2564
J. E. Dove, D. G. Jones, and H. Teitelbaum
The Mechanism of Vibrational Relaxation of Molecular Hydrogen John E. Dove,*+ David G. Jones,’ and Heshel Teltelbauml Institut fur Physikalische Chemie der Universitat Gottingen, Tammannstrasse 6, 0-3400 Gottingen, West Germany; the Department of Chemistry. Lash Miller Chemical Laboratories, University of Toronto, Toronto, Ontario, Canada M5S 1A 1; and the Advanced Research Laboratory, Rolls-Royce Ltd., P.O. Box 3 1, Derby DE2 SBJ, England (Received May 5, 1977) Publication costs assisted by the National Research Council of Canada
The mechanism of vibrational relaxation of molecular hydrogen was investigated by computer simulation of the energy transfer processes in high temperature mixtures of Hz and Ar. Transition probabilities for translation-vibration (TV),translation-rotation (TR), and simultaneous translation-rotation-vibration (TRV) transfer processes between quantized levels were calculated by the distorted wave method with inclusion of the Mies anharmonic factor. The molecular model used was a nonrigid rotating Morse oscillator perturbed by an exponential repulsion potential during collisions with Ar. Explicit account was taken of the molecular rotation throughout the computations. The transition probabilities were then used in a master equation which was solved numerically to simulate a shock wave in Hz-Ar at 3000 K. From the resulting predicted populations of the (0, J)states, the vibrational and rotational energies, relaxation times, and molecular fluxes between states were calculated as a function of time and were used to examine the relaxation mechanism. Initially, the vibrational energy relaxes slowly at a rate governed by TV processes. TR rotational relaxation is initially very rapid, but soon slows down because of the onset of transitions among upper J levels of larger energy spacing. At the same time, relatively rapid TRV transitions from the upper J states of u = 0 begin to occur, and the rate of vibrational relaxation accelerates by a factor of over 40. During the major part of the relaxation zone, vibrational relaxation of Hz is predicted to occur mainly by TRV processes among levels with J > 8. The effects of reduction in the mass of the inert collider, in the exponential repulsion parameter, and in the molecular anisotropy, and of omission of the Mies anharmonic factor, were investigated. All of these changes tend to increase the relative importance of the TV processes, but nevertheless in general TRV processes are still predicted to play a significant role in the vibrational relaxation process in high temperature H2. Introduction Until relatively recently, it was widely assumed that in order to understand the thermally induced vibrational relaxation of a diatomic molecule in an inert diluent, one needed mainly to know the transition probability or transition rate constant of the u = 1 u = 0 process for the nonrotating molecule. The rate constant for u = 0 1 could then be obtained from detailed balancing. Rate constants for transitions among higher vibrational levels, if indeed the temperature was high enough for such processes to be important, were generally assumed to have a simple relationship to the rate constants for the 0 1 and 1 0 processes.’ However there is now evidence that the vibrational relaxation process is more complex than this and involves the rotational degree of freedom, at least in the case of hydrogen-containing m o l e c ~ l e sincluding ~~~ the homonuclear species Hz i t ~ e l f .In ~ this case, the relaxation process will be significantly more complex since it will have to be regarded as involving transitions among a number of rovibrational levels of various different spacings. The approach which we have taken to understanding the vibrational relaxation process is to consider it as essentially a kinetic problem involving many of the features of a chemical reaction mechanism, with the molecules in each rovibrational level being treated as a different species. The time-dependent level populations are then calculated by solution of the chemical kinetic differential equations (“master equation”), with appropriate starting conditions, in a simulated shock wave experiment. The required rate constants for the various collision-induced interlevel transitions of the molecule are calculated by quantum mechanics. We consider here the Hz-Ar system whose
vibrational relaxation, as well as the related process of thermal dissociation, have been studied experimentally a t high temperatures by shock tube techniques. We note that the Hz molecule is of particular interest for studies of the mechanism of molecular energy transfer, because of its simple energy level structure and because many of its properties can be calculated accurately by quantum mechanics. In our computations, the Hz is assumed to be highly diluted in Ar, so that the effects of H2-H2 collisions are not important, and the translational temperature remains constant during the relaxation process. We further simplify the problem by considering here only the states of para Hz. The implied assumptions are that a t shock tube temperatures, though not necessarily a t low temperatures, the behavior of p H 2 is reasonably representative of that of normal Hz, and also that the ortho-para conversion is slow and plays no significant part in the relaxation process. The integrated master equation yields a set of predicted profiles of level populations, from which the time dependence of the internal energy and other properties of the system can be calculated, and pathways taken by the molecule during the relaxation process can be studied. In the present work, particular emphasis is given to the quantum mechanical calculation of the transition rate constants, to a study of the predicted relaxation pathways, and to the sensitivity of these pathways to a variation in the pattern of the interlevel rate constants. Our objective, a t the present time, is to obtain information about the mechanism of relaxation, rather than to achieve precise numerical agreement between calculated and measured relaxation rates.
* Address correspondence to this author at the University of Toronto. t Institut fur Physikalische Chemie der Universitat Gottingen. * Rolls-Royce Ltd.
