P. W. M. JACOBS AND A. RUSSELL-JONES
202
Sublimation of Ammonium Perchlorate by P. W. M. Jacobs] and A. Russell-Jones Department of Chemistry, Imperial College, London, England
(Received J u n e 30,1967)
The kinetics of the sublimation of ammonium perchlorate has been investigated over a wide range of temperatures a t pressures ranging from the vapor pressure up to 1 atm. A model for the sublimation of ammonium perchlorate, which takes account both of surface diffusion and of diffusion through the gas phase, has been formulated, and a new equation describing the kinetics of sublimation has been developed. This theory is in excellent agreement with the experimental data. The evaporation coefficient, /3, is strongly pressure dependent, indicating the importance of surface diffusion in the evaporation process.
I. Introduction The first comprehensive study of the thermal decomposition of ammonium perchlorate (AP) to be published was that by Bircumshaw and N e ~ m a n . ~These ?~ authors showed that at temperatures below about 300°, AP decomposes to a limited extent (-30%) leaving a residue which is itself AP, apparently identical chemically with the starting material but having a larger specific surface area.4 If the temperature is raised, sublimation of this residue occurs. Bircumshaw and Phillips6 made a limited study of the sublimation process and deduced from their data an activation energy for the rate of sublimation of -21 kcal/mole. If the pressure of an inert gas in the reaction vessel is increased, sublimation is replaced by thermal decomposition. Galwey and Jacobs6 proposed that the mechanism of this thermal decomposition was dissociation of the AP into NH, and HClOd, followed by their evaporation into the gas phase, and subsequent decomposition of the perchloric acid and oxidation of ammonia in the gas phase. From their kinetic measurements,6 an activation energy of -39 kcal/mole was deduced, but as the reaction was followed by pressure measurements, it was not clear whether the sublimation process or the gas-phase reactions were rate determining. We have therefore reinvestigated the kinetics of the sublimation and thermal decomposition of AP using the technique of weight-loss measurements. Results for the low-temperature decomposition have been described in another p~blication;~ in this paper we discuss the sublimation of the AP residue.
11. Experimental Procedure Calwey and Jacobs6 had found that the rate of the thermal decomposition of AP was unaffected by the presence of molecular oxygen, and so the rate measurements at 1 atm were performed in air using a Stanton thermogravimetric balance. Measurements of weight loss under controlled atmospheres and a t subatmospheric pressures were performed by measuring with a The Journal of Physical Chemistry
cathetometer the change in extension of a calibrated silica spiral spring from which a compressed pellet of AP was suspended. The sensitivity of the spring was 1 mm/mg. The AP was supported on a platform suspended from the spring by a straight length of fused quartz, which passed through a short length of 3-mm capillary tubing. The side arm, through which pumping was effected, was located adjacent to the capillary and this arrangement prevented the condensation of sublimate onto the spiral. All taps and joints were greased with silicone grease. The cylindrical furnace around the reaction vessel was sited so that the AP was located in the center of the zone of constant temperature and its position was adjusted periodically during the run to maintain this position. [Pellets of 40 mg were generally used so that the change in position of the platform support during a run amounted to about 4 cm.] The low-temperature reaction was allowed to proceed to completion, first, with the system open to atmosphere to reduce sublimation to a minimum. The system was then isolated, the pressure and temperature adjusted to the required values, and the change in weight of the AP followed. Reagent-grade AP from British Drug Houses (BDH) or Matheson Coleman and Bell (MCB) was used as received. Pellets were formed in a cylindrical stainless steel press of 3-mm bore.
111. Results The curves for the fractional decomposition (a) of (1) Department of Chemistry, University of Western Ontario, London, Ontario, Canada. (2) L. L. Bircumshaw and B. H. Newman, Proc. Roy. SOC.(London), A227,115 (1954). (3) L. L. Bircumshaw and B. H. Newman, ibicl., A227, 228 (1955). (4) A. K. Galwey and P. W. M . Jacobs, ibid., A254, 455 (1960). (5) L.L.Bircumshaw and T. R. Phillips, J . Chem. SOC.,4741 (1957). ( 6 ) A. K.Galwey and P. W. M. Jacobs, ibid., 837 (1959). (7) J. V. Davies, P. W. M. Jacobs, and A . Russell-Jones, Trane. Faraday Soc., 63, 1737 (1967).
