J. Phys. Chem. 1980, 84, 3050-3054
3050
Analytic Approximatlon to the Solution of Vibrational-Rotational Energy Transfer In the Dissociation and Relaxation of H2,D2, and O2 Wendell Forst Depariment of Chemistry, Universl Lava/, Qucibec, Canade G1K 7P4 (Received: April 14, 1980)
The analytical solution of the general relaxation problem in a low-pressure system with exponential transition probabilities, as given previously for vibrational energy transfer on collision, is now extended to include rotational energy transfer. The approximation is made that the rotational degrees of freedom are equilibrated and that the barrier to dissociation is linear and asymmetric. The theory is used to calculate the rate constant for dissociation, the number density incubation time, and the vibrational relaxation time in the dissociation of Hz,D2, and 02.Agreement with experiment and other more elaborate calculations is very satisfactory. 1. Introduction
where the preexponential factor is
In recent publications1s2 we have given an analytical solution of the general relaxation problem in a low-pressure system with exponential transition probabilities, on the assumption that only vibrational energy transfer is involved in the ladder-climbing process that leads to dissociation. The theory gave good account of relaxation, incubation, and dissociation rates in triatomics, but diatomic relaxation and dissociation rates could not be both fitted with the same set of parameters, although incubation times in oxygen were well reproduced. There thus appeared to be a fundamental difficulty with respect to the treatment of diatomics, which we attributed to the neglect of rotational energy transfer. The purpose of this work is to show that the discrepancy is indeed due to the neglect of rotations, by giving a simple analytical solution of the vibrational-rotational energy transfer problem and testing it on the relaxation and dissociation rates obtained from shock-wave experiments involving diatomics important in combustion processes: H2, D2, and 02.This work joins several numerical treatments of the same problem that have appeared previously.3-6 The organization of the paper is as follows. Section 2.1 gives a brief summary of the principal equations of the theory, as developed originally for collisional transfer of vibrational energy. Section 2.2 introduces vibrationalrotational energy transfer on collision, and such simplifications as are necessary to make the problem tractable; then equations of section 2.1 are modified accordingly. Section 3 gives a summary of the results, while section 4 presents a discussion, compares the results with other data, and gives an outline of the limitations of the theory. Finally section 5 gives a brief summary. 2. Theory
The thermal dissociation of a molecule is seen as a ladder-climbing process which at low-enough pressures depends only on q(x,y),the probability (per collision and per unit energy) of a transition from an initial state of energy y to a final state of energy x. We shall first give a brief r6sum6 of the vibrational energy transfer problem, for which a complete solution, including the initial transients, is available: and then introduce such modifications as are necessary to account for transfer of rotational energy. 2.1. Vibrational Energy Transfer. The transition probability that admits of an analytical solution is of exponential type which we call model B, given by1i2 q(x,y) = C(x)e-b-")IY x < y (down) - C(y)e-(x-y)Y' x > y (up) (1)
*.I
'7
=(1 + yy'e-27"w) (w = x or y) (2) 1 + YY' Here y' = l / y + l/(kT), y" = l / y 1/(2kT). The constant y is to a very good approximation the average vibrational energy lost in a collision (ref 2, eq 8) and also C(W)
+
the negative of average energy transferred in all collisions (up and down; cf. ref 2, eq 10). It is understood that x and y, at this point, refer to vibrational energy only. The probability of eq 1 is normalized in (0,m) and satisfies detailed balance in the form q(x,y)e-Y/(k"= q(y,x)e-z/(k"
(3) It can be shown1 that the solution for the time-dependent population density of the reactant requires the solution of the eigenvalue problem (4)
where ai(U) and Xi are respectively the eigenvectors and eigenvalues of q(x,y),and Eo is the energy threshold to reaction. For model B (eq l), the population density c(x,t), i.e., the fraction of particles per unit energy, at energy x and time t, is given by the eigenfunction expansion (ref 2, eq 26)
where S is the solution of = tanh - S2) and the Rj values are solutions of 2
2s
+ (r/kT)(1
(2)
(6)
Here CY = -2 tanh-l S, 0, = -2 tan-l Rj, and the constants A. and B j depend on initial conditions (ref 2, eq 27 and 28). The steady-state rate constant kJ (s-l) is given by the lowest eigenvalue Xo w[y/(kT)lZ(1 - S2) ko' = ~ (- lX0-l) = [y/(kT)I2(1 - S2)+ 411 + y/(kT)I
0022-3654/80/ 2084-3050$01.OO/0 0 1980 American Chemical Society
(8)
Dissoclation and Relaxation of
The Journal of Physical Chemistry, Vol. 84, No.
