The Journal of
Physical Chemistry
0 Copyright, 1985, by the American Chemical Society
VOLUME 89, NUMBER 20 SEPTEMBER 26, 1985
LETTERS Trajectory Study of Nonequiiibrium Effects in Diatom Dissociation Reactions George Burns* and L. Kenneth Cohen Department of Chemistry, University of Toronto, Lash Miller Chemical Laboratories, Toronto, Ontario, Canada M5S 1Al (Received: September 21, 1984; In Final Form: July 29, 1985)
Five ensembles of 3-D classical trajectories, totaling over 10’ trajectories, were used to study the dissociation of Br2 in Ar in the linear regime, i.e. with recombination neglected. It was found that for a given temperature and potential energy function dissociation occurs at a precisely determinable, unique steady state. Consequently, the steady-state nonequilibrium energy distribution functions and rate constants are also unique for a given trajectory ensemble; they differ quantitativelyfrom equilibrium distribution functions and the corresponding equilibrium rate coefficients. The findings obtained substantially tighten up the conceptual framework of the SUE method and justify its further use as a tool to determine new observables in chemical reaction kinetics.
nonequilibrium steady-state rate constants In previous and nonequilibrium energy (and angular momentum) distribution 2Br functions have been computed for the reaction Br2 Ar Ar in the linear regime, i.e. with recombination neglected. In these studies we used the classical 3-D trajectory technique and the single uniform ensemble (SUE) method. Steady-state dissociation rate constants, kPd, were obtained between 1500 and 6000 K, and for one case at 1500 K, it was shown that, for a given trajectory ensemble,’ kssd is independent of the initial diatom energy distribution to four significant figures.2 The relative nonequilibrium energy distribution (RNED) function at a steady state, P ( E ) = pm(E)/pq(E), was obtained3 as a function of the diatom internal energy E at 3500 K. The RNED was normalized by using the expression
+
+
+
S p ” ( E ) dE = S p q ( E ) dE = 1
(1)
where p(E) is the energy-dependent diatom population density. The RNED can be described3 by a simple function, F ( E ) , using (1) H. D. Kutz and G. Burns, J . Chem. Phys., 72, 3652 (1980). (2) H. D. Kutz and G. Burns, J . Chem. Phys., 74, 3947 (1981). (3) G. Burns and L. K. Cohen, J . Chem. Phys., 78, 3245 (1983).
0022-3654/85/2089-4161$01.50/0
known constants, physically meaningful constraints, and one adjustable energy distribution (ED) parameter, a. It was found which is intriguing because the that at 3500 K a = 12.00$‘;;, transition-state theory prescribes a = +a. These results*-3suggest that the SUE method can provide precise quantitative information on chemical reactions which occur in a regime very far from the equilibrium, where the use of statistical theories of reaction kinetics relies upon undesirable approximations. Therefore, it seemed reasonable to expand previous and thereby assess further the capabilities of the SUE method. In the present paper we present results for kSSd using five trajectory ensembles at four temperatures2 for the Br2-Ar system. Two initial energy distributions were employed for each trajectory ensemble. Furthermore, we study P ( E ) at 3500 K using two different trajectory ensembles. As in the previous work2 one of the initial diatom distributions was the Boltzmann distribution function characterized by the translational temperature of the gas. It yields the one-way flux equilibrium dissociation rate ~oefficient,~ The other initial distribution was the ground-state distribution in which all the (4) B. Widom, Science, 148, 1555 (1965).
0 1985 American Chemical Society
Letters
4162 The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
TABLE I: Comparison of One-way Rate Coeffkients and Steady-State Rate Constants Obtained in the Present Work (Given with Errors) with Those Obtained in Earlier's2 Work no. of T. K traiectories kcs,. L mol-l s-l 1500 22 200 (6.63 f 0.66) X lo5, 6.72 X 10' (3.37 0.38) x io', 3.35 x 105 0.51 f 0.11, 0.50 (4.83 f 0.33) X lo', 4.85 X lo7 0.19 f 0.02, 0.18 (2.61 f 0.16) X lo', 2.64 X 10' 2500 22 200 0.38 f 0.05, 0.37 (2.69 f 0.10) x 109, 2.69 x 109 (10.30 i 0.88) X lo', 10.0 X 10' 3500 22 200 (9.48 f 0.62) X lo', 9.46 X lo8 0.35 f 0.04, 0.37 (2.68 f 0.13) X lo9, 2.69 X lo9 22 200 44 4OOb (2.69 f 0.08) X lo9 (10.02 f 0.54) X lo8 0.37 f 0.03 (2.54 f 0.09) X IOIO, 2.59 X 1O1O (1.27 f 0.05) X lolo, 1.26 X 1O'O 0.50 f 0.04, 0.49 6000 22 100 "For each temperature the first value is present data and the second value is earlier1,*data. *This ensemble is the composite of the two above it.
