1639
NOTES
variety of tests conducted at the Jet Propulsion Laboratory. The induction time of ground AP was roughly 600 hr longer than that of unground AP. The decomposition rate during the acceleratory period of ground AP (Table I) is approximately l / the ~ rate of unground AP during this same period.
50
\! I
40
30
20
IO
i
0,
I
-1
1
111. Summary In this study, one of the first gases detected in the decomposition of ammonium perchlorate at 135" was nitric oxide. It was also found that nitric oxide reacted rapidly with ammonium perchlorate when it was injected in a test tube above the ammonium perchlorate. Some of the gases detected in the long-term decomposition, other than nitric oxide, were HC1, water, and certain nitrosyl chlorides. When samples of ammonium perchlorate were exposed to y radiation, it was found that the induction time decreased with increasing irradiation time up to an irradiation time of 1000 sec. The induction time was found to be directly proportional to the log of the irradiation time for samples which were irradiated less than 1000 sec. When the ammonium perchlorate was ground, it took a substantially longer time to decompose than the unground ammonium perchlorate. Acknowledgments. This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.
A Note on Optimum Parameters for the Generalized Lennard-Jones Intermolecular Potential
by Robert C. Ahlert, Gabriel Biguria, and John W. Gaston, Jr. Rutgers, The State University, New Brunswick, New Jersey 08906
TIME, hrs
Figure 3. Decomposition curves for ground ammonium perchlorate maintained a t 135'.
ground AP are shown in Figure 3. A comparison of Figures 1 and 3 reveals that the ground ammonium perchlorate takes a substantially longer time to decompose. Although the mechanism which produces these differences in stability is unknown a t this time, these results were consistent and quite reproducible in a
(Received Awil 28, 1969)
In ref 1, the use of equilibrium and nonequilibrium data in the development of optimum potential function parameters was discussed. Emphasis wm placed on use of data in the range of temperatures below Te-12* = 2, as recommended by Klein and H a n l e ~ . ~An ,~ objective function defined on low-pressure transport properties yielded numerous local minimum points for both argon and methane. An objective function based (1) W. F. Vogl and R. C. Ahlert, J. Phw. Chem., 73, 2304 (1969). (2) M.Klein, J. Res. Nut. Bur. Stand., 70A, 259 (1966). (3) H. J. M. Hanley and M. Klein, National Bureau of Standards Technical Note 360,U. 8.Government Printing Office, Washington, D. C., 1967.
Volume 74, Number 7 April 2, 2.970
1640
NOTE&
upon thermodynamic properties over ranges of both temperature and pressure, the latter to as high as 200 atm, generated a universal minimum and unique potential parameters. This was true for both argon and methane, with either compressibility or the JouleThomson coefficient as the state variable of the objective function. The obiective is t o generate a useful potential energy model for molecular interactions from a minimum amount of experimental data. A relatively high degree of flexibility in the potential model is desired also. The work of ref 1 led the authors to a conclusion that thermodynamic data were a suitable basis for development of potential functions. However, the inclusion of thermodynamic data at elevated pressures requires the inclusion of a t least the third term in the virial equation of state. Numerical evaluation of the third virial coefficient involves the assumption of pairn-ise additivity. This approach is acknowledged to be unreal is ti^,^ while the numerical evaluation of third virial coefficients is quite time consuming, even with a large computer. Present work employed equilibrium data at low (zero) pressure. The immediate objective was to question the applicability of Klein’s criterion for temperature range. In addition, the choice of thermodynamic properties and the breadth of the temperature range required to ensure a unique potential function were examined. Because of an extensive literature, as well as the experience of earlier analyses, methane was chosen as the vehicle for this investigation.