Calculation of Transition Probabilities A central problem is to calculate the collisional transition probability between two levels of (ui, J J and (uf,Jf)a t a
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The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
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2565
Vibrational Relaxation of Molecular Hydrogen
given temperature T. This is done by obtaining the transition probability as a function of E, the initial energy of approach, and integrating over a Maxwellian distribution of initial relative energies:
4 .O
P ( u f ,J f I ui, Ji;T) =
3.0
I
I
I
I
1 -
L
L c
where u(uf, Jflu,,Ji;E ) is the total cross section for collisional transitions from (ui, J,) to ( u f ,JJ at energy E , and aRb:(E) is the total collision cross section for that energy. The lower limit of integration, Et, is zero if the collision is exothermic (gives energy to translation), and is calculated from energy conservation for endothermic collisions. The method used to calculate cr as a function of E follows in many respects the description of Roberts, Bernstein, and curt is^,^ which in turn draws on the distorted wave approximation.6 The molecular and potential parameters used, and the expressions which we actually evaluated in the calculation of transition probabilities, are summarized below. Molecular Parameters. The calculation requires information on molecular energy levels and bond lengths of H2, and an H2-Ar interaction potential. The molecular parameters were obtained as follows. The vibrational motion of H2 was described by a Morse oscillator. The principal reason for choosing this model is that an analytic expression is available for the matrix elements linking two vibrational states when perturbed by an exponential interaction potential. The rotational motion was included by adding a centrifugal repulsion term to the intramolecular potential, following a prescription of Her~berg,'~ so that each rotational state J has an effective potential curve:
V,(r) = D,{I - exp[-p(r - re)])*+ hJ(J + 1)/87i2cpr2
(2)
where r is the internuclear separation. It should be noted that these effective potential curves automatically incorporate the influence of the varying moment of inertia as the molecule extends when rotating. The treatment is therefore equivalent to the expression of the rotational energy terms, F,(J), as power expansions7bin J ( J + 1). However an additional correction factor, the vibrational-rotational coupling which is often included in expressions for molecular energy levels, is omitted in this treatment. This model predicts 257 bound and quasibound states for H2, significantly less than the true number of states for this molecule.s However the differences between our predicted energies and the actual molecular energies are quite small, generally less than 3 YO,for the lower states which are expected to be the main contributors to the vibrational relaxation mechanism. In this paper, we confine our attention to the 131 states of even J , corresponding to p H 2 . Intermolecular Potential. The potential of interaction between H2 and Ar was expressed in the form N
V(r,R ,0 ) = V , exp(-OR) E u ~ P , ( ~ ) n=0
(3)
where R is the HZ-Ar intermolecular separation, r is the internuclear distance in HP, 0 is the angle between the molecular axis and the Ar atom, and CY is the steepness parameter of the intermolecular repulsion. The coefficients a, of the Nth order Legendre polynomial express the
Flgure 1. Computed values of the anisotropy parameters a 2 and a, (see text) for H2 in collisions with Ar, plotted as a function of the internuclear separation of H2in A. Intermolecular repulsion parameter, a = 4.0 A-'. Solid lines are fitted for collision energies in the range O.5kTto 5kTat 3000 K. The dashed lines are fitted for fixed collision energies of 0.5kTand 5.0kT, and indicate the sensitivity of a 2 and a, to changes in the collision energy.
deviations from sphericity of the H2-Ar equipotentials. The choice of potential parameters is described below. Careful attention was paid to the choice of a, to which transition probabilities for molecular energy transfer are very sensitive. Even though exact prediction of relaxation rates is not a primary objective here, clearly it is desirable to use a reasonably realistic value of CY.No a priori quantum mechanical potential is yet available for Ar-H2, so that various sources of experimental data were studied. Transport property data,g in conjunction with combining rules,g give CY = 5.8-6.0 A-l. However our earlier calculationslO of energy transfer rates indicated that this potential was too steep to account for experimental relaxation data. Moreover molecular beam measurementsll indicated a softer potential with CY about 3.0 A-l. Preliminary calculations using an SSH expression for vibration-translation energy transfer indicated that a compromise value of CY = 4.0 A-1 would account for experimental relaxation rates, so this value was adopted initially, but lowered in a later computation. To determine the anisotropy parameters a,, the method of Roberts, Bernstein, and Curtiss5 was used. For a given energy V, the distance of closest approach can be calculated using pairwise additivity of interactions between the three atoms. This was done for a number of values of 8, and for values of V in the range 0.5kT to 5kT at 3000 K, and the results were then least-squares fitted to a Legendre polynomial. In this way, values of a2 and u4 were calculated for r between 0.75 and 3.0 A, and are shown as the solid lines in Figure 1. Values of coefficients beyond u4 were not used in our computations. The coefficients are not very sensitive to the exact value of V used in the fitting; the dashed lines in Figure 1 show the values of a2 and a4 which would be obtained by fitting at fixed energies of V = 0.5kT and V = 5kT (at 3000 K), respectively. Under our conditions, the attractive part of the interaction potential speeds up the collisions somewhat but is not expected to have a major effect on the probability of energy transfer. However we took it into account by using the standard correction factor, exp(4/kT), where 4/k = 66.4' for Hz-Are9 The Journal of Physical Chemistry, Vol. 8 I, No. 25, 1977
2586
J. E. Dove, D. G. Jones, and
H. Teitelbaum
Expressions Evaluated. The inelastic cross section, used in eq 1, for the transition from state (ui, J i ) to (uf, J f )at initial relative energy E is given by Sn U(Uf,J f ! U i ,
Ji; E ) = -q21 k; i
+ l)P,(ki)
(4)
hi = 2nhui/h is the initial wave vector, with ui and w being the initial velocity and reduced mass of Ar relative to H2. Fl(ki) is the probability of transition from state i to state f for the partial wave of orbital angular momentum quantum number 1. The summation over partial waves 1 essentially accounts for collisions of different impact parameter. s is a steric factor equal to 1 for Av = 0 and to 113 otherwise. The calculation of transition probabilities is greatly simplified by use of the modified wavenumber approximation of Takayanagi,I2 which allows the transition probabilities for the higher partial waves to be related to Lhat for the s wave (1 = 0). The modified wavenumber, hi, is defined by (5) where R, is the distance of closest approach on collision. The practical application of this approximation requires that the initial and final modified wavenumbers be real. Summations over 1 are therefore taken up to l,, which is the largest integer for which
Figure 2. Computed transition probabilities (log scale) for energy transfer collisions of Ar with H2at 3000 K, plotted as a function of inltial Jvalue. Mies anharmonic factor included in the computations. Repulsion parameter, a = 4.0 A-'. a 2 and a4 values taken from data of Figure 1: (TR) V , J = (0, J ) (0,Jf 2); (TV) (0,J ) (1, J); (TRV) (0, J f 2) (1, J).