diatoms were assigned to the lowest energy and angular momentum bin of the SUE method (eq 2, ref 2). In all cases, there were 250 equally distributed energy bins and 10 equally distributed angular momentum bins.' The development of the second initial energy distribution to a steady state describes reasonably well, albeit within the classical approximation, the chemistry and physics behind a shock front where molecules accumulate internal energy and then begin to react. In the present work, SEL 55 and 75 minicomputers were used. Earlier trajectory and SUE programs,'V2 written for the IBM 7094 computer, were revised to make them suitable for the SEL minicomputers. Initially, a total of 1080 new Br,-Ar trajectories were obtained at 3500 K by using the stratified and importance sampling5 techniques. Information secured from these trajectories yielded kqd = (3.6 f 1.1) X lo9 L mol-' s-I in agreement with earlier calculations',6 at 3500 K which yielded 2.7 X lo9 and 3.7 X lo9 L mol-' s-', respectively. The uncertainties in the present work are expressed as a standard error of the mean.' Once it was established that the computer program for the SUE method was reliable, it was applied to ensembles of Br,-Ar trajectories accumulatedli2 earlier. Over 22 000 trajectories2 were used to generate each trajectory ensemble (Table I). At 3500 K, two independent ensembles of 22 200 trajectories each' were available. In the present work we also used the combined ensemble (44400 trajectories) a t 3500 K. Each S U E computation was terminated after the ensemble had undergone 200 collisions. In this regime well-defined steady-state rate constants were consistently obtained (Table I). Except for typographical errors, now removed (Table I), earlier data2 obtained on an IBM computer agree, within standard error, with the present calculations. This agreement in absolute values of k was, generally, within three significant figures (Table I). Whenever calculations were performed by the same operator and on the same computer, it was possible to obtain somewhat more precise results. Thus, at 1500 K the nonequilibrium ~teady-statel-~ correction factors,' f = kssd/kqd, obtained from the two initial distributions agreed, as previously,2with each other within four significant figures. The largest difference in thef factors from the two distributions occurred in the present work at 2500 K where lAfl = St -F"laml d 0.0007. Furthermore, at 3500 and 6000 K the two f factors were within four significant figures of each other. Thus, in the S U E method the four significantfigure accuracy for the k"8d is a general result (for a given ensemble of a certain size) (Table I). In Figure 1 we present f factors as a function of collision number, N , for three ensembles at three temperatures. In each case, although the initial values o f f factors vary rapidly with N , they become quite indistinguishable in the steady state. When the two different ensembles of trajectories at 3500 K were used, the kssddiffered' by 8% (Table I) which is approximately the average value of the standard error obtained in this work (Table I). In Figure 2 we present f factors as a function of collision number N for these independent ensembles. When these two trajectory ensembles were combined, the resultant ensemble yielded the f factors shown in Figure 1. Note that the ( 5 ) W. H. Wong and G. Bums, Proc. R.SOC.London A , 341, 105 (1974). ( 6 ) D. T. Chang and G. Burns, Can. J . Chem., 55, 380 (1977). (7) J. Topping, "Errors of Observation and Their Treatment", Chapman
and Hall, London, 1975, p 62. (8) A. Gelb. R. Kapral, and G. Burns, J . Chem. Phys., 56, 4631 (1972).
1/' 1/
0 ' I - . -q
n
'
I
I
\
I
,
I
,
1 1
t \'\
Figure 2. Nonequilibrium correction factors as a function of the collision number N. Two trajectory ensembles, each consisting of 22 200 trajectories, are calculated for 3500 K.
two individual trajectory ensembles yield (Figure 2) two different steady-state f factors, one for each ensemble. However, the difference in f values for these two ensembles is smaller than the calculated error of the mean for the individual ensembles (Table I). For each individual trajectory ensemble thef factors agree within four significant figures. In previous work3 the relative nonequilibrium energy distribution function, as(,!?), was obtained at 3500 K using the Boltzmann function as the initial distribution. The combined ensemble of 44400 trajectories' was e m p l ~ y e d . In ~ the present work, this trajectory ensemble was used to calculate dSS(E),now using the ground-state initial distribution. It was found that this distribution approaches steady state considerably slower than for the case in
Letters
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985 4163
I n i t i a l Distribution-All molecules in the lowest bin
Br,: Ar
LL V
I n i t i a l Distribution-Boltzmonn N
5 0 m
1.2
5
0.8
>0 i LL V
a'
- 0.4 n
0
-6.0
0 2 .o -20 DIATOM MOLECULAR ENERGY ( K T )
-4.0
4 .O
Figure 3. Relative nonequilibrium distribution function d'O(E) for the Br2-Ar system at 3500 K after the trajectory ensemble underwent 10 collisions (44 400 trajectories).
