Thermal Dissociation of t i 2
by
He
The Journal 01‘ Physical Chemistry, Vol. 83, No. 1,
1979 127
An ab initio Calculation of the Rate of Vibrational Relaxation and Thermal Dissociation of Hydrogen by Helium at High Temperatures John E. Dove” and Susanne Raynor Lash Miller Chemical Laboratories, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1A 1 (Received August 8, 1978)
The master equation for the thermal dissociation of para-H2infinitely dilute in He, was solved for temperatures of 1000-10000 K. Transition probabilities, used in the master equation, were obtained, in the case of energy transfer transitions, from distorted wave and quasi-classicaltrajectory calculations and, for dissociative processes, from trajectory calculations alone. An ab initio potential was used. li’rom the solution, values of the dissociation rate constant, vibrational relaxation times, and incubation times jfor dissociation and vibrational relaxation were calculated. The sensitivity of the calculated results to variations in the transition probabilities was examined. Vibrational relaxation is most sensitive to simultaneous transitions in vibration and rotation (VRT processes); pure rotational (RT) transitions also have a substantial effect. Dksociation is most strongly affected by RT processes, but changes in VRT and groups of dissociative transitions also have a significant effect. However complete suppression of all dissociative transitions except those from levels immediately next to the continuum lowers the dissociation rates only by a factor of about 2. The location of the dissociation “bottleneck” is discussed.
Introduction The dissociation of a diatomic gas in an inert diluent is a simple prototype for all thermal dissociation reactions and has beein studied very extensively. The essential experimental findings are that the measured rate is greater than expected for a simple one-step dissociation and that the Arrhenius activation energy is less than the bond dissociation A great amount of theoretical work has gone into understanding these experimental results. While the magnitudes of the observed rates can be accounted for in terms of a multistep process, e.g., vibration-rotation ladder climbing, the lowering of the activation energ!? has proved less easy to understand. Most studies of this lowering have concentrated on two effects, namely, the iinfluence of internal degrees of freedom on the dissociation process, and the effects of thermal disequilibrium (sometimes discussed as a “bottleneck” phenomenon5). In recent theoretical studies, the H2 molecule has played an important role. A major reason for this is that H2 has about 370 internal (o,J) levels, a number which is small enough for the master equation governing the populations of these levels to be solved directly by modern numerical techniques. The size of the problem can be further reduced by coiifinirig attention to, e.g., the 176 levels of para-H,. A substantial number of master equation studies of the H2-He and I-L-Ar systems have been made by Pritchard and co-workers,6 and also by o u r s e l ~ e s . ~ ~ ~ Essential input for a master equation study of dissociation, or of the related process of relaxation of the populations of the internal levels, is a set of transition probabilities or rate constants for transitions among bound levels and from bound levels to the continuum. Many of the master equation studies have not attempted to evaluate these transitlion probabilities very accurately. Instead, plausible sets of transition probabilities have been used to explore the general behavior of such systems. It is arguedP6reasonably, that most or all diatomic molecules appear to show the same general features in their dissociation kinetics. The overall kinetic behavior therefore cannot be dependent on the finer details of the pattern of transition probabilities. Generally speaking, the master equation studies indicate 0022-3654/79/2083-0127$01 .OO/O
that both rotation and vibration are important in the dissociation dynamics, and that the lowering of activation energies cannot be understood without taking rotation explicitly into account, which was not done in the earlier studies. Nevertheless, Johnston2 has differed from this conclusion and has suggested that, with certain assumptions, the olbserved dissociation kinetics can still be accounted for without considering molecular rotation. Recently we have made a careful master equation calculationgof the rates of relaxation and dissociation in the para-H2/He system; the transition probabilities used were calculated ab initio. A major motivation for this work was as follows. While studies using assumed sets of transition probalbilities have yielded much valuable information about the general behavior, inevitably some doubts remain. The use of ab initio transition probabilities should help to remove doubts about the mechanisms of relaxation and dissociation, especially about the role of molecular rotation. Also, an accurate study will assist in understanding aind interpreting the activation energies found a t high temperatures; because the range of 1 / T is often quite limited, both experimental and theoretical activation energies can be sensitive to relatively sinal1 uncertainties in the rate. The H2/He system was chosen for these calculations because of the availability of ab initio interaction p o t e r ~ t i a l s . ’ ~ - ~ ~ An important feature of systems such as those represented by the vibration-rotation master equation is the existence of “network effects”, which have been discussed by Pritchard6 in m excellent review article. One feature of these effects is that they tend to damp out the influence, on the calculated dissociation rate, of variations in the transition probabilities. Of course, such variations do affect the calculated rates, but the network effects make it hard to predict in advance how large the influence will be, and the consequences of these effects are sometimes quite surprising. The objective of the present paper is to analyze the quantitative influence of various transition probabiliities on the calculated relaxation and dissociation behavior of H2/He. Such an analysis is important for several reasons. It will throw further light on the network behavior. It will assist us in understanding how accurate our calculated rates are likely to be, and what firm conclusions can be 0 1979 American
Chemical Society
128
The Journal of Physical Chemistry, Vol. 83,
No.