-
+
+
The translational factor TL(kfIki)was calculated from the distorted wave expression first given by Jackson and Mot@ for the initial and final partial waves. In our computations, I, was typically of the order of 100. Pl(hi) in eq 4 is then replaced by Pl(hi) which may be conveniently written as
Pl (gi)= R (Jf! Ji)A V z( u f! ui)T2(&
lxi)
(7)
Here, R2(JflJi)is the rotational matrix element5 linking states Ji and Jfand was evaluated from
1 2 J ( J - 1) R2(JflJ,) = 4 ( 2 J t 1 ) ( 2 J - 1) 6 ( J - 2 ) ( J + 1) 9(25- 5 ) ( 2 ~ +3ja4
-
which is valid for J J - 2. Equation 18 of Mies13 was used directly to calculate the vibrational matrix element V(vf1ui)linking the states of a Morse oscillator when perturbed by an exponential repulsion potential. The MiesI3 anharmonic factor A is given by
where X2 = Vff/Vii, i / 2 = ~ ' 7 1 2 2F1 , is the Gauss hypergeometric series
The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
Values of R,,,(E) required in eq 1were calculated as the distance of closest approach for a direct collision at initial relative energy E. Using these methods, we calculated the transition probabilities for the following types of transition: VT RT TRV
A V = -1, A J = 0 AV = 0, A J = -2 A V = +1,A J = - 2
Transition probabilities for the reverse transitions (TV, TR, and VRT) were then calculated using detailed balancing. Results. The dependence of the calculated TV (0, J l,J),TR(O,J-O,J+2),andTRV(O,J-l,J-2) transition probabilities, thermally averaged at 3000 K, on initial rotation state is shown in Figure 2. It will be seen that, in the present model, rotation of H2does not enhance the TV transition probabilities; in reality, one would expect dynamical effects to cause these transition probabilities to increase with J. The T R transition probabilities are large a t small J , but decrease with J as the size of the energy gap rises. (However note that the spacings of the J levels begin to decrease again above J = 22.) At low J , the TRV transition probabilities are smaller than the corresponding TV ones, but this order is later reversed and there is considerable enhancement of TRV processes a t high J .
2567
Vibrational Relaxation of Molecular Hydrogen
The distorted wave method is essentially a perturbation procedure, and (within the limitations of the potential) would be expected to calculate the small TV transition probabilities well.14J5 The TR transition probabilities among the lower J levels are large, in some cases exceeding 0.1, and are expected to be overestimated. Classical trajectory calculationsi6 indicate that they are about a factor of 3 too large. This difference would not influence our conclusions about the relaxation mechanism, as discussed below. Concerning the TRV transition probabilities, our opinion is that they are underestimated, very possibly substantially so. In order to achieve the separation of factors in eq 7, several approximations have been made which will have had the effect of reducing the coupling of rotation to the other degrees of freedom. Support (admittedly indirect) for this statement that the TRV transition probabilities are underestimated is provided by close-coupling calculations on energy transfer in H,-He; these calculations3 indicate that the transition probability for the (1, 0) (0, 2) transition is already comparable to or greater than that for (1,O) (0,O). This contrasts with our computations where the transition probability for (1,O) (0, 2) is smaller by a factor of 20 than for the VT process.