HP,D,, and O2
where w is the collision frequency. The relaxation times T~ (I) in dimensionless form are given by the higher eigenvalues X j 0' = 1,2, ...)
For threshold that is not too small (Eo> kT), it can be shown (ref 1, eq 79 and 80) that eq 8 simplifies to
From eq 5-9 it is piossible to deduce the entire time history of the system, including the time-dependent properties prior to establishment of steady state. In the present context we shall need among these properties only the so-called number density incubation time tincwhich measures the time delay in the onset of steady-state decay of the reactant. We have (ref 2, eq 37) CY
-2kTAOe-Eo/2kT cosh - sinh 2 (11)
Equations 5-11 apply to a hypothetical molecule having a constant (Le., energy-independent) density of states, as indicated by the simple form of the detailed balance in eq 3. We now assume that the exact detailed balance condition, i.e., one including energy-dependent density of states, need be satisfied only within kT of threshold, which can be shown amounts to defining an effective threshold Eo* given by (ref 21, eq 67) Eo* = Eo - kT In [N(Eo)kT/Q] (12) where N(Eo)is the density of states at Eo and Q is the partition function. Equation 12 is the special form, applicable to diatomic reactant, of a more general relation derived in ref 2. In actual application to real systems, Eo is taken to be the (0 K) bond dissociation energy of the bond undergoing rupture, and (in diatomic dissociation) Eo* is calculated from eq 12. The thxeshold that appears either explicitly or implicitly in eq 4-11 is then interpreted to mean the effective threshold Eo*; in other words, Eo* is used for Eo throughout. In particular, the steady-state rate constant becomes, from eq 10 and 12 we-dEo*/(kT ) wk TN(EO)e-EoI(kn k0t =: - =(13) (1
+~ T / T ) ~
(1
assumptions are necessary to make the problem tractable: (1) The vibrational-rotational transition probability factorizes into vibrational and rotational parts, subject to the detailed balance condition of eq 3 ~ ( x , Y=) ~ v ( ~ v , ~qr(Xr,Yr) v ) (14) where subscripts v and r serve to distinguish between vibrational and rotational energies. Both qv and qr will be given by the model B exponential transition probability. Thus qv is given by eq 1, with x,, yv and yv replacing x,y, and y, respectively. (2) The rotational part qr is given by the strong-collision version of model B probability (ref 2, eq 12): In other words, it is assumed that qr is the Boltzmann distribution for the rotational energy of the final state, and that it is therefore independent of the energy of the initial state. The justification for this is the well-known fact that rotational energy transfer is very much faster than vibrational energy transfer, so that to a good approximation the rotational degrees of freedom may be considered to be equilibrated at all times. (3) The boundary that separates bound and dissociated states is given by eq 16. In other words, the energy barrier x, + B*xr = Eo (16) to dissociation is assumed to be linear and asymmetric,9 the parameter B* I 1 being the measure of asymmetry. B* < 1 imples that rotational energy is less efficient in promoting dissociation than vibrational energy. As a result of assumptions 1-3, integration over yr in eq 4 can be performed separately, and consequently the problem becomes effectively one-dimensional:
where @(x,,y ) of eq 4 has been replaced with $(xv,yv)e-(rv+xr) 7(kn(kT)-2. The function $(x,,y,) measures the fractional deviation of @(xv,yv) from the (quasi-diatomic) equilibrium distribution exp[-(x, + x,)/(kT)] (kT)". Using standard procedure,l the integral equation, eq 17, can be transformed into the differential equation
+/ZT/~)~