1, 1979
drawn from our calculations a t their present level. Also, it will show which processes are most important in determining overall rates, so that they need especially careful treatment in future studies.
J.
Chart I
Method of Calculation The overall calculation can be divided into four stages: (a) selection of an interaction potential, and fitting to a suitable form. (b) calculation of transition probabilities, and rate constants, at various temperatures, for transitions among bound states and from bound states to the continuum. (c) solution of the master equation to simulate shock wave experiments in which the translational temperature of the gas, initially in equilibrium a t 298 K, is suddenly raised to a chosen temperature in the range 1000-10000 K. The results of these calculations show how the populations of the 176 (u,J) levels of para-H2and of dissociated molecules are predicted to evolve with time. (d) variation of chosen transition probabilities, to show which processes are most important in determining the results of (c) above. Stages (a), (b), and (c) are being described fully elsewhere, and will only be summarized here. The potential energy hypersurface was a combination of the Wilson, Kapral, and Burns13 surface with H-H interaction data from Kolos and Wolniewicz,16and Bishop and Shih,16and the H-He potential of Gengenbach, Hahn, and T0ennies.l' We have discussed this hypersurface elsewhere.ls Transition probabilities among bound states were calculated by a combination of the quantum mechanical distorted wave (dw) methodIg and quasi-classical trajectory calculations. The transitions shown in Chart I were included. The quantum and quasi-classical calculations agreed reasonably well, over the temperature range of interest, for transitions of type VT1 and RT1 (Table I). The dw results were used unmodified in these cases. However, as expected, the dw method seriously underestimates multiquantum transition probabilities, and it also underestimates the probability of many of the transitions in which u and J change simultaneously. Our earlier work8,20indicated that the transition type VRTl is especially important in relaxation and probably also in dissociation; effects of VRT2 can also be quite significant. We therefore used the results of extensive quasi-classical trajectory calculations for ten initial states to develop correction formulae for the dw results in these cases. These results were subsequently checked by making additional sets of calculations for three more initial states. With these correction factors, the temperature averaged standard deviation for the fit of the corrected dw results to the trajectory calculations of the interlevel rate constants was 88% as compared with 7600% for the uncorrected dw results for VRT1, and 55% as compared with 2 X for the uncorrected dw results for VRT2. It was not found necessary to correct transitions of type VRT3 because the fit between dw and trajectory results is fairly good for the lower J states; the fit is poor for higher states, but the rate constants for type VRTl are then so much larger that they dominate anyway. RT2 transition rate constants were used uncorrected (Table I). In every case, the rate constant in one direction was obtained by the procedure outlined above, and that for the reverse process was obtained by detailed balance. Rate constants for dissociation out of the level (u,J) were
E. Dove and S.Raynor
name
AV
AJ
VT 1 RT1 RT2 VRTl VRT2 VHT3
21 0 0
0 i2 t4 r2 T4 F4
tl
i2 I1
TABLE I: Average Deviations between Trajectory and Distorted Wave Calculated Values of Transition Rate Constantsa
_ _ _ _ _ l _ _ _
transition used in comparison 1000 ~ VT-dw (all) 1.75 * 0.90 RT1-dw (impt) 0.49 t 0.46 RT1-dw (all) 1.17 i 0.74 VRT1-dw (impt) 1.90 i0.53 VRT1-dw (all) 2.43 i 0 . 8 3 VRTl-corrected 0.28 ?: 0.16 (impt) VRTI-corrected 0.34 i 0.19 (all) RT2-dw (impt) 1.13 ?: 0.46 RT2-dw (all) 2.00 + 1.07 VRT2-dw (all) 8.74 t 3.10 VRT2-corrected 0.21 i 0.18 (all) VRT3-dw (all) 3.44 t 0.88 dissociative 0.16 i0.15
3000
10000 1 _ 1 -
0.98 ?: 0.78 0.40 t 0.11 0.45 i 0.31 1.46 + 0.36 1.92 i. 0.70 0.23 t 0.08
0.50 i0.34 0.51 t 0.08 0.43 i 0.24 1.03 t 0.27 1.31 -c 0.71 0.19 t 0.10
0.27
0.23
0.26
i
0.22
* 0.26
i
0.29
t
1.73 0.12
