerged in an ice bath (Figure 1). The voltage from the thermopile was balanced against a reference voltage from a dry cell approxiinately equal t o that from the thermopile. The voltage difference was then "chopped" and amplified, and t,hen fed t o four GAS; vacuum tubes which were used as current amplifiers t o control the heating element. This amplified signal controlled approximately 0.1 of the heater current in a degenerat,ive manner. The unit was sensit,ive to a temperature change of 0.01" C. An iron-constantan thermocouple (G) was hard soldered t,o the surface of t,he copper t'ube (D).This thermocouple vim used t o ascertain the exact temperature in the sample. To det,ermine the t,emperature yithin the copper tube in t'he region occupied by the sample of explosive, an iron-constantan thermocouple probe which had h e m carefully calibrated to read the temperature of pure molten lead (melting point', 327.4" C.),
1090
Jane 1956
1091
INDUSTRIAL AND ENGINEERING CHEMISTRY THERMOPILE WELL
1 rBAFFLE5
(0
I
NlCHROME HCATING ELEMENT
pj lj
11
'I,
TO
GAS ANALYSIS APPARATUS
QUARTZ SPRING
CATHETONETER
(E) 'ER
TUBE (Dl
THERMOCOUPLE
SAMPLE BUCKET
POTE
THERMOCOUPLE
(GI
BALANCE UNIT
Figure 1.
Schematic diagram of isothermal decomposition apparatus
tin (melting point 231.9' C.), and the boiling point of water was inserted axially through the furnace into this region. Readings were corrected t o give the temperature of the probe. The quartz springs used in this apparatus had constants ranging from 0.1000 t o 0.1300 gram per inch. The spring was calibrated in its operational position by elevating rod ( B )(Figure 1). T h e cathetometer had a least count of 0,0001 inch; however, due t o occasional small oscillations of the spring during a reading, the precision was reduced to approximately 0.0005 inch. This corresponded t o an accuracy of 0.1 mg. in the weight of the sample in the bucket, equivalent t o analytical balance accuracy. T h e samples used were of the highest purity obtainablemelting points coincided within 0.2" C. with the melting point of the pure compound. Where the accurate melting points were not known, the samples were purified by repeated recrystallization. Preliminary studies of the decomposition of T N T showed t h a t appreciable vaporization occurred simultaneously with decomposition. Thus the total weight loss was not representative of decomposition alone. T o separate the rate of decomposition from the total rate of weight loss in this case, a reaction bulb consisting of a borosilicate glass flask of 96.7-m1. capacity, including the volume of the stem up t o the level of the high temperature bath, was used. The sample was placed in a delivery spoon mounted in the stem, some 7 em. above the level of the bath, and was introduced into the reaction bulb by rotating the spoon in its ground glass housing. The stein was chosen as small as possible (5 mm. diameter) consistent with trouble-free delivery of the sample. The high temperature bath consisted of a mixture of 45% of sodium nitrate and 5570 of potassium nitrate by weight which melts a t 220' C. The bath was heated by two lOOO-watt, 115-volt stainless steel immersion-type heaters positioned in a spiral around the reaction bulb. The outsputof the heaters was controlled by a variable transformer. Fiber glass insulation, approximately 3 inches thick, was placed on the bottom and around the sides of the bath container, and a 1/4-inch-thick sheet of asbestos was placed over the top. Provisions were made initially t o control the furnace temperature of the bath by means of the same electronic device used t o control the furnace temperature in the weight-loss apparatus described above. However, with the nearly 20 lb. of the NaN08-KNOs mixture in the bath the temperature could be held constant within 0.2" C. for the duration of a run by merely adjusting the opening of the asbestos cover. The bath temperature was
Figure 2. Schematic diagram of furnace in isothermal decomposition apparatus
measured by means of a precision grade partial immersion mercury thermometer placed in the reaction bulb. The pressure measurements were read from a mercury manometer connected to the reaction bulb through 30 inches of capillary tubing. RESULTS
The temperature variation of the reaction rate data obtained in this study in general followed the Arrhenius equation The data were also expressed in terms of the Eyring absolute reaction rate theory equation