which the initial distribution was Boltzmann. Thus, even after 10 collisions most of the molecules acquired very little energy (see dIo(E)shown in Figure 3). This histogram is radically different from the dl0(E)generated from a distribution which was initially Boltzmann (Figure 3), reported in earlier9 studies. The P ( E ) function3 was quite complex, exhibiting many sharp peaks at well-defined energies. Yet, it was found that at the steady state the histograms initiated from either one of the distributions agreed with each other at all energies, Eo d E d Eh (i.e. at all 250 energy bins), the difference being lAP(E)I Q 0.0006 for all E . Here Eo is the classical potential minimum for Br2, and Eh is the highest possible energy of a classically stable Br2. Since the histogram generated from the ground-state distribution was visually indistinguishable from that shown in Figure 1, ref 3, it is not reproduced here. The inequality lA&(E)I d 0.0006 is the expected cumulative machine error of the present calculations. Note that convergence for &'(E) is more significant than that for kssd,because dN(E) is the histogram function from which kdNcan be generated and because the agreement in P ( E ) was obtained for each one of the 250 (ref 1) histogram energy values from which only one k"d value can be computed. Whenever two different RNED trajectory ensembles are used at the same temperature, the P ( E ) appear to have similar general characteristics (Figure 4). However, the details of the two RNED ensembles are quite different (Figure 4). Note especially high peaks near Eh present in the upper ensemble, and relatively low noise in the same energy region in the lower ensemble (Figure 4). Such differences in detailed structure of P ( E ) are apparently due to the large weights associated with large energy transfers in rare, nonrepresentative trajectories9 The principal computational finding of our work is that both the kSSd and the entire @'(E) histograms, in all their details, are independent of the nature of the initial state used to generate the timeinvariant steady state. Since the two initial states were chosen arbitrarily, and in fact to be most different, one may expect that our findings for the kind of systems hereby studied are quite general. However, the results are valid only for a given ensemble and a given temperature, and are subject to the normalization condition (eq 1). Furthermore, if eq 1 were not included in SUE calculations, all molecules would dissociate after a sufficient number of collisions; this would make it impossible to calculate a rate coefficient. There is a question as to why different trajectory ensembles yield different k"d and P ( E ) . It should be kept in mind, however, that the trajectory ensembles used, albeit large even by today's
DIATOM MOLECULAR ENERGY ( K T )
Figure 4. Relative nonequilibrium distribution functions for the Br,-Ar system at steady state, P ( E ) , obtained at 3500 K for two different trajectory ensembles, consisting of 22 200 trajectories each.
standards, are still sufficiently small to introduce an appreciable error into both k"'d and @ ( E ) (Table I and ref 3). However, as was pointed out above, the standard error in kssdis of the order of the difference between the k"d values. This implies that different because ensembles are trajectory ensembles yield different kSSd, not sufficiently large to be truly representative. Classical mechanics was used throughout our calculations; if a constraint were to be imposed on it, both kssdand h ( E ) would be indeedg affected. One can question the validity of our assumption that the reverse reaction can be neglected. Intuitively this assumption is reasonable: there should be a point, at the very early stages of a reaction such as dissociation in a shock tube, when the back-reaction could be neglected because products are still absent. There has been calculation supporting our assumption. Thus, Lim and Truhlarlo*" studied computationally the mixture effects in the nonequilibrium kinetics of homonuclear diatomic dissociation and recombination. Their system consisted of the O2 molecule approximated by a 27-state (v = 0, ..., 26) truncated harmonic oscillator. They considered the possibility of a back-reaction and showed that, for their models, a steady state is reached before the back-reaction becomes phenomenologically significant. Thus, to that extent, more recent support our assumption that dissociation can be investigated in the linear time domain, Le., with recombination neglected. The work described in this Letter was done to provide an accurate test of previous conclusions and to tighten conceptually arguments which were presented previo~sly.~,~ These results have implications to statistical quasi-equilibrium theories of chemical
reaction^.^ Acknowledgment. We thank Professors J. M. Deckers and R. Kapral for helpful discussions. This work was supported by the US.Air Force Office of Scientific Research under Grant No. A8 1-0028 and A84-0 127. (10) C. Lim and D. G. Truhlar, J . Phys. Chem., 87, 2683 (1983); 88, 778 ( 1984).
(9) G. Burns and L. K. Cohen, J . Chem. Phys., 81, 5218 (1984).
( 1 1 ) C. Lim and D. G. Truhlar, J . Chem. Phys., 79, 3296 (1983).