The Curse of Multimodality A generalized Lennard-Jones potential in the form c
4(T) = -re
+c r’-l
was chosen, leaving the theoretically acceptable attractive exponent of G invariant. Convergent series solutions for the cluster integral] representing the second virial coefficient, are found in the Equivalent convergent series can be written for the temperature derivative of the second virial Coefficient. Three objective functions were examined. The first was based on the second virial coefficient, while the second employed the product of temperature and the derivative of the second virial coefficient. The temperature-derivative product was chosen to maintain dimensional consistency. The third objective function consisted of an unweighted, linear combination of the first two. Experimental second virial data were required in the first case. Experimental or derived Joule-Thomson coefficients, second virial data, and heat capacities, at zero pressure, were required in the second and third instances. Din’s compilations was chosen as the initial source of data. Zero-pressure heat capacities were based upon The Journal of Physical Chemistry
the experimental data of Rossinilgas reduced t o polynomial representation by Din.* T* is defined as the temperature divided by depth of the potential well for the Lennard-Jones model, reduced by the Boltzmann constant] i.e., T / e / k . An acceptable value of the 6-12 Lennard-Jones well depth for methane is approximately 145”Ii.l Thus, T6-12* = 2 corresponds to a teniperature of about 290°K. This study was initiated with G data points at 20” intervals over a range from 200 to 300”K, i.e., a range of 100” located for the most part below the limit of T6-12* = 2. Complete analyses viere carried out at repulsive exponents of 15 and 48. These represented extremes of interest, supplementing earlier work at 6 = 12.1310 The derivative objective function exhibited unique minima for the case of 6 data points. The virial objective function was bimodal at 6 = 15 and unimodal at 6 = 48. The combined objective function exhibited this same bimodal-unimodal behavior, as a function of the repulsive exponent. To test the effect of the number of data points chosen as reference, this analysis was expanded to include the same temperature range by intervals of 10” leading to manipulation of 11 data points. In all three cases, the effect of increased data over the common temperature range was relatively insignificant. The only effect was an increase in absolute magnitude of the objective functions. The location and number of minima \yere not influenced. Comparable calculations were made with data from the same source at 20” intervals over a temperature range corresponding to approximately 2 < T6-12* < 3, Le., 300-460°K. The combined range, involving temperatures between 200 and 4G0°1i, was investigated, also. The virial objective function displayed unimodality in both the high-temperature and combined temperature ranges. The derivative objective function continued to display only unimodality, but the hightemperature data group produced an objective function that lacked a clearly defined minimum. In all cases the combined objective function paralleled the virial objective function. Inconsistencies developed with this single source of data were the most significant result of the initial analysis.
(4) J. S. Rowlinson, Discuss. Faraday Soc., 40, 19 (1965). (6) J. 0. Hirschfelder, C. F. Curtiss, and R . C. Bird, “l\Iolecular Theory of Gases and Liquids,” John TTiley and Sons, S e w York, N. Y . , 1954. (6) T . Kihara, Rev.Mod. Phys., 25, 39 (1952). (7) R. C. Ahlert, G. Biguria, and J, Gaston, Jr., A.1.Ck.E. J., 14, 5, 816 (1968). (8) F. Din, “Thermodynamic Functions of Gases,” T’ol. I and 11, 1 s t ed, Butterworth and Co. Ltd., London, 1961. (9) F. Rossini, “Selected Values of Properties of Hydrocarbons,”
American Petroleum Institute, Project 44, Carnegie Press, Plttsburgh, Pa., 1952. (10) R. C. Ahlert and W, F. Vogl, A.I.Ch.E. J., 12, 1025 (1966).
NOTES
1641
Evaluation of Source Data To define the role of choice of experimental data, values of the virial coefficient and the zero-pressure Joule-Thomson coefficient from several sources were compared.*~~~-’7 Analysis of the virial data indicated a high degree of agreement between the several sources. Hanley and Klein pointed out that some error can be tolerated in experimental data without destroying the capacity of the data to define a potential function. For the virial coefficient, this error is given by 0.04. [ B ( T ) a] t T6-12* = 2 or about 2.4 cc/mol for methane and is not a function of temperature. Virial data from the several sources generally meet this specification. Joule-Thomson data were quite the opposite; it was highly scattered and exhibited variations between sources of as much as 25% of the average value at a particular temperature. It was demonstrated that the variation of Joule-Thomson coefficients in the literature produced large variations in the location of the optimum attractive coefficient. These observations were in agreement with requirements on Joule-Tliomson data according to Hanley and Klein. They claimed errors greater than 1% make it impossible to distinguish between members of a family of potentials. Because of this problem, the literature Joule-Thomson coefficient data for methane were discarded as a source of information on the intermolecular potential.
Final Treatment of Second Virial Data
A polynomial expression for all of the virial data used to generate an 18-point data set with the following distribution: 12O-18O0K, 4 points; 200300”I300°K, 3 points. Low-temperature and high-temperature points (the high-temperature points each at 500 and 600°K) were used to broaden the range substantially. At the upper end, the raiige extended to a 116--12* of approximately 4 and at the lower end to a T6-12” of about 0.8. The initial part of the investigation led to several instances of multimodality or lack of uniqueness in the objective function of the second virial coefficient, even when the Hanley and Klein criteria were satisfied. To test whether this lack of uniqueness resulted from using too small a temperature range, it was decided to use 4. the extended range from Ts-12* 0.8 to T6-12* As shown by the distribution given, the interval lower than T6-12* = 2 was more heavily weighed to satisfy the Hanley and Klein criteria. Figure 1 shows clearly that the set of virial data selected led to unique minima at all repulsive exponents. Each curve of constant repulsive exponent (6) differs significantly from the others as well. The locus of “constant 6” optimum (c, d) pairs exhibits a “global” minimum at 6 1: 21. This set of parameters gave the best overall fit. Model constants apociated with this point are: c / K = 1.04 X lo6O K , A6and d/K = 2.01 WRR
I
m
j
w
\LOCUS
I
OF LOCAL OPTIMA
300200 3
5
/
loo4
4 . x 10-6 k
Figure 1. Local optimum potential function parameters for methane.