0.42 1.22 7.22 0.15 2.54 0.05
i
i i i
0.62 2.39 0.18
0.36 0.55 4.28 0.21
t i
0.78 0.04
1.70 t 0.98 0.39 t 0.20
* 0.36
Tabulated values are evaluated Erom (l/N)Z llog(htrW/ N transitions, ktraJ is the rate constant for the transition ( u , J ) (v',J') evaluated directly from the quasiclassical trajectory results and hd,-.d is that calculated using the distorted wave (dw) or corrected formulae (corrected). In the case of dissociation, kcdCg is calculated from our general formula for cross sections for transitions from ( u , J )t o the continuum, The designation impt indicates that the cases compared were those for which the trajectory evaluated rate constant for the transition of the specified type was within a factor of 0.1 times the maximum rate constant (excluding dissociation) out of that initial level. The designation all indicates that all transitions for which calculations were made were included. a
kcalcd) 1 where the summation is made over
--f
obtained from quasi-classical trajectory calculations, using procedures described elsewhere.'8~20*21 Complete calculations were made for 1 2 levels, and the results were fitted to a simple set of expressionsZ1which also served as interpolation formulae to generate rate constants for levels other than those for which trajectory calculations had actually been made. In Table I we have tabulated the average absolute deviations of the logarithms of the rate constants evaluated from the quasi-classical trajectory calculations with those evaluated from dw and from the corrected formulae. Having generated transition probabilities for each of the types of transitions discussed, we then solved the master equationz2 for this system at each temperature to be considered: dn (U ,J )/dt = C,,,s(h(u ',J'+u,J; T )~ Z ( U',J 1 h (u,J-u ',J';T)n(u,J)) - h (u,J-*cont;T)n(u,J) (1) where n(u,J)is the population of level (L',J) at time t and k(u',J'-u,J;T) is the rate constant for the transition (u',J? ( u , J ) a t a temperature T . There is one such equation for each of the 176 levels of para-H2. Approximately 2000 rate constants were used in the calculation. The numerical integration of the coupled differential equations was done using Gear's algorithmz3including modifications for efficiently handlingz4 sparse Jacobian matrices. The equations were integrated until steady dissociation was
-
Thermal Dissociation of H, by He
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 129
TABLE 11: Cadculated Dissociation Rate Constants, Vibrational Relaxation Times, and Incubation Times I _ _ -
transitions included kda I. V T l , dime 1,03(--10) 1000 2.67( I) 2000 3000 1.44( 5) 4.97(8) 6000 1.72(10) 10000 11. VY”,RTl,diss 2.39(-9) 1000 1.20(6) 3000 1.56( 11) 10000 111. VT1,RTl ,VRTl,diss 8.90(-9) 1000 3000 6.61( 6) 4.12(11) 10000 IV. VT‘l,RTl,RT2,VRTl,VRT2,VRT3,diss 9.99(-9) 1000 1.62(-3) 2000 1.58(6) 3000 2.32( 1 0 ) 6000 10000 4.77(11) V. limited dissociationf 3.56(-9) 1000 3000 3.45( 6 ) 2.73( 11) 10000 VI. equilibriurng and SHOh 1.62(-7) 1000 2000 3.65(4) 3000 2.08(8) 9.95( 11) 6000 1.87(13) 10000
TV
b
__
ti .d
ti.“ d
4.01(-1) 9.19(-3) 1.25(-3) 5.21(-5) 6.40(-. 6 )
1.22(-2) 1.60(--3) 7.17(-5) 8.41(-6)
3.94(-1) 1.1.7(-- 3) 2.44(-6)
4.98(-1) 1.45(-- 3 ) 9.54(-7)
7.59(-3) 6.12(-5)
3.86(-5) 1,15(-6) 2.73(-7)
1.25(-4) 2.66(-6) 4.05(-7)
2.36(-6) 4.82( -7) 1.52(-7)
2.82(-5) 2.23(-6) l.Ol(-6) 4.15(-7) 2.25(-7)
5.58(-5) 5.46(-6) 2.33(-6) 7.61(-7) 3.46(-7)
2.93(-6) 7.23(-7) 4.7 3(-- 7 ) 2.1 0(-7) 1.34(- 7 )
2.82(-5) l.Ol(-6) 2.60(-7)
2.00(-4) 2.44(-6) 4.64(-7)
6.79(- 5 ) 4.71(-7) 1.81(-7)
.-
4.02(-1) 9.38(-3) 1.36(- 3) 8.85(-5) 1.9O( -- 5 )
kd is the rate constant for quasi-steady dissociation in mol s units. T\, is the vibrational relaxation time in units ti,” is the incubation time for vibration in units ti ,d is the incubation time for dissociation in units of atm s. Limited of atm s. e Terminology used for transition types is explained in the text; e.g. VTI means A u = 11, A J = 0. dissociation indicates that calculations were performed as for item IV but with dissociation only from levels nearest the Values of rV were continuum. g Values of k d were evaluated using thermal equilibrium populations in all levels. calculated using a simple harmonic oscillator model. a
of atm s.