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-
-+
Solution of the Master Equation The Equation. The master equation for this system is a set of simultaneous differential equations expressing the rates of change of the populations of the energy levels (u, J) dn(u, J ) = E: { k ( u , JIU’, J‘)n(u’,J ’ ) dt u t ,J’ k(U’,J ’ l u , J ) n ( u , J ) } (12) where n(u,J)is the concentration of molecules in that level and k(u’, J’Iu, J)is the pseudo-first-order rate constant for transitions from (u, J)to (u’, J’) by collision with Ar. The rate constants for a given temperature T were obtained from the transition probabilities described in the previous section, using
k(U’,J’Iu,J ) = Z , , ~ P ( UJ‘Iu, ‘ , J; T )
(13)
where Z”J= i~R,2[Ar](8kT/np)’/~ is the collision frequency of an Hz(u, J)molecule infinitely diluted with argon at [Ar]. R,2 is calculated for collisions at an initial relative kinetic energy of kT, while [Ar] may be expressed as p / k T . The master equation included 131 differential equations for the bound levels, plus equations for a set of pseudolevels representing dissociated molecules formed by transitions from particular levels near the continuum. Dissociation was included in our model (but not recombination, since the H z was assumed infinitely dilute in Ar); this was done in order to be able to investigate also the dissociation reaction using the same methods. All of the computations reported in this paper were carried out a t 3000 K, at which temperature the effect of dissociation on the vibrational relaxation process itself is very small. We may define T ( S H O ) ,the relaxation time of a simple harmonic oscillator having the same value of P(0, 011, 0; T ) as our anharmonic molecule, by 1/7(SHO) = Z,,,P(O, 011, 0 ; T ) [ 1- e-hv/hT] = k ( O , O I 1 , 0 ) [ 1 - e-hv/hT] (14) where hv is the energy of the (1, 0) level. We found it convenient to divide the terms in the master equation by k(O,OIl, O)no,where nois the total concentration of H2 at t = 0, to obtain the dimensionless form
[ 1 - ,-hv/hT
dY(u, J ) d( t 17 (SHO))
Y(U’,J ’ ) k(U’,J ’ l u , J ) k(O,O11,0)
vu,
Y is a fractional level population. Solutions. The master equation can be solved by matrix methods, as has been done by Pritchard in a number of elegant studies of relaxation and dissociation in H2-He systems.17 Alternatively it may be solved by direct numerical integration. We chose numerical integration, mainly because this method can, in principle a t least, readily handle nonlinear differential equations such as those which occur when vibration-vibration transfer processes are included. In either method difficulties occur, either in the form of parasitic eigenvalues or as stiffness of the rate equations, when processes having characteristic rates of very different orders of magnitude are included, as is the case here. These difficulties were largely overcome in our studies by using the variable step length algorithm of Gear.18 The starting conditions for the computation were Y(0,O)= 1, and Y(u, J)= 0 otherwise, at t = 0, and the integration was allowed to proceed until the fractional populations of the bound levels were steady. The integration yields the quantities of Y and dY/ d(t/@HO)) as a function of the reduced time variable t/T(SHO)).A useful feature of the particular model of H2 which we used is that rotational and vibrational energies can be separated in a consistent manner by
Erot =
E: f ( 0 ,J)Y(U,J )
(16b)
u,J
where ~ ( uJ) , is the energy of that state. Then the total internal energy is Ebt = E d b E,t. From these quantities, phenomenological relaxation times ~i can be defined at any time t by analogy with the Bethe-Teller law1
+
T~/T(SHO)[Ei”- Ei]/[diEi/d(t/~(SHO))]
(17)
where i denotes vibration, rotation, or total internal energy as the case may be. In this equation the energies are all expressed per mole and are corrected where necessary for dissociation. Note that Ei” differs slightly from the equilibrium molar energy because dissociation depletes the uppermost levels. Incidentally relaxation times defined in this manner may themselves be time dependent, as indeed we find them to be in this study. In addition to the above information, the program also computed periodically the net flux of molecules between all pairs of levels between which transitions could occur. This information was very useful for tracing out relaxation pathways and in interpreting the relaxation mechanism. Results. Figure 3 shows the computed time dependence of ?yib, Trot, and T~~ (Note that the time scale is expressed in terms of T ( C O ) ,the final value of the energy relaxation time; this relaxation time is characteristic of the zone that would be observed in a standard high temperature vibrational relaxation experiment.) We can distinguish three relaxation regions: initial, intermediate, and final. In the initial stage ( t / ~ ( m ) < we find that the rotational energy is relaxing quickly at a rate largely determined by the fast TR processes among the lower rotational states of u = 0. T , , ~remains nearly constant until The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
2568
J. E. Dove, D. G. Jones, and H. Teitelbaum
-- 1 0
I
v)
IO3
b
\
10-4.
t IO-*
10-4
10-6
loo
tl‘t(m)
IO-^
10-4 tlf
10-2
100
(00)
Figure 3. Computed characteristic energy relaxation times (see text) as a function of time, obtained from master equation solution for H2 in Ar at 3000 K: vibrational relaxation time, T , , ~ (---); rotational
Figure 4. Computed time history of rates of change of internal energy (log scale) for H, in Ar at 3000 K: dE,,/d(t/T(a)) (---); dE,dd(t/T(a)) (--); d€,,,/d(t/T(m)) (-). Data as in Figure 3. Mies anharmonic factor included in transition probabilities.
relaxation time, T~~~ (-); total energy relaxation time, T~~~(--). Computed transition probabilities included the Mies anharmonic factor and used 01 = 4.0 A-’, with a 2 and a 4 values of Figure 1. Relaxation times (log scale) are divided by the SHO relaxation time as defined in the text. The time scale t (log scale) is expressed in units of ~ ( m )the , final energy relaxation time of H2 in this computation.