The remaining parameter that must be specified is y, the average energy transferred per collision. It is obtained from the experimental relaxation time ~~~~~l which is taken to mean T ~ the , first relaxation time; thus eq 9, with j = 1, is solved for y by letting T~ ~ ~ We~denote ~ the ~ y i so obtained by yeXpd.If yemais used in eq 13, hot is between one and two orders of magnitude too small for most diatomic dissociations. In other words, it is impossible to reproduce both relaxation and dissociation rates with the same y. 2.2. Vibrational-Rotational Energy Transfer. When rotational, as well as vibrational, energy is transferred on collision, the basic eigenvalue equation is still eq 4, except that x and y now refer to total (vibrational plus rotational) final and initial energies, respectively. The problem is thus two-dimensional and more difficult to handle. An approximate solution for model B transition probability of this two-dimensional problem has been given by Penner,' whose argument we shall follow. The following simplifying
23, 1980 3051
(18)
where the subscript v on x, y, and y' has been dropped since now only vibrational energy is involved. For B* = 0 this equation reduces to the equation characteristic of vibrational-energy-only transfer problem (cf. eq 70 of ref 1.). The substitution z = exp[-(Eo - x)/(2B*kT)], $(XI = , f(z), transforms eq 18 to z2f"(z) + (1- 2B*)zf'(z)
-
(2B*a)2 [I h(1 - 22)]f(z) = 0 (19)
where u2 = (kT/y)(l + kT/y). Equation 19 can be solved analytically only if we let X = 1,which in turn requires that we delete the boundary condition at x = 0. This destroys in effect the eigenvalue character of the original problem, as has been pointed out by Penner,7 and consequently no relaxation times can be obtained, only the steady-state distribution of populations, and from it the steady-state rate constant KO+. For X = 1 and integral B*, the solution of eq 19 can be written in terms of modified Bessel fun~tions.~ However
3052
The Journal of Physlcal Chemlstry, Vol. 84, No. 23, 1980
for X = 1 and B* = l/z, eq 19 simplifies to the point where the solution is a simple e~ponential:~ a y sinh a + k T cosh a f(z) = cash uz sinh az a y cosh a + k T sinh a
I
(20)
Note that although the boundary condition at x = 0 has been deleted, f(z) remains well-behaved at the origin [f(z) 13 provided Eo/kT is not too small. This is not a seriom restriction, for Eo is the bond-dissociation energy and as such is therefore usually quite high. Hence E o / ( k T )remains large even at temperatures of interest in the shock tube (2000-5000 K). At such temperatures a is quite large also, since, typically, y 100 cm-l or so, and consequently sinh a cosh a to a very good approximation. Hence for all practical purposes f(z) cosh az - sinh az = e-az (21) We shall develop the remainder of the argument entirely in terms of B* = l/z. The motivation for retaining this particular value of B* and the corresponding solution is threefold (1) the solution is extremely simple; (2) there are reasons to believe that the solution is relatively insensitive to B* as long as B* is not too close to unity; (3) real systems have B* that is not far from For example, it can be deduced from the exact calculations of LeRoys that for both H2 and Dzthe value of B* is 0.81, with a barrier that is linear over much of the range. For inverse sixth-power attractive p ~ t e n t i a lB* , ~ = 2/3, and for Lennard-Jones (6-12) potential, B* = 0.56 (this follows from the inflection point which is atlo 0.8 e), again with fairly linear barriers. For Oz,in the Morse potential representation,“ the vibrational ground state dissociates when the rotational quantum number is J = 220, which leads to B* = 0.58 if we assume that the rotational constant id2 1.4457 cm-l, independent of J. Once the steady-state population distribution function @‘se is known, the steady-state rate constant kJ is evaluated as “upward flow” from13
-
-
kot =: w
4
-
-
dx, dx,
,-lc,/(kTI x kT
!hroevsfiold
Forst