SUBLIMATION OF AMMONIUM PERCHLORATE
203
time (mid
0
80
40
120
160
time (mid
Figure 1. Test of the applicability of eq 1 and 2 to the sublimation of AP a t 304": 0, eq 1; 0 , eq 2.
Figure 3. Test of the applicability of eq 1 to the vacuum sublimation of AP at 247.5".
;
I -3
9.0 4.8 1.5
1.6
1.7
OYT'K
I
Figure 2. Arrhenius plots for the sublimation of AP, using rate constants derived from eq 2: 0, MCB AP (40-mg pellets); 0,BDH AP (40-mg pellets); V, twice recrystallized BDH AP (40-mg pellets); A, BDH A P (70-mg pellets); 0, BDH AP (100-mg pellets).
- CY)"' = kt
(1) Figure 1 shows that this equation only applies over the first two-thirds of the reaction. Empirically, the exponent was altered from to l/z when a much better representation of the data, up to CY = 0.93, was obtained. The modified equation 1
-
(1
- CY)"' = kt
400 600 pressure (torr)
Figure 4. Variation of the sublimation rate constant k [from eq (l)]with the ambient pressure of inert gas, P.
residue against time ( t ) under 1 atm of air were deceleratory in the temperature range 304-375". If a reaction occurs on the surface at a rate proportional to the surface area and propagates inwardly at a rate aok, where a. is the initial value of some characteristic size parameter (e.g., the radius for a sphere of reactant), then the kinetics should conform to the contracting-volume expression 1 - (1
200
0
(2)
was therefore used to analyze the high-temperature data. Plots of log k vs. 5"-l are shown in Figure 2. These illustrate that k is independent of the source and chemical purity of the AP, but that the rate constant decreases as the mass of the reactant is increased. The activation energy, E, is 30.6 kcal/mole. The higher value for E, found from pressure measurements, indicates that gas-phase reactions are rate determining in this temperature range and, therefore, that the rate process being followed by measurements of weight loss, is sublimation under 1 atm of air. Sublimation rates under vacuo were measured, using the quartz spiral spring. The data are well fitted by eq 1 up to, at least, CY = 0.90. The applicability of this equation is illustrated in Figure 3. The activation energy is 30.0 k.cal/mole. The pressure dependence of
the sublimation rate was measured a t 270". The results (Figure 4)show that there is a dramatic decrease in the rate constant [obtained from eq 11as the pressure, P, of nitrogen is increased, but that at higher pressures the rate tends toward a constant value. The total decrease in rate as the total pressure is increased from the equilibrium pressure of AP to 1 atm is about 200fold
IV. Theory The sublimation of ammonium perchlorate has been found empirically to fit either of two equations, both being of the general form 1
- (1 - CY)"'
= kt
(3)
For vacuum sublimation, y = 3, but as the pressure, y = 2 gives a better description of the data. We now examine the theoretical basis for these equations. The AP pellets were small cylinders, 3 mm in diameter and about 3 mm high. Since these dimensions are much smaller than those of the reaction vessel, no great error should result from approximating the shape to a sphere of radius a. Let n be the concentration (in molecules/cm*) of the subliming species at a distance r from the center of the AP, and let n1 be the (uniform) concentration of inert gas. Sublimation of AP is dissociative but, in the steady state, the flux of ammonia through the gas phase must equal that of perchloric acid: J A = - DA(bnA/ br) = J P = -Dp(bnp/br). It is convenient, therefore, to think of an effective mean flux, J , mean concentration gradient, bn/br, and mean diffusion coeffi-
P, of inert gas is increased,
Volume 76,Number 1 January 1068
P. W. M. JACOBS AND A. RUSSELL-JONES
204 cient, D. Then because of the assumed spherical symmetry
I n the steady state, bn/bt = 0, so that the diffusion equation (4) has the solution n = A + -