Erothas reached about 20% of its final value; the distribution over J states of u = 0 is still far from Boltzmann. At the same time, the vibrational energy is relaxing at a very low rate determined by the slow TV processes. The intermediate region extends to about t / ~ ( a=)loF2. Trot decreases rapidly as transitions into upper J levels begin to play a significant role in the energy relaxation. As the upper J levels of u = 0 become populated, the relatively fast TRV processes begin to influence the relaxation, and TGb decreases by a factor of 45 from the initial value. Erothas reached about 99% of its final value, but the vibrational energy has scarcely increased and is still less than 1%of Evibm. The end of the intermediate region therefore can define an induction time for relaxation of the vibrational energy of H2. Values of dE/dt are plotted as a function of time in Figure 4,which shows clearly the acceleration of dE,b/dt toward the end of the intermediate region. In the find stage of the relaxation process, vibration and rotation are completely coupled, and Trot, Tvib, and Ttot become indistinguishable. This is the region where mainly vibrational energy is relaxing at the accelerated rate of the TRV processes, which would be directly observable in a vibrational relaxation experiment in a shock tube. Figure 5 shows the main net fluxes between the energy levels at a time when the final vibrational relaxation process is established. At equilibrium at 3000 K, more than 60% of the vibrational energy is accounted for by the J = 2 , 4 , 6, and 8 states of u = 1. Hence it is of particular interest to study how they are populated. It will be seen that the predicted route is to climb the rotational “ladder” of u = 0 up to the J = 10-14 region, and then to make a TRV transition to u = 1. The J = 2-6 states of u = 1 are then populated mainly by transitions down the u = 1 rotational ladder. An interesting prediction is that whereas the major flux during relaxation in u = 0 is toward higher J , in u = The Journal of Physical Chemlstry, Vol. 31, No. 25, 1977
15000
I V = ~
7
-
10000
5 .
L
I
1
0
2
4
6
8
J-
IO
12
14
16
Flgure 5. Computed major interlevel fluxes during relaxation of H2 by Ar at 3000 K. Horizontal solid lines represent v , J energy levels for v = 0 to 3, with J increasing to the right and energy increasing upward. Width of arrows is proportional to net flux. Net fluxes smaller in magnitude than 6 % of the largest net flux have been omitted. Data as in Figure 3. Mies anharmonic factor included. t / ~ ( m=) 1.79. It will be seen that transitions involving simultaneous changes in v and J(arrows sloping diagonally upward to the left) dominate the flux from v = 0 to 1, and from v = 1 to 2.
1 and 2 it is from high J downward. This does not, of course, mean that there will be a population inversion among J states for u > 0. However, whereas for u = 0 the ratios of the populations of successively higher J states correspond to a temperature slightly less than the heat bath temperature, 3000 K, for u > 0 the population ratios of many of the J states correspond to a temperature somewhat greater than 3000 K. Discussion Using this model of the relaxation process, we conclude that the main mechanism of vibrational relaxation is by jumps involving simultaneous changes in u and J from levels with J > 8 in u = 0. This mechanism is predicted to be over 40 times faster than that given by transitions in which the rotational degree of freedom does not par-
2569
Vibrational Relaxation of Molecular Hydrogen
lO"t,,
0
, , , 10
,
,
. J
,
, 20
,
,
,
,
,
,
I
10-31
,
,
10-6
,
,
,
,
,
10-2
10-4
30
,
, 10'
100
t IT(a3)
Figure 6. Computed transition probabiliies (log scale) for energy transfer collisions of H2 with Ar at 3000 K, plotted as a function of initial Jvalue. Mies anharmonic factor not included. Other data as in Figure 2.
ticipate. The main relaxation of vibrational energy occurs after an induction time which is determined by the time required for the upper J levels of u = 0 to become sufficiently rotationally excited to contribute to the u = 0 1 rate by a TRV process. Similar considerations apply to u = 2 which is populated mainly by TRV transitions from J = 10-16 of u = 1, followed by downward transitions among the J levels. In setting up our model, we restricted the number of types of transitions allowed, and we should now consider the probable effects of these restrictions. In the case of TV transitions, we included only Av = fl in the master equation. However we have calculated the transition probabilities for Au = f 2 , using the methods outlined above, and find that they are too small to change the relaxation mechanism significantly. In the case of T R transitions, we could include jumps with AJ = f4 and A J = k 6 in our master equation, which would somewhat increase the rate of rotational relaxation. However, as discussed above, the T R transition probabilities for AJ = f 2 should be reduced, probably by a factor of about 3. The net result is expected to be some slowing down of the initial rotational relaxation process. However the TR transitions do not have a major rate-determining effect on the vibrational relaxation, and our conclusions about the mechanism would not be changed significantly. In the case of TRV transitions, we included in our model the two types, for lAul = 1 and lAJl = 2 , that were most nearly resonant, i.e., Au = -1, A J = + 2 and its inverse, and that would therefore be expected to have large transition probabilities. Other TRV transitions certainly occur and could be included. Their inclusion would undoubtedly speed up the vibrational relaxation somewhat but would not change the qualitative conclusion about the mechanism, namely, that TRV transitions are already very important. We have also considered whether other parameters which we have used in our model, such as a,, a , and the mass of the collision partner, would change our main conclusions. The effects of these parameters, and also of the anharmonic factor A , are considered below. E f f e c t of Anharmonic Factor. The Mies anharmonic factor A can be thought of as taking account of the coupling of the collision trajectory with the detailed vibrational motion of the anharmonic oscillator. While there appear to be good grounds for using this factor,13 it has in fact often been omitted from distorted wave calculations of molecular energy transfer. Since the effect of including
-
Figure 7. Computed characteristic energy relaxation times (log scale) as a function of time (log scale) for H2 in Ar at 3000 K. T , , ~(---); 7,,t 7tol(-). Mies anharmonic factor not included in transition probabilities. Other data as in Figure 3. (.-a);
15000
- 10000 6
.