k'(T) = A'Te-AsURT (5) (The Arrhenius form of k'( T ) is included along with the Eyring form for convenience in making comparisons with data in the literature. The Arrhenius form can be transformed into the Eyring form with sufficient accuracy simply by setting the Arrhenius frequency factor A equal to A'T using the mean value of the experimental temperature range for T . The data presented in this article were all handled in this manner.) The experimentally determined values of A , A', AE,, and A H $ for all explosives studied are given in Table I; they were obtained in all cases by a least-squares fit of the experimental loglo k'( T ) us. 1/T curves. All weight-loss measurements were made in the presence of air a t atmospheric pressure. Ammonium Nitrate. Samples of ammonium nitrate from 0.1234 t o 0.1778 gram were used. Measurements were made from 217.5 to 267.1' C. As shown in Figure 3 the decomposition was first-order throughout with no autocatalysis. Pentaerythritol Tetranitrate (PETN). The decomposition of PETK was examined over the temperature range 137.4 to 157.0' C. using samples ranging in size from 0.0783 to 0.1103 gram. The rate of decomposition below and above the melting point (139" C.) accelerated autocatalytically at the start, and subsequently became first-order. Above the melting point considerable frothing and NO2 evolution were observed during the autocatalytic portion of the decomposition. The isothermal decomposition curves are shown in Figure 4. Values of k ' ( T ) obtained from the slope and intercept of the autocatalytic portion of the decomposition are also given in Table I.
1092
INDUSTRIAL AND ENGINEERING CHEMISTRY
Ethylenedinitramine (EDNA). The kinetics of the decomposition of EDNA were studied over the temperature range from 144.4 to 163.5" C. All runs were made below the melting point (177-179' C.). The samples ranged in size from 0.1266 to 0.2368 gram. At the higher temperatures the samples were observed to darken to a reddish-brown color with the evolution of NOg. At the lower temperatures only a slight discoloration was observed. The decomposition v a s first-order a t the start (Figure 5 ) . After a certain period, ho-rl-ever,the decomposition rate decreased and merged into another apparent first-order segment of lesser slope. This apparent first-order segment prevailed for considerable time then gradually merged into a stage of steadily increasing rate. Above 161.0" C. only one firstorder slope was obtained corresponding to the initial slope of all the other curves.
iI
Figure 3.
Experimental plots of log u; cs. time for isothermal decomposition of I\TH[4NQ3
The general nature of the decomposition curves (Figure 5 ) indicated that the initial slopes correctly represent the true decomposition rate of EDNA. The fact that this slope lasted only a relatively short time suggests the formation of a stabilizer instead of the usual autocatalyst as one of the products of decomposition. Trinitrophenylmethylnitramine (Tetryl). The decomposition of tetryl was studied over a temperature range from 131.9" to 163.7" C. The samples ranged in size from 0.1121 t o 0.2218 gram. A plot of loglo (I - W ' / U O ) gave curves showing niarlted autocatalysis a t all temperatures (Figure 9). Decomposition eventually decreased in rate after remaining autocatalytic for an appreciable time a t temperatures below 152' C. The slopes of these curves were extrapolated to zero time to obtain the initial rate of decomposition. Hydrazine Nitrate. The thermal decomposition of hydrazine nitrate was studied over the temperature range 188.7' to 220.4' C. The material used for these experiments was dried in a desiccator over concentrated sulfuric acid for several days. This drying process was not adequate, however, to dry the 0.0563 t o 0.1523 gram samples completely. A plot of loglo (1 - W ' / W O ) us. t gave straight lines (Figure 6 ) . As seen from these curves there was an initial rapid loss of weight in the samples aa the water still present was being expelled. This was followed by a straight segment in the curve which was maintained until the onset of autocatalysis. The experiment run a t 188.7' C. was measured over a period of 6 hours. (Only a small portion of this curve is shown in Figure 6.) The decomposition remained firstorder for the entire period with no evidence of autocatalysis. Approximately 0.5 hour was required to drive off t'he water in the sample. The experimental run measured a t 220.4' C. required 4 minutes t o expel the moisture from the sample and 10 minutes later became noticeably autocatalytic.