n = Oatr =
B r
(5)
If the boundary conditions
tuted in eq 11, the rate of sublimation is given by
Q
are imposed, then the flux is
(7)
= Dano/r2
and the rate of sublimation becomes =
_-divdt
=
_--1 dw = 4rDano mo dt
where no is the concentration of subliming species in the gas phase in equilibrium with the solid, N is the number of molecules of AP in the solid, mois the mass of one molecule of AP, and w is the mass of the AP pellet. Equation 8 is essentially the sublimation equation of Stefan,s Maxwell,9 and Langmuir.’O It is well known that eq 8 leads to an absurdity for spheres of very small radii; the rate of sublimation per unit area is
Q/4aa2 = Dno/a (9) as a -t 0. The cause of this cataswhich tends to trophe is the assumption that the macroscopic diffusion equation (4) holds right up to the surface of the subliming solid [see (6)]. This difficulty was resolved by Fuchs,ll who modified the Stefan-Maxwell formula (8) in the following way. If the molecules evaporating from the solid surface were unaffected by the molecules in the gas phase in any way, then the rate of sublimation, once equilibrium was attained, would be
where /3 is the evaporation coefficient and E is the mean velocity of the subliming molecules in the gas phase. Let the concentration in the gas phase at a distance A from the surface be n* < no; the diffusion flux at r = a A is then
+
J
n=n*atr=a+A
(6)
n = noatr= a
Q
co
If eq 8 and 9 are solved for n* and the result is substi-
n=Oatr=oJ
J
The Fuchs model thus considers that molecules will evaporate from the surface as though unaffected by the molecules in the gas phase, but after they travel a distance A, of the order of the mean free path, they start to undergo collisions and the macroscopic diff usion equation (4) applies, but with the modified boundary conditions
= Dn*/(a
+ A)
(11)
and this must equal the net rate of arrival, per unit area, of subliming molecules at r = a A
+
ri
The Journal of Physical Chewviatry
1
= 47r(a
+ A)2J =
4 .IrDano D a apv a A
-+- +
(13)
where v has been written for c/4. Note that eq 13 degenerates to eq 10, when A >> a, and to eq 8 when a >> A and also a >> D/pu. If a >> A, but no assumption is made as to the validity of the second inequality, a >> D/pv, then eq 13 reduces to
We anticipate that A should be of the same order as the mean free path of the subliming molecules
A%--(-) 37r D 16 2v
The substitution (15) reduces eq 14 to
Apart from the factor 3r/16 ‘v 1/2, eq 16 is the sublimation equation of Monchick and Reiss,12 who used a nonequilibrium velocity distribution function. The experimental evidence does not permit an unequivocal decision on the correct form for A, although an analysis by Wrightla suggests that A = D/2v without the factor 3n/16 is marginally the better expression for the evaporation of liquids. Therefore, for the present, we shall retain A as an unidentified parameter.
(8) J. Stefan, Wien. Be?., 68,386 (1874); 83, 645,943 (1881); 98, 1410 (1889). (9) J. C. Maxwell, “Scientific Papers,” Vol. 2,Cambridge University Press, London, 1890,p 638. (10) I. Langmuir, Phys. Rev., 12, 368 (1918). (11) N. A.Fuchs, Physilc. Z.Sowjetunion 6, 224 (1934); R. S. Bradley, M. G. Evans, and R. W. Whytlaw-Gray, Proc. Roy. Soc. (London), A186,368 (1946). (12) L. Monohick and H. Reiss, J . Chem. Phys., 22, 831 (1954). (13) P. G. Wright, Discussions Faraday Soc., 30, 100 (1960).
SUBLIMATION OF AMMONIUM PERCHLORATE Let a. denote the initial radius a t t = 0, and fraction sublimed at time t so that 1-
CY
= as/ao8
205 CY
the a
(17)
Substituting (-4mzpm)(da/dt) for Q in eq 13 gives a differential equation which is integrable by elementary methods; on using eq 17, the result is 10
0
20
40
30
50
60
time (min)
Figure 5. Test of the applicability of eq 18 to the sublimation of AP under 1 atm pressure of nitrogen at 375": , theory; 0, experiment.