> W a w 5000 z w
0
0
2
4
6
8
IO
12
14
16
JFigure 8. Computed major interlevel fluxes during relaxation of H2 by Ar at 3000 K. Mies anharmonic factor not included in transition probabilities. f / 7 ( m ) = 2.29. Other data as Figure 5. It will be seen that while the largest net fluxes from v = 0 to v = 1 and v = 1 to v = 2 involve simultaneous changes in vand J, TV transitions ( A v = +I, AJ = 0, vertical arrows) now also contribute substantially.
A is to decrease transition probabilities by a factorlg which can be as much as 100,or even more, it would in some cases remove the agreement with experiment. It is therefore of interest to compare the results of the calculations described above with another set of calculations (which were in fact made earlier) which were similar except that A was omitted from the computation of the transition probabilities. Figure 6 shows that on omission of A the TV probabilities are increased by a factor of 400). The TRV probabilities are also increased, though by a smaller factor, egg.,by 9.3 for P(0, 1211, 10). R T probabilities are unaffected. (See also Table I.) Figure 7 shows the time dependence of 7Yib) T ~ , , ~and , 7tot when the resulting master equation is integrated. It will be seen that the overall qualitative behavior is similar to that in the calculation which included A , but that there are substantial quantitative differences. The vibrational incubation time is increased relative to ~ ( m ) . Vibration-rotation coupling still causes 7,ib to decrease below the SHO value, but the decrease is only by a factor of 2.2 as compared with 44 when A is included. Flux diagrams during the vibrational relaxation period (Figure 8) show that TV and TRV processes contribute about equally to The Journal of Physical Chemlstty, Vol. 81, No. 25, 1977
2570
J. E. Dove, D. G. Jones, and H. Teitetbaum
0.1
L
~
01’
’
’
10-2
10-6
ioo
IO*^
1 / 7 : (to)
Figure 9. Computed time history of net flux, in arbitrary units, from v = 0 to v = 1 for H2 in Ar at 3000 K. Mies anharmonic factor not included in transition probabilities (cf. Figure 6). Note that initially the main flux is by TV processes ( A v = f l , AJ= 0), but after t l s ( m ) = TRV processes (simultaneous changes in v and J ) begin to contribute significantly. The largest total net flux (TOT = TV f TRV)
-
is more than double the TV contribution.
the u = 0 1flux with the maximum TV flux being from the heavily populated J = 4 state and the maximum TRV flux being from (0,lO) to (1,8). As previously, the net flux among the J states of u = 1 and 2 is downward, except at very high J. The relaxation behavior is illustrated also by Figure 9, which shows the time dependence of the TV and TRV fluxes for u = 0 1. Initially only the TV flux is significant. As the higher rotational levels become populated, the TRV flux becomes important, and the total flux doubles before declining to zero as equilibrium is established. E f f e c t s of Collision Partner Mass, Anisotropy, and Exponential Repulsion Parameter. In other sets of computations, the values of three molecular parameters were varied separately to study their effects. (In these computations, A was omitted from the transition probabilities. The effects of the molecular parameters on the relaxation behavior is most readily seen when the TV and TRV routes contribute roughly equally, as they do when A is not used.) The collision partner mass m was changed from 40 to 4.0 amu, the steepness parameter a of the exponential repulsion potential was changed from 4.0 to 3.45 A-l (a value from molecular beam data) while initially keeping a, unchanged, and then the anisotropy parameters a, were changed by refitting the equipotential curves generated by a = 3.45 A-l. The results are summarized in Table I. We note that changing m affects the TV and TR probabilities, a, changes TR and TRV probabilities, while a and A affect mainly the TV processes. The main features of the resulting flux network during relaxation manifest themselves in the quantities initial), 7 (m), the incubation time tint, and s ( m ) / ~ ( S H o )In . general it can be said that none of these parameter changes produces a spectacular alteration in the pattern of the relaxation flux. Reducing the mass of the collision partner speeds up the overall relaxation, and somewhat decreases the importance of the TRV route. Reducing a, of course reduces the importance of processes involving rotation, and at sufficiently low a, the relaxation of vibrational energy will become a pure TV type. However with low a,, the rotational relaxation process itself slows down, and this could lead to a situation where rotational and vibrational relaxation occur on the same time scale and become practically indistinguishable experimentally. The effects of reducing a alone, i.e., of making the repulsion potential
-
The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
Vibrational Relaxation of Molecular Hydrogen
257 1
to near the lower end of the range of the values which we softer, are particularly interesting. The TV transition have considered. The indicated mechanism, with correct probabilities are, as expected, reduced. However TRV inclusion of the anharmonic factor A, would remain the processes increase in importance, so that the overall resame. Incidentally we obtained the best numerical laxation rate decreases less than the TV probabilities. agreement with experiment for cy = 3.45 A-1 and a2 = 0.245 However if both a and a,, are reduced, the relaxation rate for (u = 0, J = 0). However since A was omitted from the is substantially reduced, and TRV diminishes in imporcorresponding transition probabilities, the significance of tance. this agreement is doubtful. Other Studies, Pritchard has used matrix methods to solve the master equation for Hz relaxation and dissociConclusions ation." By judicious variation of the transition probabilities, which were not calculated a priori, he concluded, We have investigated the mechanism of relaxation of as we do, that TRV processes are important.20 R a b i t ~ ~ ~internal energy of Hz by solving the vibrational-rotational solved the master equation for H2-He vibrational relaxmaster equation for H2 in Ar behind a shock wave at 3000 ation at relatively low temperatures, using transition K. Quantum mechanical transition probabilities for this probabilities from an effective potential coupled-channel computation were obtained by the distorted wave method quantum mechanical calculation, and concluded that