the diatomic steady-state rate constant, based on the assumption that collisions transfer both vibrational and rotational energy and that rotational energy transfer is sufficiently fast to maintain equilibrium distribution of rotational energies. Unfortunately, as a consequence of the approximation that had to be made in the course of the derivation, information about pre-steady-state properties of the system (relaxation times in particular) has been lost. We can assume, however, in line with previous argument, that the bulk properties of the relaxing system depend essentially only on effective threshold, of which the steady-state rate constant is the most sensitive measure, for only the rate constant depends on threshold exponentially, while all other properties (relaxationtimes, for example) are much weaker functions of threshold. We should therefore expect to obtain a useful approximation to pre-steady state properties of the vibrational-rotational system if we proceed as in the vibrational-onlycase (section 2.1) but defiie an effective threshold Eo* in such a way as to obtain for kot the “correct” expression, i.e., eq 24. The condition on Eo* is therefore we-Eo*f(W - wkTN,(Eo)e-Eo/(kn - (1 + k T / r ) Q , (25) (1 + k T / y ) 2 from which
Equation 26 thus replaces eq 12 when collisions transfer both vibrational and rotational energy. The term that remains to be made explicit is N,(Eo). We adopt the Morse representation, in which the density of vibrational states of a diatomic is given by13
(
L L)‘z De - E
N,(E) = hv
where v is the frequency of the Morse oscillator, and De = Eo + hv/2, the dissociation energy measured from potential minimum. For two rigid rotors, the density of vibrational-rotational states is given by the conv~lution~~ qn 1 / 2 Y U e ’
which yields, using eq 21, i.e., B* = ‘/z
This result is to be compared with eq 10 since the treatment so far has been analogous to that leading to eq 10, in that constant density of states has been assumed. By comparing eq 23 and 10, one can see that the effect of rotational energy transfer is therefore, approximately, an increase in the steady-state rate constant ko+by a factor of 1 + k T / r . In order to convert the rate constant kot of eq 23 into one based on energy-dependent density of states, we can make use of the same argument employed in deriving eq 13, with the result
where N,(Eo) is now the density of vibrational-rotational states near threshold, and Q, is the vibrational-rotational partition function. Equation 24 is the analytical form of
Q:
7 ID:/’
- (De -
(28)
where Q,’ = 87?I/h2 (I = moment of inertia). In evaluating the partition function Q,, we write Q, = Q,Q,, where we take Q, kT/hv, the classical high-temperature limit on account of the high temperatures in a shock tube, and Q, = kTQ,’. Thus
-
and hence Eo* = Eo -
,
1
2(1
\
I
+ kT/r)D,1j2[De1/‘- ( h ~ / 2 ) ~ / ~(30) ] kT
3. Results 3.1. Low-Pressure Dissociation of Hydrogen and Deuterium, M Xz- M 2X (X = H or D,M = He or Ar). The “best” bond dissociation energy of Hzis8Eo = 36117.5
+
+
The Journal of Physical Chemistty, Vol. 84, No. 23, 1980 3053
Dissociation and Relaxation of H,, DP,and 0,
TABLE I: Values of Parameters in Eq 31 and 32 system A B C D H, t He 41.35 8.984 1.6625X 10' 95.2119 H, t Ar 45.09 8.956 2.2584 X lo3 103.824 D, t Ar 51.25 9.0 1.476 x lo3 118 54.29 9.42 5.609 X 10' 125
-~
-
ref 15 15 16 17
cm-l, and its zero-point energy id2E, = (hv/2) = 2198 cm-l; for D2,the corresponding values are12*14 Eo,= 36 746 cm-l, E, = 1559 cm-l. Experimental relaxation times are available in the form log pr = A T 1 / 3- B atm s
(31)
usually valid between -1300 and -3000 K, which for the present purpose are transformed into the dimensionless form UTexptl
= CT-'/S exp(DT'/s)
(32)