reached. The calculated rate constant for dissociation, kd, was evaluated from the net flux to the continuum
1:h (u +-.con t ;T )n (u,J) U,J
and the vibrational relaxation time, T,, was evaluated by analogy with the Bethe-Teller law: (Etot(m) 7,
=:
-
Et,&))
(3)
&,&)/at
where
E,,, = l m J , 4 n ( u , 4 / Z n ( u , 4 L,J
(4)
L,J
and E(u,J) is the internal energy of level (u,J). Values of 7 , discussed here are “final” relaxation times as t m. The dissociation reaction and the vibrational relaxation process both exhibit incubation times,8~20~26 t1,d and ,t,,J,, which were evaluated a t the end of the time integration: -*
ti,d
= t
+ (1 / h & In [l - Z n ( u , 4 ]
(5)
u,J
and
was used to evaluate the vibrational energy of the system. We then studkd the sensitivity of the calculatioin of these four quantities, kd, T,, t l , d , and t,,,, to the transition probabilities (or transition rate constants) used. The brute force method wtcluld be to vary each of the transition probabilities, to find which of them have the largest effect on the calculation. With approximately 2000 transition probabilities to consider, even a single solution of the master equation is a large undertaking, so that this method is not a t all feasible. Shuler and c o - ~ o r k e r shlave ~~ demonstrated a much more efficient algorithm for studying the sensitivity of the calculation to these changes, but wen their method is not feasible for such a large system. We therefore worked as follows. First we studied the effects of progressively adding various types of transition to the system, for a number of different temperatures. We then made a more detailed study, varying groups of transition probabilities systematically, in order to establish regions where the calculated results were most sensitive to changes; this latter set of calculations was made a t 3000 K only. All calculations were made for para-H, infinitely dilute in He. Because of the infinite dilution condition, collisions other than those between H2 and He, and recombination processes, were not included.
Results a n d Discussion
where
(7)
The essential results of this study are contained in Tables I1 and II[ and in Figures 1-5. These tables and figures show how the calculated values of hd, T,, tl,d,and t,,”depend upon the transition probabilities used. In the first seq!uence of computations, carried out over the range 1000-30000 K, we studied the effect of ]progressively adding; various types of transition. In a certain
130
The Journal of Physical Chemistry, Vol. 83, No. I , 1979
J. E. Dove and S. Raynor
TABLE 111: Rate Constants, Relaxation Times, and Incubation Times a t 3000 K (Effects of Changes in Selected Groups of Transition Probabilities) notes ti&‘ ti,” VT1, diss 0.0218 1100 600 VT1, R T I , diss 0,181 1020 540 127 VT1, V R T l , diss 0.0605 13000 9800 52000 VT1, R T I , VRT1, diss 1,000 1.00 1.00 a 1-00 V T l , R T I , VRTX, prediss, diss 1.000 1.00 1.00 1.00 b VT1, RT1, VRT1, VRT3, diss 1.026 0.94 0.96 1.00 VT1, RT1, RT2, VRT1, VRT2, VRT3, diss 1.146 0.87 0.88 0.98 C VT1, RT1, RT2, VRT1, VRT2, VRT3, diss, prediss 0.87 0.88 0.98 d 1.146 V T I , RT1, RT2, VRT1, VRTP, VRT3, limited diss 0.522 0.88 0,92 0.98 e VT1, RT1, VRTl(dw), diss 0.367 82 42 11.2 f V T I , RT1, V R T l (mixed), diss 0.989 77 30 4.2 g VT1, RT1/2, VRT1, diss 0.752 1.38 1.56 1.40 h VT1, RT1/10, VRT1, diss 0.403 4.28 5.0 4.9 h VT1, RT1, VRT1/2, diss 0.859 1.55 1.34 1.30 h VT1, RT1, VRT1/10, diss 0.593 5.13 3.52 2.2 h VT1/10, RT1, VRT1, diss 0.954 1.00 1.01 1.00 h VT1, RT1, VRT1, diss/2 0.811 1.00 1.01 1.00 1 VT1, RT1, VRT1, diss/lO 0.521 1.00 1.04 1.00 i VT1, RT1/10 for u = 0, VRT1, diss 0.991 2.41 2.00 4.5 j VT1, RT1, V R T l / l O for u = 0 1, diss 0.999 4.74 2.28 1.61 k VT1, RT1/2 and VRT1/2 for J = 10 + 12, diss 0.942 1.16 1.16 1.19 1 VT1, R T l / l O and VRT1/10 for J = 1 0 + 1 2 , diss 0.893 1.60 1.52 1.57 I VT1, RT1, V R T l i 2 for max flux diagonal, diss 1.000 1.00 1.00 1.00 rn VT1, RT1/2 and VRT1/2 for max flux levels, diss 0.901 1.00 1.00 1.00 n, 0 VT1, R T l / l O and VRT1/10 for max flux levels diss 0.775 1.00 1.01 1.00 n, 0 VT1, RT1, VRT1, diss/2 for max flux levels 0.942 1.00 1.00 1.00 4P VT1, RT1, VRT1, diss/lO for max flux levels 0.878 1.00 1.01 1.00 n,P a The values of k d etc. obtained in all other calculations have been normalized by division by the values obtained in this calculation, no. 4. The absolute values, for calculation no. 4, are given in Table 11. Predissociation out of quasibound states was included in this calculation; a diluent pressure of 0.01 atm was assumed. This is the “full” calculation, including all types of transition described in the text, except predissociation. As c , but including also predissociation, assuming a diluent