-
This is an equation describing the kinetics of sublims tion, which is of the form a ( X l , X z , X 3 , t) = 0, where X I , X z , Xaare the three parameters
Xi Xz
=;
Ala0
(D/@A)
0.2
-1
(19) 0
X a = Dmno/ao2p
50
150
100
time
Equation 18 is not amenable to testing by plotting some simple function of CY against t, except in certain limiting situations. Therefore a computer program was written which, using the usual least-mean-squares criterion, found the values of X I , X z , X3,giving the best fit of the equation to the experimental data. Comparison with Experimental Data That eq 18 can provide an excellent fit to the experimental data is illustrated by Figures 5 and 6 . Unfortunately it iti not possible, however, to determine the individual values of the three parameters with equal precision over the whole range of pressure and temperature. The third term on the left-hand side of eq 18 only contributes significantly when A is of the same order as ao. This means that, except over a narrow pressure range, only the product X I X z can be determined, the least-squares procedure being insensitive to the individual values of X 1 and X z . Furthermore, a t very low pressures, A >> ao,X I X z>> 0.5, and eq 18 degenerates into the "contracting volume" formula, eq 1, with
V.
-
At moderate pressures of inert gag, P 1 atm, A > 1, then eq 20 reduces to
from which p may be calculated. The group of terms XI/XIX2 may be determined with greater precision than the individual parameters, and hence there is less scatter in p, the mean value for vacuum sublimation being 0.041 f 0.006. The error introduced by the approximation (about 2%) is thus much less than the mean deviation and so a more accurate calculation, involving a substitution for A, is hardly justified. The data for sublimation under various pressures of nitrogen at 270" were also well fitted by eq 18. Values of the evaporation coefficient determined by both of the Table I: Values of the Evaporation Coefficient, p , for (a) Vacuum Sublimation and (b) Sublimation under Various Pressures of Nitrogen at 270"
49 1 496 502 507 514 520 528 536 543
0.0398 0.0417 0.0412 0.0368 0.0476 0.0392 0.0500 0.0414 0.0294
7.75 11.0 22.0 38.0 105 312 760
Mean = 0.041 f 0.006. The Journal of Physical Chemistry
0.01007 0.00844 0.00504 0.00278 0.00129 0.00087 0.00020
0.00883 0 00824 0.00509 0.00274 0.00128 0.00082 0.00042 I
above methods appear in Table I. The measure of agreement is encouraging since, while neither method requires more than an order-of-magnitude knowledge of A ( p being small enough to justify the neglect of unity in comparison with DlvpA), X1X2 involves D, but not the vapor pressure, whereas the reverse is true for XdXlX2. Values of X3 were also calculated a t 270' from eq 19 and these are compared in Table I1 with the computed values obtained by fitting the experimental data to eq 18. At low pressures, XlX2 >> 0.5 and so excellent agreement cannot be expected. Even so, this being an absolute calculation with no unknown parameters, the measure of agreement is impressive.
Table 11: Values of Xa Calculated from Eq 19 and from the Experimental Sublimation Data X 104, min-1-
-Xa
P,torr
Calcd
Exptl
7.75 11.0 22.0 38.0 105.0 312.0 760.0
67.6 47.8 24.1 13.9 5.05 1.70 0.69
75.1 61.1 31.4 15.5 4.97 2.07 1.16
VI. Discussion Analysis of experimental data described in section V shows the sublimation of AP can be described very adequately in terms of Fuchs' ideas.'l The new kinetic equation (18) represents the data satisfactorily and explains why the empirical laws (1) and (2) fit the experimental data in certain pressure and temperature regimes. The evaporation coefficient varies from about 4 X loF2for sublimation in vucuo to about 5 X for sublimation under 1 atm. This can be understood in terms of the following model for the sublimation process. The process commences by the transfer of a proton from an NH4+ ion to a c104- ion at surface kink site. The NH3 and HC104 molecules formed remain adsorbed a t these positions for a finite time, but unless they can diffuse away to become adsorbed on separated sites away from the surface step, the proton-transfer process will reverse itself. The evaporation process is then completed by the desorption of NH3 and HC104 molecules into the gas phase. This whole process may be represented by (16) S. H. Inami, W. A. Rosser, and H. Wise, J . Phys. Chem., 67, 1077 (1963). (16) A. P. Hardt, W. M. Foley, and R. L. Brandon, Astronaut. Acta, 11,340(1965).