using the Lennard-Jones collision number13 ZL-j = 21'(2/3)Z[e/(kT)]1/3where Z is the hard-sphere collision number and e is the (depthof the L-J potential well. Table I gives a summary of the relevant parameters. The energy-transfer parameter yexptl is obtained by solving eq 33,
where orexptlis given by eq 32 and R1 is the first solution of eq 7. The rate constant hot then follows from eq 8, and the incubation time from eq 11. The threshold energy is taken throughout tal be the effective threshold Eo*of eq 30. The calculated results are summarized in Table 11, where the last column gives ko(exptl),the experimental value of the rate constant for dissociation. For H2 + Ar, the recommended'* value is ko(exptl)= 3.7 X exp(-48300/!0 cm3 molecule-" s-l, valid between 2500 and 5000 K. For D2 + Ar, the calculated results are those using relaxation data of ref 17 (see Discussion), while ko(exptl) is that of Appel and Appleton,l@who give ko(exptl) = 2.4 X exp(-47006/T) cm3molecule-l s-l, valid between 1800 and 4000 K. The column headed or1in Table I1 represents no new information, merely a check that the calculation
does give back the experimental relaxation time given by eq 32. 3.2. Low-Pressure Dissociation of Oxygen, Ar + O2 Ar + 20. The 0-0bond dissociation energy in O2isx2Eo = 40985 crn-', and E, = 790 cm-l. Relaxation times in Ar-02 mixtures were measured between 1200 and 7000 K by CamacZ0and Whitez1from whose results one obtains, using ZL-j (cf. ref 2, eq 77) wrexptl = 1.4577 X 1- e~p(-2228/T)]-~e~p(218.279Tl/~) (34) The calculations then proceed as in section 3.1. The results are given at the bottom of Table 11. The experimental rate constant ko(exptl) is available from BaulchZ2who gives ko(exptl) = (3.0 X lO"/T) exp(-5.938 X lo4/?? cm3 molecule-l s-l, said to apply from 3000 to 18000 K. 4. Discussion
Comparison of hot and ho(exptl) in Table I1 shows immediately that the vibrational-rotational calculation outlined section 2.2 successfully reproduces (within a factor of -2) the experimental rate constants, using a yexpathat also gives the correct experimental relaxation times, something that cannot be accomplished when only vibrational energy is assumed to be transferred on collision.2 At the same time, incubation times are also well reproduced, as shown in the case of oxygen, for which experimental data are available (Figure 1). In fact, these incubation times are virtually the same as those obtained in the previous calculation2 on the basis of vibrational energy transfer only. The same is true of yexptl,as can be seen by comparing yexptlfor O2 + Ar in Table I1 and in Table IV of ref 2. This may serve as further proof of our contention that incubation and relaxation times are very weak functions of effective threshold. There are no experimental measurements of incubation times in hydrogen, but Dove and Raynor* have published a rather elaborate ab initio numerical calculation on the H2 + He system which may be used for comparison. On the basis of a calculation that includes rotational transitions AJ = f2, f4, and vibrational transitions Av = f l , f 2 (ref 6, Table 11, calculation IV), they give, after conversion into dimensionlessform, uthc= 1.55 X 104 and 4.73 X lo3 at 2000 and 3000 K, respectively, which compares
TABLE 11: Calculated Results for Low-Pressure Dissociations of H,, D,, and OzU Yexpfl, k,?, cm3 T,K cmmolecule s w finc H, t He 2000 35.1 4.15(- 21) 2.51 (4) 3000 89.0 2.85(-17) 4.48(3) 4000 145.8 2.09(- 15) 1.56(3) 5000 181.0 2.48(- 14) 7.69(2)
w71
5.64(3) 1.55(3) 6.66(2) 3.60(2)
H, t Ar
2000 3000 4000 5000
21.2 55.3 89.8 107.6
3.22(- 21) 2.29(- 17) 1.68(- 15) 1.99(- 14)
2000 3000 4000 5000
18.4 53.6 95.0 123.1
1.35(-21) 1.25(- 17) 1.07(- 15) 1.41(-14)
2500 3000 4000 5000
20.5 35.1 70.2 99.5
1.78(-20) 1.11(-18) 1.77(- 16) 3.31(-15)
6.59(4) 1.07(4) 3.58(3) 1.71(3)
1.62(4) 3.81(3) 1.56(3) 8.08(2)
8.89(4) 1.18(4) 3.40(3) 1.48(3)
2.02(4) 4.13(3) :I.47( 3) 6.94(2)
D,+ Ar
a
Powers of ten in parentheses.