Vol. 48, No. 6
Trinitrotoluene (TNT). As mentioned above ,initial studies of the decomposition of T N T by the weight-loss method showed that t,he total weight loss was not a measure of decomposition alone since vaporization accompanied the decomposition. Since weight loss measurements were not applicable pressure measurements were used. However, any over-all pressure measurement would not necessarily differentiate between vapor-phase and liquid-phase decomposition, neither would it distinguish between a pressure increase resulting from decomposition as contrasted to one resulting from vaporization. According to Robertson (IS) the vapor pressure of T N T is given by the equation log,, pv(mm. Hg) = 9.11
3850
-T
for T in O K. The vapor pressure of T N T was determined from this equation a t the particular temperat'ure of the experiment. This, along with a knowledge of the volunie of the spparatus, enabled appropriate sample sizes to be selected such that complete vaporization was assured. For example, a t 250" C. (523.1"K.) the vapor pressure of T N T is 56.2 nun. Hg. -1 sample n-eighing 0.0095 gram would, upon vaporization, exert B pressure of only 14.1 mm. Hg in t'he react,ion flask used assuming the vapor to behave ideally. The apparatus was evacuated in order to vaporize the sample as quickly as possible. .Ifter vaporization was complete the system was pressurized by various amounts. This was done to observe possible effects due to pressure as well as to reduce the rate of diffusion in the stem of the reaction flask. The vapor-phase decomposition was studied over a tempera,ture range from 250.0' to 301.0" C. in the presence of air a t pressures ranging from 35 to 540 mm. Hg. Sample sizes ranged from 0.0074 to 0.0145 gram. The interesting result was that the total pressure change in each instance was just that necessary to account for the vaporization of the sample. There was no measurable decomposition of the T N T vapor over the above temperature and pressure ranges (Table 11). The above results indicated the possibility of studying the decomposition of T N T strictly in the liquid phase with no contribution whatsoever from the vapor phase in the ranges of temperature and pressure mentioned. Thus, if a sufficiently large total sample Tvere delivered into the reaction flask, part would vaporize and part would remain as a liquid. The part in t'he liquid state undergoing decomposition would be the t o m 1 original sample less the amount in the vapor state after equilibrium xa8 reached a t the temperature of the experiment. The amount in the vapor state could be determined from the volume and temperature of the reaction flask, and the vapor pressure of T N T a t that temperature as given by Equation 6. For first-order liquid phase decomposition
- d71 - k'(Tjn dt
(7)
If the increase in pressure due t o decomposition is proportional to the decrease in the number of moles of liquid T N T present, then dp = - Cdn, or p = C(no-n). Hence n = ( C n o - p ) / C , and
Cno represents the total pressure which would be obtained were the entire liquid sample t o decompose-i.e., the pressure ohtained for large t if the decomposition remained first-order. The evaluation of Cn0 requires some understanding of the composition of the products of decomposition of TNT. Assuming that the decomposition products of T N T are CO, C(s), hydrogen, and nitrogen, about 10 moles of gaseous decomposition products should be produced per mole of T N T decomposed. Any error in this assumption will influence only the frequency factor and
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1956
Table I. A, Sec.-l
Explosive Bmmonium nitrate
PETN
Autocatalytic Extrapolated EDNA Initial Autoinhibitory Tetryl Hydrazine mononitrate TNT Vapor Liquid
Experimentally Determined Kinetic Parameters
A', See.-' Deg.
Prob. Error in, A and A See. - 1 D e i .
Prob. Error in A E and AH$. A E and A H $ , 8al./Mole d)al./Mole 38,300 i- 300
c.
217.5-267.1
0.1234-0.1778
i-1500 &loo0
137.4-157.0 137.4-157.0
0.0783-0.1103 0.0783-0.1103
30,800 71,700 34,900
12000
~O*O.SO
*11600 1500
144.4-163.5 144.4-162.1 131.9-163.7
0.1266-0.2368 0.1266-0.2368 0.1121-0.2218
10+0.'8
38,100
ill00
188.7-220.4
0.0563-0.1523
* 800
250.0-301.0 237.5-276.8
0.0074-0.0145 0.1032-0.1862
108.67
10"O."