SUBLIMATION OF'AMMONIUM PERCHLORATE
G NHaHC104(ak)
NH4+C104-(1/2c) NH3(1)
+ HCI04(1)
2 -2
3 -3
NHa(a)
"a(d
207
+ HC104a) HC104(g)
where I/zc denotes the half-crystal site, ak the state in which the molecules are adsorbed a t the kink site, 1 where the molecules are adsorbed at ledge sites, and a where they are adsorbed on any site on the plane surface. The importance of the ledge sites arises from the fact that it will be easier energetically for the adsorbed molecules to leave the half-crystal position by diffusing along the ledge rather than by being ejected directly onto the plane surface. The various sites involved are shown schematically in Figure 7 . Clearly the reverse steps will be involved in condensation. The importance of surface diffusion is shown by the observed pressure dependence of p. As the pressure of nitrogen is increased, nitrogen molecules will be adsorbed on the AI' surface and the first sites to be affected will be just those sites (e in Figure 7 ) which are adjacent to kink sites. 'The diffusion of NHa and HClO4 molecules from kink sites to ledge sites will thus be impeded. It therefore seems natural to write
P
= kl(i
- e) + kZ
0
50
100
ICYP ( t o r t ' )
Figure 8. Test of eq 27 for the pressure dependence of 8. The means by which the constants kl and ks are evaluated is explained in the text.
(25)
where kl relates to the diffusion to ledge sites and kz (kl >> kz) to the direct diffusion from kink sites to a sites (Figure 7). The fraction of e sites covered by adsorbed nitrogen molecules is e. Then as P + 0, e -P 0, and p 4 kl k,, while as P 4 a ,0 + 1, and P --+ kz. TheUconstants kl kz and kz can therefore be found from the values of p for vacuum sublimation and for sublimation under high pressures of inert gas. The latter value is 4 X as found by a short extrapolation of the plot of p us. P-l to infinite pressure, and as P = 0.0294 for vacuum sublimation at 543"K, kl ~ 1 ! 0.029. Rearranging eq 25 and assuming a Langmuir adsorption isotherm for e, P gives
+
Figure 7. Schematic representation of the evaporation of AP: a, molecules adsorbed on the plane surface; b, molecules adsorbed at ledge sites; c, kink site (the half-crystal position); d, ions; e, favored sites for adsorption.
+
where x ( T ) is the product of the vibrational partition function of the adsorbed molecule p(T), Xo(T)the absolute activity & nitrogen molecule in the gas phase at the standard pressure, and e-uQ/lzT,where uois the potential energy of a nitrogen molecule adsorbed on an e site referred to that of a similar molecule a t rest in the
infinitely dilute gas as zero. Notice that 0 refers to adsorption on e sites only and that uo (which is a negative quantity) will be several times larger in magnitude than the corresponding quantity for a sites. Equation 26 predicts that [l - (P k J / k ~ ] should -~ be a linear function of P I , and this is confirmed by the data plotted in Figure 8. With endothermic processes such as sublimation, one has to bear in mind the possibility of self-cooling. I n connection with the thermal decomposition studies,6 it became necessary to correct for the weight loss due to sublimation, and so the sublimation rate under 730 torr of nitrogen was measured down to very low tempem tures. These data agreed well with the extrapolation of the high-temperature data reported here and the constant activation energy for a variation in log k covering more than three decades is not to be expected if self-heating is affecting the data significantly.
-
Acknowledgment. We are grateful to the United Kingdom Ministry of Aviation and to the Canadian National Research Council for their support of this work.
Volume 78, Number 1 January 1968