pressure of 0.01 atm. e The only dissociative transitions included were those from levels immediately next t o the continuum. f This calculation uses “uncorrected” V R T l transition rate constants obtained directly from the distorted wave calculations. The quasi-classical trajectory calculations indicate that these rate constants are too small by a factor of typically 100. g This calculation uses “uncorrected” V R T l transition rate constants (see note f ) for transitions between u = 0 and u = 1, but corrected values for all others. In calculations 1 2 t o 16, all transition rate constants of a certain type have been divided by the factor indicated. E.g., R T l / 2 means that all rate constants of type RT1 have been divided,by 2. In calculations 1 7 and 1 8 , dissociation rate constants out of all levels were divided by 2 and 10, respectively. KT1 transition rate constants divided by 1 0 , for u = 0 levels only. VRTl transition rate constants divided by 1 0 for transitions between u = 0 and I I = 1 only. Transition rate constants for RT1 and V R T l transitions between J = 1 0 and J = 1 2 were divided by 2 in calculation 21 and by 1 0 in calculation 22. V R T l transition rate constants were divided by 2 for transitions from levels with u = 9 n , J = 1 2 F 2n. Level (9,12) in calculation 4 had the maximum flux t o the continuum. Maximum flux levels indicates the level having the maximum flux to the continuum, together with eight other levels whose fluxes t o the continuum were larger than 0.5 times the maximum flux. O In calculations 24 and 25, the R T 1 and V R T l transition rate constants into and out of the maximum flux levels were divided by 2 and 10, respectively. P In calculations 26 and 27, the dissociation rate constants from k d is the rate constant for quasi-steady dissociation. the maximum flux levels were divided by 2 and 10, respectively. r V is the vibrational relaxation time. ti is the incubation time for dissociation. ti,v is the incubation time for vibrational relaxation, The terminology use$for the transition types is explained in the text; e.g. VT1 means A u = il, A J = 0. “diss” means that dissociative transitions out of all levels were included in the computation. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
transitions included’
dq
TV
J
+
sense, the calculation of kd is a variational one. Adding extra transitions to the calculation will presumably speed up equilibration and so would be expected, on the present model, to increase the calculated value of hd. If the transition probabilities are accurate, then the calculated hd will always be less than the true one, but will. approach the true value as more and more transitions are added to the calculation. (It is, of course, implied that when any transition among bound states is added, the corresponding reverse transition is also included.) The behavior on adding transitions is demonstrated in Figure I and in Tables I1 and 111. As one progresses from a computation with only VT1 and dissociative transitions to one which includes also IiTl and VRT1, kd increases by almost two orders of magnitude. Addition of further transition types (cf. calculations 4-9 in Table 111) produces only small additional increases. The final value is less than, but close to, hd for H2-Ar. (No experimental data are available on hd for H2-He at these temperatures.) In a similar way, 7, shortens progressively as transitions among bound states are added to the calculation. (The
results of the most complete calculation are a factor of 2 slower than our experimental dataz7for T”.) The apparent anomaly between calculations 1 and 3 probably arises from the fact that rotational excitation cannot occur in calculation l; in a sense, therefore, these calculations are performed for different models. The incubation time for dissociation roughly parallels 7,; that for vibrational relaxation is very long in calculation 3, but is almost independent of the finer details of the calculation (calculations 5-9). In most of our calculations, all classically bound states were considered as fully bound. In reality, some of these are quasi-bound statesz8 which can predissociate by tunneling through the rotational barrier. The ratio of predissociation to collision-induced dissociation varies inversely with the concentration of collision partners. Predissociation was included in calculations 5 and 8, assuming a gas pressure of 0.01 atm; this is already a fairly extreme case, since 0.01 atm is well below the shock pressure of most shock tube experiments. Even though some of the quasi-bound states have extremely short
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 131
Thermal Dissociation of H, by He
i i 1 I I1 1.0-
0.8-
..