0,
+ Ar
9.00(4) 3.23(4) 7.81(3) 3.09(3)
h,(exptl), cm3 molecule-' s-I
2.39(4) 1.04(4) 3.20(3) 1.42(3)
ref 18 1.20(-20) 3.77(-17) 2.11(- 15) 2.36(-14) ref 19 1.49(- 20) 3.76(-17) 1.89(- 15) 1.98(-14) ref 22 5.81(- 20) 2.53(-18) 2.68(- 16) 4.17(- 15)
3054
The Journal of Physical Chemistry, Vol. 84, No. 23, 7980
Forst
region below threshold is in fact n k T . 7 Itn ifhasthebeenimportant suggested23that depopulation below threshold
h lo
-
lo3
s
t
I
\ Bv 0
5 10‘
0 0
40 loo
L 2
4
6
8
10
~ 1 1 0O K~ Flgure 1. Incubatlon time in ps for dissociation of oxygen In argon. Circles below 6000 K are data of Watt;24the rest are data of Wray.*‘ Full line is calculated from eq 11, uslng E,’ of eq 30 for E,.
well with our results shown in Table 11. Numerically, the diatomic Y~~~~~values in Table I1 are comparable to those obtained in the N20 + Ar and C02 + Ar systems,2which is what one would expect, namely, that argon should remove roughly the same amount of energy per collision from a “small” molecule, be it diatomic or triatomic. However the temperature coefficient of diatomic Yexptl is somewhat larger, which translates into non-Arrhenius behavior of kat, Le., a decline of activation energy as temperature increases. For example, in the case of H2 + He, calculated activation energy declines from 105.7 kcal/mol in the 2000-3000 K range to 98.2 kcal/mol in the 4000-5000 K range. (The same is true of the cited calculation of Dove and Raynor! where the decline is from 100.8 kcal/mol in the 2000-3000 K range to 95.7 kcal/mol in the 3000-6000 K range.) Part of the difficulty may be that above 3000 K the present calculations make use of experimental relaxation data extrapolated beyond their temperature domain of validity. In contrast with the previous calculation on triatomics? where a useful a priori estimate of y could be obtained from the Schwartz-Slawsky-Herzfeld (SSH) theory (called YSSH), in the case of diatomic5 7ssHis much too large (by a factor of 5-10, relative to Yexptl) to be useful. The principal drawback of the present treatment is the limited temperature range over which calculations may be performed. This has to do with the requirement that E,*/(kT) should not be too small (%), which limits the accessible temperature range to no higher than -5000 K in most cases. If E,*/(kT) is smaller, i.e., if temperature is higher than -5000 K, eq 33 has no physically meaningful solution, and the treatment breaks down. In the case of D2+ Ar, this breakdown occurs already above 4000 K if relaxation data of ref 16 are used, which shows some sensitivity of the results to the relaxation times input. Since the relaxation data of ref 17 gave a better-behaved solution, they were used in preparing Table 11. The choice made in deriving eq 12, 26, and 30, that detailed balance need be satisfied only within k T of threshold, is reasonable but arbitrary, meaning that the calculated rate constants k,t are arbitrary to within a factor
-
in some cases may extend to 2- or 3 k T , which implies n 2 or 3. The choice n 2 would indeed improve agreement with experiment, but not the physical insight, given the simplicity of the basic model on which the present treatment of transients rests. If we had chosen B* = 1, and carried through the calculations using Penner’s solution,’ the calculated rate constants ko+would have been larger by about two orders of magnitude compared to the results with B* = 1 / 2 . This is because a symmetric barrier (B* = 1) represents the maximum possible efficiency of rotations in promoting dissociation.