1023.1 1015.3
1020.5
1Oi0.70
1012.0
10'0.63
1011.1
105.5
10*'.01
1031.6
10u.9 10lQ.a
1Oi0.83
~012.0
10'2.1
108.6
No measurable decomposition 109.47
Sample Size, Gram
Temp. Range,
52,300 38,600
1012.28
1012.18
1093
43,400
10'0.52
able (Figure 8). No difference in the rate was detectable when the decomposition was carried out in the presence of nitrogen in place of air. DISCUSSION OF RESULTS
Robertson studied the decomposition of AN, PETN, and E D N A by pressure measurements in the presence of air a t atmospheric pressure. Similar measurements were made in nitrogen. The values of the Arrhenius constants reported for amnionium nitrate were ( 1 1 )A = sec.-l and AE,= 40.5 kcal./mole, corresponding t o a k ' ( T ) a t 250" C. about four times larger than that calculated from the data given in Table I. This is due primarily to the larger value of the frequency factor reported by Robertson-namely, 1013.8 see.-' compared with our 10'2.28 sec. - 1 . Since no autocatalysis was reported for either the pressure or the
I
'
Figure 4.
I
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1
TIME ( H O U R S )
Experimental plots of log w us. time for isothermal decomposition of PETN
the activation energy will remain unaffected. Therefore the value of Cno should be given approximately by the equation
With the liquid sample sizes used Cno was of the order of 2000 mm. Hg. Since the measured pressure increase, p , was not more than 10 mm. Hg, p was small compared with C m and the lefthand side of Equation 8 may thus be written --In (1 - p / C n ~ ) =p/Cno ignoring second and higher order terms. Thus it is seen that for Cno large compared with p , Equation 8 may be written p = Cnok'(T)t
1635%
(10)
A plot of p us. t for first-order decomposition should therefore yield a straight line with slope equal to Cnok'( T ) . The decomposition of liquid T N T was studied over a temperature range from 237.5' t o 276.8" C. in the presence of air a t slightly less than atmospheric pressure. The samples varied in size from 0.1032 to 0.1862 gram. After an initial rapid Table 11. Results of TNT Vapor Phase Decomposition Approxipressure increase due to vaporimate Time zation, the decomposition reVapor Total Required TemperaPressure of Pressure Sample Pressure for Vapormained first-order until the ture Apparatus, of T N T , Size, ization, Mm. Hg Gram r%?."fI!i Sec. 0 c.' Mm. Hg onset of autocatalvsis. The 250.0 124 56.2 0.0095 14.1 80 plot of p us. t suggested by 250.7 37 57.5 0.0074 11 .o 40 128 162 0.0127 20.1 90 Equation 10 gave consistently 284.5 288.4 540 234 0.0145 23.5 60 straight lines before auto301.0 354 251 0.0122 19.9 50 catalytic effects became notice-
Duration of Run, Min.