- 0.6tz
-
\
5
\
I
'.
0.4-
0.2 - //
E
LI
CT
O0
0.2'
0.4
i 0.6 FRACTION
i 0.8
I I.o
Figure 3. Relative vibrational relaxation times as a function of the fraction of various tra.nsition types included. See Figure 2 for explanation of symbols used. 2
L
i
0.4
0.3
0.2
0.I
io3/
0.5
T
Figure 1. Arrhenius plots of rate constants for dissociation of H, by He: (-) master equation calculations with all transition types included; (- - -) calc:uiations from thermal equilibrium populations; (0)results from limited dissociation calculations (see text); (A) calculations with VT1, RT1. and dissociative transitions only; . (. 0 )calculations with VT1 and dissociative transitions only.
I 1
I
I
i
1.0-
V - L U
0.8 -
OO
0.2
0.4
0.6
0.8
1.0
FRACTION
,..
0.6-
Figure 4. Relative incubation times for dissociation as a function of the fracture of various transition types included. See Figure 2 for explanation of symbols used.
W
L
n 1
0.4 -
I
I.0i
+
..........................
I
........e......................
I
I
0.2
0
0.2
0.4
0.6 FRACTION
I .o
0.8
Flgure 2. Relative rate constants for dissociation, showing effect of changing the values of a specified set of transition probabilities used in the calculation. "Fraction" is the factor by which that set of rate constants has been multiplied. "Relative" in this and subsequent figures indicates that the?calculated value has been divided by the corresponding value for the standard calculation: (-) variation of VRTl transition probabilities: (---) variation of RT1 transition probabilities: variation of dissociative transition probabilities (the lines join the points for which calculations were made, in order to guide the eye); (0) VRTl transition probabilities varied for levels with v = 9 f n , J = 12 =F n (cf. footnote rn to Table 111): ( X ) RT1 and VRTl varied for J 10 12 (cf. footnote I to Table 111); (4")RTI and VRTl varied for maximum flux levels (cf. footnote m to Table 111): ( 0 )dissociative transitions varied for maximum flux levels (cf. fmootnote p to Table 111); (0) RT1 varied for v = 0; (A) VRTl varied foir v = 0 1.
Figure 5. Relative incubation times for vibrational relaxation as a function of the fraction of various transition types included. See Figure 2 for explanation of symbols used.
predissociation lifetimes, we find that, because of network effects, predissociation has a completely negligible effect on the calcul.ated rates (Table 111). In the "limited dissociation" calculation (no. 10, Table 111), dissociation was suppressed from all levels except those immediately next to the continuum. Again because of network effects, th.e effect is not very large; kd is lowered
by factors of 2.8, 2.2, and 1.7 for temperatures of 1.000, 3000, and 10000 K, respectively. This result appears a reasonable justification of the earlier use7 of this model, Le., with dissociation occurring only from levels next to the continuum, to c,alculate kd. It is particularly intereating to note that the effects of using limited dissociation are least a t high temperatures. This is a case where network
(-a)
0
1 0
0.2
0.4
1 0.6 FRACTION
I
1
0.8
1.0
132
The Joutnal of Physical Chemistry, Vol. 83, No.
I, 1979
effects under nonequilibrium conditions produced a result which was, a t least to us, quite unexpected. Since dissociation from low-lying levels will clearly occur more readily at high temperatures, one might reasonably expect the effects of preventing such dissociation also to be largest a t high temperatures. The contrary effect may be due to the greater ease of motion through excited levels at high temperatures. In the next sequence of calculations, groups of transition probabilities were varied separately. The salient results are given in Figures 2-5. Note that the “standard” calculation for these results was the relatively simple one employing only VT1, RT1, and VRTl transitions, together with dissociation out of all levels. Particular groups of transition probabilities were multiplied by a factor less than unity (the abscissa of Figures 2-5). and the master equation was then re-solved. Note also that in Figures 2-5, the ordinates show relative values, i.e., the calculated values of kd etc. were divided by the corresponding values from the standard calculation. Considering first the results for hd, we see that the largest effect is produced when the RT1 transition probabilities are varied as a group. This indicates that, in any further refinement of our calculations of hd, special consideration should be given to the evaluation of these transition probabilities. Moderate changes in the dissociative transition probabilities have somewhat less influence than RT1, while VRTl transitions have somewhat less influence still. This last result is somewhat surprising. The VRT transitions provide a very efficient access to higher u states, from the rotationally excited states of u = 0, and one might expect them to be particularly effective in promoting dissociation. Their somewhat reduced influence may be due to the fact that they tend to move molecules parallel to the dissociation limit, Le., the total internal energy tends not to change very much in a VRT transition, whereas rotational excitation moves a molecule directly toward the continuum. The following calculations were designed to provide information about the location of a possible “bottleneck”. The dissociation rate was