H2 with explicit inclusion of the rotational degree of freedom. also relaxes via TRV routes under these conditions. InThe main conclusion is that, after an initial transient cidentally, the cost of coupled-channel computations seems period, simultaneous vibrational and rotational transitions likely to be prohibitive under high temperature conditions play a substantial role in the vibrational relaxation of Hz as in the present work. by Ar. The influence of this type of transition is apparent Studies by Billingz1and Alexanderz2have also taken the in all of our computations but is strongest in what we rotational degree of freedom explicitly into account in believe to be the most reliable computation in which the calculation of the quantum transition probabilities, and Mies anharmonic factor was incorporated into the calhave used a rotational equilibrium assumption in calcuculation of transition probabilities. In this case, the vilating vibrational relaxation rates. We have also evaluated brational-rotational transitions increased the relaxation a thermal equilibrium vibrational relaxation time from rate by a factor of 45 above that which would be calculated transition rate constants Boltzmann-averaged over initial without taking rotation explicitly into account. J and summed over the final J', from We also find that inclusion of rotation considerably modifies the effect of use of the Mies anharmonic factor 1 / r = Z[X(l, J)Z:h(O, J'll, J ) A on the magnitude of the calculated relaxation times. Use J J' of A can reduce translation-vibration (TV) transition X ( 0 , J ) Z l z ( 1,J ' IO, J ) ] (18) probabilities by factors of 100 or more, and will have a J' corresponding effect on TV relaxation times. However the where X ( u , J)is the equilibrium fractional population of transition probabilities for simultaneous changes in vithe level ( u , J). For the conditions of our calculations, 7 bration and rotation are reduced to a much smaller extent. is found to be extremely close to the phenomenological Hence, when rotation is included in the calculations, use ~ ( m found ) from our master equation calculations. Thus, of A depresses the computed vibrational relaxation time when using transition probabilities containing A, T / ~ ( S H O ) considerably less than has been found in earlier work. = 1/44.8 for the rotationally equilibrated case, as compared with 1/44.0 from the master equation. It turns out, Acknowledgment. This work was supported in part by the National Research Council of Canada and the Scitherefore, that 7(m) could have been found directly from entific Affairs Division of the North Atlantic Treaty an equilibrium calculation, though naturally then all information about the transient behavior and about the Organization. We are grateful to Professors T. E. Hull and actual extent of deviation from thermal equilibrium would W. Enright for discussions of problems of numerical integration. J. E. Dove thanks the Alexander von Humboldt be lost. Foundation for the award of a Special Fellowship, and Comparison with Experiment. Our main objective was Professor J. Troe for his cordial hospitality in Gottingen. stated to be the study of the relaxation mechanism, rather than achievement of exact agreement with experimentally References and Notes measured relaxation rates. Nevertheless the actual rates (a) J. F. Clarke and M. McChesney, "The Dynamics of Real Gases", which we have calculated should be briefly discussed. In Butterworths, London, 1964. (b) E. W. Montroll and K. Shuler, J . a previous we have used shock tube methods to Chem. fhys., 26, 454 (1957). measure the relaxation time of Hz by Ar, and obtained 1.5 (a) T. L. Cottrell and A. J. Matheson, Trans. faraday SOC.,58, 2336 X lo4 atm s at 3000 K which agrees very well with a (1962); (b) T. L. Cottrell, R. C. Dobbie, J. McLain, and A. W. Read, ibid., 80, 241 (1964); (c) T. L. Cottrell, and A. J. Matheson, ibid., measurement by Kiefer and L ~ t z In . ~ our ~ calculation 59, 824 (1963); (d) froc. Chem. SOC.,114 (1961); (e) ibid., 17 using the anharmonic factor, we found T = 6.3 X lo4 atm (1962); (f) J. H. Klefer, W. D. Breshears, and P. F. Bird, J. Chem. s, about 4 times slower than experiment. In view of the fhys., 50, 3641 (1969); (9) W. D. Breshears and P. F. Bird, J. Chem. f h y s . , 52, 999 (1970); (h) ibid., 50, 333 (1969). uncertainties about the potential, the agreement must be (a) H. Rabitz and G. Zarur, J . Chem. f h y s . , 62, 1425 (1975); (b) considered reasonable. Several improvements could be M. H. Alexander, ibid., 56, 3030 (1972); (c) P. McGuire and J. P. suggested in the calculations. Removal of some approxToennies, ib& 62, 4623 (1975); (d) M. H. Alexander and P.McGuire, ibid., 64, 452 (1976). imations which restrict vibration-rotation coupling, and J. E. Dove, D. G. Jones, and H. Teitelbaum, Symp. (Int.) Combust., incorporation of additional TRV processes, would un[ f r o c . ] , W h , 177 (1973). doubtedly speed up energy transfer and further increase R.E. Roberts, R . B. Bernstein, and C. F. Curtiss, J . Chem. Phys., 50, 5163 (1969). the importance of rotation in the vibrational relaxation J. M. Jackson and N. F. Mott, Proc. R . SOC.London, Ser. A , 137, mechanism. However at the same time, a and a, should 703 (1932). possibly be decreased, which would have the opposite G. Herzberg, "Molecular Spectra and Molecular Structure", Vol. 1, effect. After these computations were completed, T r ~ h l a r ~ ~ Van Nostrand, Princeton, N.J., 1964: (a) pp 425-426; (b) p 106; (c) p 109. published an Hz-Ar potential obtained in part from an T. G. Waech and R. B. Bernstein, J . Chem. fhys., 46, 4905 (1967). electron gas calculation. Comparison of his potential and J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory ours suggests that cy and a2 should indeed both be reduced of Gases and Liquids", Wiley, New York, N.Y., 1954. The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
2572 (10) (11) (12) (13) (14) (15) (16) (17) (18)
R. Shaw and F. E. Walker J. E. Dove and D. G. Jones, J . Chem. Phys., 55, 1531 (1971). I. Amdur, Adv. Chem. Phys., 10, 29 (1966). K. Takayanagi, Adv. A t . Mol. Phys., 1, 149 (1965). F. H. Mies, J . Chem. Phys., 40, 523 (1964). R. E. Roberts, J . Chem. Phys., 49, 2880 (1968). R. L. McKenzie, J . Chem. Phys., 63, 1655 (1975). J. E. Dove and H. Teitelbaum, to be submitted for publication. H. 0. Pritchard, Chem. SOC., Spec. Rep., 1, 243-290 (1975). C.W. Gear, Commun. A m . Comput. Mach., 14, 176, 185 (1971).