35 15 30 45 45
INDUSTRIAL AND ENGINEERING CHEMISTRY
1094
7
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The values of A and A E p given by Robertson (12) for E D N A m-ere A = 1012.8 sec. and PEP= 30.5 kcal./mole. The latter is in excellent agreement with the 4 E pvalue reported in Table I ; hon-ever, Robertson's A mas greater by a factor of 50. This may be associated with the autocatalytic effects involved a t the higher temperatures, as mentioned above. As seen from Figure 5 the decomposition of E D S d was wholly autocatalytic a t temperatures above 163" C. The range covered by Robertson was 184" to 254" C. Tomlinson ( 1 7 ) also studied the thermal decomposition of E D N A but primarily in aqueous solutions. Some of his work did include pure E D S A , but he gave no activation energy and frequency data for pure EDNA decomposition. Several investigators have studied tetryl decomposition: AEp, Temperature A, Kcal./ Range, Investigators (Sec. -1) Mole O c. Rideal and Robertson ( 1 0 ) 1015.4 38.4 211-260 1027.6 60.0 ... Roginsky (16) Liquid 1024.5 52 0 ... Solid 102'.5 52 0 Farmer ( 7 ) .. 39 0 iiili20 This invkstigation (Table I ) 10'2.9 34 9 131 9-163 7 I n addition, Hinshelwood (9) showed that the decomposition of tetryl was autocatalytic with picric acid as the catalyst. The data in Table I11 of Rideal and Robertson were in closest agreement with those obtained in this investigation, ho\\ ever Rideal and Robertson values for k ' ( T ) data were about cight times greater a t 190" C. Since tetryl decomposed autocatalytically from the start, k'( T ) data evaluated from the data in Table I are
I
Vol. 48, No. 6
any effort made to distinguish between the pressure increase due t o vaporization and that due to decomposition. For instance, the data reported by Robertson reflect some combination of vaporization and decomposition, and a t 275" C., k ' ( T ) is GOO times that computed from the data in Table I. Of the six explosives described in this article only pure ammonium nitrate decomposed strictly first-order over the entire deconiposition range. T h a t ammonium nitrate showed no anomalous ther-
temperature studied. Hence, none of these products can accumulate in the explosive in sufficient amount to influence significantly the rate of decomposition. It is clear from the shape of the thermal decomposition curves for PETT that previous theoretical treatments based wholly on t'he linear (aut'ocatalytic) section of the decomposition curves are inadequate. The true rate constant must be determined before appreciable accumulation of products of decomposition. This presents a problem, however, since it is difficult to evaluate k'( T j a t zero time. The value a t zero time was approximated bjmeans of a mirror as follows. A curve was carefully passed through each of the points obtained during an experimental run. A front-surface mirror was placed on edge a t the origin of the curve so t h a t the plane of the mirror was perpendicular to the plane of the paper. The mirror was pivoted about the origin of the curve until the curve and its reflection in the mirror formed one continuous curve. A straight line was then scribed along the edge of the mirror through the origin of the curve. A perpendicular was erected from this line a t the origin, and t8heslope of this perpendicular was taken t o be the initial slope of the curve. This technique v a s subject to random errors of considerable magnitude, but it was felt that it v a s better than any purely visual technique if for no other reason than that of removing prejudice on the part of the investigator. This technique v-as also used in the tetryl study. It was fortunate in the case of EDNA t h a t the true rate constant could be measured before t,he accumulation of the rate inhibitor became sufficient to mask the decomposition. The decomposition of hydrazine mononitrate presented no difficulties in the analysis of the data obtained. Autocatalysis became appreciable a t the higher temperatures of the experiment, but only after sufficient time to permit a direct measurement of the uncatalyzed portion of the decomposition.
June 1956
I
INDUSTRIAL AND ENGINEERING CHEMISTRY I
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5
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1
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5
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1095
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980
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E
7 960
5
h
;BT
940
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s i920
0
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900
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k
I 20
I 40
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I 60
80 TIME
I
100
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140
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160
(MINI
’
WS.
+
(12)
re = 0.6745
(n X )’’ and*D - 2
= nZz,* - ( Z Z , ) ~
and di are the residuals, i.e., the differences between the observed and calculated values of y, z is any point not necessarily a measured value, ra is the probable error in a, rb t h a t in b, and r y t h a t in y. Here the term probable error has its usual significancei.e., that deviation which, taken on each side of the mean of a normal distribution curve, would include half of the area under the curve. This corresponds to confidence limits of 50%. The probable errors listed for the frequency factors and activation energies for the six explosives studied were computed by means of Equations 11 and 12. NOMENCLATURE
w
w’
= original sample weight = weight of sample present a t time = weight decomposed a t time t
F i g u r e 8.