found to be fairly sensitive to variation of the RT1 and VRTl transitions into and out of the level ( u = 9, J = 12) and eight other levels nearby: the level (9,12) has the largest flux to the continuum, and these other eight levels have fluxes to the continuum of not less than half of the largest flux. Rather surprisingly, k d is somewhat less sensitive to the values of the actual dissociation rate constants out of these maximum flux levels, The region around J = 10 and J = 1 2 appears to be somewhat important in general, since the dissociation rate is moderately sensitive to variations in the RT1 and VRTl transition probabilities in this region. Further details of the results are given in Table 111. Turning now to the vibrational relaxation times, we see that variation of the VRTl transition probabilities as a group has the largest effect. Comparison of calculations 15 and 20 (Table 111) shows that it is mainly the VRT transitions between u = 0 and L) = 1 that are important. However the calculation is almost as sensitive to variations of the RTI probabilities, indicating that access to upper rotational levels is important to the relaxation process. Nevertheless, comparison of calculations 4, 13, and 19 shows not only the RT1 transitions for 1) = 0, but also those for higher u , art. significant. Evidently, downward rotational transitions of rotationally excited molecules, with u > 0 which have made transitions from u = 0, play a significant role in the vibrational relaxation process. For a refinement of the calculation of T,, it is important to have good RT, as well as VRT, transition probabilities.
J. E. Dove and
S. Raynor
The incubation time for dissociation behaves in a manner generally parallel to that of T , (Figures 3 and 4). However it is more dependent, than T,, on energy transfer among highly excited states, as expected from theory.26 The incubation time for vibrational relaxation is very strongly influenced by RT1 transitions, especially in u = 0, and is also quite sensitive to VRT processes (Figure 5 ) .
Summary By means of master equation techniques, we have studied the dissociation of para-H, infinitely dilute in He at shock tube temperatures, using transition probabilities calculated ab initio. We have examined the sensitivity of the calculated dissociation rate constant, vibrational relaxation time, and incubation times for vibrational relaxation and dissociation, to the input transition probabilities. The salient finding is that collisions involving rotation-translation (RT) energy transfer and simultaneous changes in rotation and vibrational energy (VRT processes) both play a major part in determining the system behavior. The dissociation behavior is also, of course, significantly influenced by transitions to the continuum. However, the sensitivity to moderate variations in the probabilities of these transitions is less than the sensitivity to R T transitions and comparable to the sensitivity to VRT. Even the relatively drastic change of eliminating all dissociative transitions except those from levels immediately next to the continuum reduces the overall dissociation rates only by a factor of about 2. We discuss “bottleneck” effects and some surprising effects due to network behavior.
Acknowledgment. This work was supported by the National Research Council of Canada. References and Notes J. Troe and H. Gg. Wagner, Ber. Bunsenges. Phys. Chem., 71, 930 (1967). H. S.Johnston and J. Birks, Acc. Chem. Res., 5 , 327 (1972), and references therein. W. D. Breshears and P. F. Bird, Symp. ( h t . ) Combust., [Proc.], 14th, 211 (1973). D. Appel and J. P. Appleton, Symp. (Int.) Combust. [Proc.], 15th, 701 (1975). H. 0. Pritchard, J . Phys. Chem., 66, 211 1 (1962). H. 0. Pritchard, Acc. Chem. Res., 9, 99 (1976); Chem. SOC.,Spec. Rep., 1, 243 (1975); and references therein. J. E. Dove and D. G. Jones, J . Chem. Phys., 55, 1531 (1971); Chem. Phvs. Lett.. 17. 134 (1972). J. E. Dove, D. G. Jones, and H. Teiteibaum, Symp. (Int.) Combust. [Proc.], 14th, 177 (1973). j . E. Dove and S. Raynor, manuscript in preparation. M. D. Gordon and D. Secrest, J . Chem. Phys., 52, 120 (1970) M. Krauss and F. H. Mies, J , Chem Phys., 42, 2703 (1965). T. Tsaoiine and W. Kutzeinioo. Chem. Phvs. Lett.. 23, 173 (1973). C W Wilson Jr., R Kaprairgnd G. Burns, Chem. Phys Lett, 24, 488 (1974) -. Raczkowski and W. A. Lester, Jr., Chem. Phys. Lett., 47, 45 (1977). W. Koios and L. Woiniewicz, J. Chem. Phys., 43, 2429 (1965); 49, 404 (1968); Chem. Phys. Len., 24, 457 (1974); J. Mol. Spectrosc., 54, 303 (1965). D. M. Bishop and S. Shih, J. Chem. Phys., 64, 162 (1976); 67, 4313 (1977). R. Gengenbach, Ch. Hahn, and J. P. Toennies, Phys. Rev, A , 7 , 98 (1973). J. E. Dove and S. Raynor, Chem. Phys., 28, 113 (1978). R. E. Roberts, R. B. Bernstein, and C. F. Curtiss, J . Chem. Phys., 50, 5163 (1969); J. M. Jackson and N. F. Mott, froc. R. Soc London, Ser. A , 137, 703 (1932). H. Teiteibaum, P h D Thesis, Toronto, 1974. J. E. Dove and S. Raynor, manuscript in preparation. E. W. Montroll and K. E. Shuier, Adv. Chem. Phys., 1, 361 (1958). C. W. Gear, Comm. A.C.M., 14, 176, 185 (1971). J. W. Speiimann and A. C. Hindmarsh, UCID-30116, Lawrence Livermore Laboratory, 1975. R. I. Cukier, J. H. Schaibly, and K. E. Shuier, J . Chem. Phys., 63, 1140 (1975); R. I. Cukier, C. M. Fortuin, K. E. Shuier, A. G. Pelschek, \
A:h.