X 1014OK, Wal. These constants are within i10% of the true minimum. The dispersion coefficient, c6/k, can be calculs-vted theoretically. The value of Cg/h obtained by Dalgarnols is: ce/k = 1.04 x lo6 OK, W6, which is identical with the value predicted by the optimization on virial data. This extraordinary agreement is somewhat deceiving. Dalgarno suggested that the theoretical dispersion coefficient is probably accurate to better than &lo%. Thus, theoretical calculation of the dispersion coefficient predicted a value consistent with that obtained from the search scheme and within the limits of uncertainty on both results.
Discussion The optimum set (6 = 21) and two suboptimum sets (6 = 12, 6 = 48) of potential parameters obtained from the virial optimization were employed to predict theoretical values of at temperatures between 150 and 350°K. Figure 2 shows the results of these calculations with respect to the uncertainty of experimental data. All three sets of parameters yield curves that fall within the spread of the latter. Thus, no distinction can be drawn on the basis of this data population. The pattern of the u and E / Kfor various local optimum 6’s parallels closely the result Hanley and Klein (11) G. Thomaes and R. Van Steenwinkel, Nature, 187, 230 (1960).
(12) D. Douslin, “Progress in International Research on Thermodynamics and Transport Properties,” Princeton University, 1962, p 135. (13) M. R. Jones, M. A. Byrne, and L. A. K. Staveley, Trans. Faraday Soc., 64, 1747 (1968). (14) M. L. Jones, Jr., D . T. M a p , R . C. Faulkner, Jr., and D. L. Katz, Chem. Eng. Progr. Symp. Ser., 44, 52 (1963). (15) C. 6. Mathews and C. 0. Hurd, Trans. Amer. Inst. Chem. Eng., 42, 55 (1946). (16) B. A. Budenholzer, B . H. Sage, and W . N . Lacey, Ind. Eng. Chem., 31, 369 (1939). (17) R . C. Ahlert, “Joule-Thomson Coefficients and Equations of State for Mixtures,” Lehigh University, Bethlehem, Pa., 1964. (18) A. Dalgarno, “Interniolecular Forces,” J. 0. Hirsohfelder, Ed., John Wiley and Sons, New York, N. Y . , 1967.
Volume 742Number 7
April 8 , 1970
NOTES
1642
eters for the generalized Lennard-Jones potential. Using data in this region does not guarantee uniqueness in the intermolecular potential parameters. A necessary condition for uniqueness appears to be the use of experimental d a t a over a broad temperature range. The authors found that for the virial objective function, a temperature range of 0.8 < Tfi-Iz* < 4 was satisfactory. The range between 200 and 300”Ii was inadequate for specification of the virial objective function, although this region was within the temperature limits of the Hanley and Klein criteria.
Hydrogen-Bond Effect in the Radiation Resistance
of Chloral Hydrate to I50
200
250
T E M P E R l i IJRE
by F. kc. Milia and E.
C’
-I METHANE - R U T G E R S
i
ARGON-HtK
0
8:lZ
0
8.13
i
0
IO0
150
250
200 c/k,
300
O K
Figure 3. Comparison of well depths and residual volumes for local optimum parameters.
obtained for argon. As 6 -+ 48, the potential well depth becomes greater and the repulsive wall becomes steeper (harder). For 6 --t 12, well depth decreases and the repulsive wall becomes softer. The optimum condition, 6 = 21, occurs at an intermediate condition. These results are described in Figure 3. At present, there is no method that can be used to determine directly the dimerization energy and thus, the well depth, independently and without a priori selection of a potential energy model. The well depth resulting = 218°K. from the optimum parameters is Multimodality in the virial objective function only disappeared when the temperature range was increased. The final set of data selected was heavily weighed the sensitive temperature region criteria* It appears according to the Hanley and that i t is necessary, but not sufficient, to use data in the temperature-sensitive region specified by the HanleyKlein criteria in order to obtain the best set of paramThe Journal of Physical Chemistry
Rays
350
30C
, ’K
Figure 2. Comparison of selected Joule-Thomson coefficient predictions with t h e results of experiment.
so
y
I