(19) (a) F. H.Mies, J . Chem. Phys., 41, 903 (1964); (b) ibid., 42, 2709 (1965). (20) E. Kamaratos and H. 0. Pritchard, Can. J. Chem., 51, 1923 (1973). (21) (a) G. D. B. Strensen, J. Chem. Phys., 57, 5241 (1972); (b) G. D. Billing, Chem. Phys., 9, 359 (1975). (22) M. H. Alexander, Chem. Phys., 8, 86 (1975). (23) J. E. Dove and H. Teitelbaum, Chem. Phys., 6, 431 (1974). (24) J. H. Kiefer and R. W. Lutz, J . Chem. Phys., 44, 668 (1966). (25) N. C. Blais and D. G. Truhlar, J . Chem. Phys., 65, 5335 (1976).
Estimated Kinetics and Thermochemistry of Some Initial Unimolecular Reactions in the Thermal Decomposition of 1,3,5,7-Tetranitro-I ,3,5,7-tetraazacyclooctane in the Gas Phase Robert Shaw" Chemistry Department, Lockheed Missiles & Space Company, h e . , Lockheed Paio Alto Research Laboratory, Palo Alto, California 94304
and Franklln E. Walker Lawrence Livermore Laboratories, University of California, Livermore, California 94550 (Received May 27, 1977) Publication costs assisted by Lockheed Missiles & Space Company, Incorporated
A survey of the literature has shown that there are eight possible initial unimolecular steps in the thermal decomposition of HMX. Of the eight steps, Arrhenius parameters have been estimated for four of them: reaction, log As (where As = A/s-'), E/(kcal/mol); N-NO2 fission, 16.4, 46.2; homolytic C-N fission, 18, 60; five-center elimination of HONO, 10.8, 238; four-center elimination of HN02, 10.8, 238. The depolymerization of HMX to 4CH2NN02(a fifth possible step) was estimated to be 35.4 kcal/mol endothermic. The Arrhenius parameters are such that N-NO2 fission predominates at around 550 K. The estimated rate constant, lo-' s-l at 500 K, is one power of ten less than previously measured rate constants for the overall decomposition.
Introduction
4CH2NN02,5reaction 4; N-N02 fission,6-8 reaction 5 ;
The following initial steps in the thermal decomposition of HMX, 1,3,5,7-tetranitro-l,3,5,7-tetraazacyclooctane, have been postulated: transfer of an oxygen atom from a n -NOz group to a neighboring -CH2- group,lS2reaction 1; heterolytic C-N ~ l e a v a g ereaction ,~ 2; elimination of
/N\,,
T2 Y 02N-N\ CH2 N,
0
4
NP2
TH2 Y "\CH
02N
\
O ,
cp
\
(l)
C H N 2,C H ,2
- NI
I
I
NO2
NO2
02N
iH2 YrNo2 \" - I / 0-CH2
I \
0
CH
i
CH
2\N/CH2
%CH2
I N02
(3)
CH2 ,N/CH2
I N02
The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
"(32
i-"02 -
\
/ CHZ
CH2,
I N02
/HZ
OZN
I NO2
I N .
2
N
i"''\" Y rNo2 - 02"-1 -7
I
0 N-N
"pz .
CHZ,"/CHZ
NO2
(2)
CH2NN02,4 reaction 3; concerted depolymerization to 2
OZN
"p2
NO 7
y
rNo2 - -"\
/ CH2
rNo2iH'N\cH Y iH2 "?
2N ,C / H2
I
, "
homolytic C-N ~leavage,~,' reaction 6; and five-center
N\CH
N-NN(t)
\
\
Nf2 (-'
702
2H C,N,2H C
\
CH2
HUY
"\CH
rNo2
I
,NCN-742
02N-N
CH2N ,C / H2
02N-N
/CH2
rNo2 rNo2
N
"p?
Np?
02N-N\
CH2
,
/CHZ
i-"02
(6)
I N02
elimination of HON0,6,7-9reaction 7 . Recently Benson" has postulated that nitroalkanes may decompose by a four-center elimination of HN02, reaction 8. By analogy, elimination, HMX may also undergo four-center "02 reaction 9. It is not clear at present how to estimate the thermochemistry or the kinetics of the first three processes. However, we record here estimates of the kinetics of