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Thermal Dissociation of H2 by He and J. H. Schaibly, bid., 59, 3873 (1973);J. H. Schaibly and K. E. Shuler, ibid., 59, 3879 (1973). (26) C. A. Brau, J. C. Keck, and G. F. Carrier, Phvs. NUUS,9, 1885 (1966). (27) J. E. Dove and H. Teitelbaum, Chem. Phys., 6, 431 (1974). (28) R. J. Le Roy, WIS-TCI-387, Theoretical Chemistry Institute, University of Wisconsin, Madison, Wis., 1971.
Discussion J . TROE(Institut fur Physikalische Chemie der Universitat Gottingen). Just’ and I2 have developed an analytical solution of the steady-state two-dimensional (u-J) master equation. We find that the eclui-depletion lines in the (u,s) plane are essentially parallel to the centrifugal barrier boundary in the plane in nice agreement with this calculation. It would be interesting to compare the present numerical calculation with our analytical solution. (1) Th. Just, to be published. ( 2 ) J. Troe, J . Chem. Phys., 66, 4745 (1977).
J. E. DOVE. For the quasi-steady dissociation of H2by He, we find that the (:qui-depletion lines in the (Evlb, E,,J plane are essentially parallel to the centrifugal barrier except at high Evib. At high Evlb,the lines tend to curve away from the centrifugal barrier, to become more nearly parallel to the Erotaxis. If the results are replotted in the ( u , J) plane, this curvature is accentuated. This subject will be discussed in detail in a subsequent paper. DONALD G. TRUHLAlZ (University of Minnesota). There are several encour,iging similarities between the results reported by Dove and our recently reported work on H2 collisions with Ar. Like Dove anti Raynor we used an accurate potential energy
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 133
energy surface and the quasi-classical trajectory method.lJ At 4500 K we found an equilibrium energy of activation for dissociation of 97.2 kcal/mol, 6.1 kcal/mol below Dove and Raynor find a similar equilibrium value for He. We concluded that high rotational states of low vibrational levels as well as high vibrational levels are selectively depleted by nonequilibrium effect^.^ This prediction is confirmed by the work of Dove and Raynor. During the discussion period the question of the accuracy of the quasi-classical trajectory method for 0 1 vibrational excitation was raised. For H2collisions with Ar we computed this energy transfer rate at 4500 K using both the quasi-classical trajectory method4 and a semiclassical forced quantum oscillator method which should be more accurate for highly quantal transitions. We found that the quasi-classical trajectory result exceeded the semiclassical forced quantum oscillator one, but the difference was onky a factor of 1.7.5 (1) N. C. Blais and D. G. Truhlar, J . Chem. Phys., 65, b335 (1976). (2) N. C. Blais ,and D. G. Truhlar, J. Chem. Phys., 66, 772 (1977). (3) D. G. Truhlar and N. C. Blais, J. Am. Chem. SOC., 99,11108 (1977). (4) N. C. Blais and D. G. Truhlar in “State-to-State Chemistry”, P. R. Brooks and E. F. Hayes, Ed., American Chemical Society, Washington, D.C.. 1977, p 243. (5) N. C. Blais and D. J. Truhlar, J. Chem. Phys., 69,846 (1978).
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J. E. DOVE.The similarities between H2/He and H2/Ar are interesting, since tlhey indicate that, at least for these two cases, the equilibrium disriociation kinetics are not strongly dependent on fine details of the intermolecular potential. The results for the 0 1transition are also interesting since they show that, even for a case where quantum effects might be expected to be significant, quasiclassical trajectories give quite good results.
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