5. Conclusions The theory of initial transients, based on collisional transfer of vibrational energy, has been successfully-and simply-modified to take into account the transfer of rotational energy as well. Considering that the calculations presented here can all be performed on a programmable pocket calculator, the theory yields results that compare quite well with very much more elaborate calculations, and-more importantly-with experiment. Calculations to be reported elsewhere show that shock-tube dissociations of N2,CO, and Br2 can be just as successfully dealt with as those of H2,D2,and O2reported on here. In fact, a system is yet to be found where the present theory fails. Acknowledgment. I am indebted to the National Sciences and Engineering Council of Canada for financial assistance and to Alvin Penner for several enlightening discussions.
References and Notes Penner, A. P.; Forst, W. J. Chem. Phys. 1977, 67, 5296. Forst, W.; Penner, A. P. J. Chem. Phys. 1980, 72, 1435. Yau, A. W.; Pritchard, H. 0. Proc. R . SOC.London, Ser. A 1978, 362, 113; J . Phys. Chem. 1979, 83, 134. Stace, A. J. Mol. Phys. 1979, 38, 155. Kiefer, J. H.; Hajduk, J. C. Chem. Phys. 1979, 3 8 , 329. Truhlar, D. 0.;Bials, N. C.; Hajduk, J. C.; Kiefer, J. H. Chem. Phys. Left. 1979, 63, 337. Dove, J. E.; Raynor, S. J. Phys. Chem. 1979, 8 3 , 127. Penner, A. P. Mol. Phys. 1978, 36, 1373. Le Roy, R. J. Madison, WI, 1971, Theoretical Chemistry Institute, University of Wisconsin, Report WIS-TCI-387; J. Chem. phys. 1971, 54, 5433. Penner, A. P.; Forst, W. Chem. Phys. 1975, 7 7 , 243, Flgure 3. Hirschfekier, J. 0.; Curtlss, C.F.; Bird, R. 8. “Molecular Theory of Gases and Llquids”; Wiley: New York, 1967; p 544. Bauer, S. H.; Tsang, S. C. Phys. Fluids 1963, 6 , 182. Herzberg, 0. “Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules”; Van Nostrand: Princeton NJ, 1957; p 502, Table 39. Forst, W. “Theory of Unimolecular Reactions”; Academic Press: New York, 1973. Darwent, B. deB. Nag. Stand. Ref. Data Ser. (US., Natl. Bw. Stand.) 1970, No. 3 7 , 32. Dove, J. E.; Teitelbaum, H. Chem. Phys. 1974, 6, 431. Klefer, J. H.; Lutz, R. W. J . Chem. Phys. 1966, 44, 858. Moreno, J. B. Phys. Flolds 1986, 9 , 431. Bauich, D. L.; Drysdale, D. D.; Horne, D. G.; Lloyd, A. C. “Evaluated Klnetlc Data for High Temperature Reactions”; CRC Press: Cleveland, 1972; Vol. 1, p 309. Appel, D.; Appleton, J. P. Symp. (fnt.) Combust. [Proc.] 1974, 75, 701. Camac, M. J. Chem. Phys. 1961, 34, 448. White, D. R.; Miilikan, R. C. J. Chem. Phys. 1963, 39, 1807. Reference 18, Vol. 3., p 11, Butterworths, London, 1978. Chang, D. T.; Burns, G. Can. J. Chem. 1977, 55, 380. Watt, W. S.; Myerson, A. L. J . Chem. Phys. 1989, 51, 1638. Wray, K. L. J. Chem. ph . 1982, 37, 1254; 1963, 38, 1518; Symp. (Int.1 Combust. iPr0c.Y196S, 70, 523.