1JT
I n the case of T N T , pressure measurements became unambiguous after the pressure increase due t o vaporization was properly accounted for. Autocatalysis always eventually appeared at the higher temperatures, but sufficient time was available for a direct measurement of the decomposition before it became noticeable. Since the uncertainties involved in the above measurements were indeterminate in nature a n analysis based on random errors is appropriate. Consider a number of points through which, by least squares, the most probable straight line is to be passed. The first important fact concerning the method of least squares is t h a t it applies strictly only t o sets of measurements whose errors follow the Gaussian error curve. It may be proved t h a t this curve should be followed if the errors are purely random in nature, are equally likely t o be positive or negative, and are more lilcelytobe small than large, with very large errors entirely lacking ( 2 9 ) . Following the discussion of Birge ( I ) , if the equation of bx, the the straight line obtained by least squares is y = a probable errors in a, b, and y-Le., the function itself, assuming all points to be equally weighted-are
wo
I
I
I
I
I
5
10
15
20
2s
30
TIME (mlo ]
Figure 7. E x p e r i m e n t a l plots of loglo (1 - w’/wo) time for isothermal decomposition of t e t r y l
where
I
I
t
E x p e r i m e n t a l plots of pressure vs. t i m e for the isothermal decomposition of TNT
20’
= -
wo
- fraction of
sample decomposed at time t
k’( T ) = specific reaction rate constant K = thermal conductivity = average temperature of the apparatus T,, T = absolute temperature TO = (effective) initial temperature (absolute) p = density c = specific heat = heat generated per unit mass of explosive decomposed Q = dimensionless constant expressing the order of reaction f 7 = time lag between impact and initiation a, /3 = theoretical constants involving the heat and entropy of activation A = Arrhenius frequency factor k eASS/R A’ =h k = Boltzmann constant h = Planck constant A s $ = entropy of activation AH’; = heat of activation AEp = experimental activation energy t = time pv = vapor pressure = pressure increase due to decomposition p n = moles = original number of moles no C = pressure increase per mole of explosive decomposed V = volume LITERATURE CITED
(1) Birge, R. T., Phys. Rev. 40, 207 (1932). (2) Bowden, F. P., Gurton, 0. A . , Proc. R o y . SOC.A198, 337 (1949). (3) Bowden, F. P., Mulcahy, M. F. R., Vines, R. G., Yoffe, A , , Ibid., A188, 291 (1947). (4) Cook, M. A., Horsley, G. S.,Partridge, W. S., Ursenbach, W. O., J . Chem. Phys. 24, 60 (1956). (5) Cook, M. A., Mayfield, E. B., Partridge, W. S.,J. Phys. Chem. 59, 675, Table I V (1955). (6) Copp, J. L., Napier, S. E., Nash, T., Powell, W. J., Skelly, H., Ubbelohde, A. R., Woodward, P., Phil. Trans. Rou. SOC. A241, 197 (1949). (7) Farmer, R. C., J. Chem. SOC.117, 1603 (1920). (8) Henkin, H., McGill, R., IND. ENG.CHEM.44, 1391 (1952). (9) Hinshelwood, C. N., J . Chem. Soc. 119, 721 (1921). (10) Rideal, E. K., Robertson, A . J. B., Proc. R o y . SOC.A195, 135 (1948). (11) Robertson, A. J. B., J . SOC.Chem. I n d . (London) 67, 221 (1948). (12) Robertson, A. J. B., Trans. Faraday SOC.44, 677 (1948). (13) Ibid., p. 977. (14) Robertson, R., J. Chem. SOC.119, 1 (1921). (15) Roginsky, S., 2. physik. Chem. B18, 364 (1932). (16) Roginsky, S., 2. phusik. Sowjet. 1, 640 (1932). (17) Tomlinson, W. R., J . Org. Chern. 17, 648 (1952). (18) Urbanski, T., Rychter, Cornpt. rend. 208, 900 (1939). (19) Whittaker, E. T., Robinson, G., “The Calculus of Observations,” Blackie and Son, Ltd., London, 1924. (20) Yoffe, A. D., Proc. Roy. SOC.A208, 188 (1951). RECEIVED f o r review August 13, 1955. ACCEPTED February 10, 1956. This investigation was supported by Office of Naval Research, contract number N7-onr-45107, project number 357239.
