Chemical Physics 104 (1986) 107-122 North-Holland, Amsterdam
VIBRATIONAL RELAXATION OF N,O-Ar AND CH,-Ar MIXTURES. SHOCK TUBE EXPERIMENTS AND MASTER EQUATION CALCULATIONS Zein BAALBAKI, Ottawa-Carleton
John E. DOVE
University of Ottawa Campus, Ottawa, Canada KIN 9B4
and Wing S. NIP *
Department of Chemistry, Lash Miller Chemical Laboratories, Toronto, Ontario, Canada MSS 1Al Received
1984; in final form 21 November
University of Toronto,
The vibrational relaxation of N,O-Ar mixtures has been studied by laser schlieren measurements in shock waves and by numerical integration of the master equation for vibration-translation and vibration-vibration energy transfer. The system shows some deviations from the linear mixture rule, particularly at high temperatures and low mole fractions of N,O. The expression log,,(pr/atm us)N20_Ar = - 1.317 + 13.655T-‘/3 appears to fit our measurements and literature data to within 20% over the range 300-1600 K, but because of mixture rule non-linearity this result should be used with caution, especially at high temperatures. Laser schlieren measurements gave log,,(pT) = -2.680+19.235T-‘/3 for pure CH, at 340 to 700 K. Relaxation in mixtures containing from 0.5 to 50% CH, in Ar was also measured, but here the observed deviations from the linear mixture rule were so great that it was not possible to extrapolate the measurements reliably to obtain a relaxation rate for CH, at infinite dilution in Ar.
1. Introduction The interconversion of molecular energy by vibrational-translational (VT) and vibrationalvibrational (VV) processes is important in many non-equilibrium chemical systems, especially unimolecular reactions in the fall-off and low-pressure regimes [l-4]. The thermal decomposition of nitrous oxide in dilute mixtures with inert gases is considered to be an’important test case for theories of unimolecular reaction [5-181. It is therefore of interest to measure the rate of vibrational energy transfer in mixtures of nitrous oxide with argon, which is the inert gas used for most of the unimolecular reaction studies. In making such
measurements, we would particularly like to be able to distinguish the effects of N,O-N,O collisions, in which inter- and intra-molecular W as well as VT processes can occur, from those of N,O-Ar collisions in which only VT and intramolecular W transfer can take place. When analyzing the results of vibrational relaxation measurements on gas mixtures, the linear mixture rule (LMR) is almost invariably used in order to separate the effects of the different collision partners. Thus for a mixture of a relaxing molecule R with an inert gas A, the LMR asserts that the inverse relaxation time at unit total prescan be written as sure, l/(P),ixty 1/(P~)mixt=XR/(P7)R-R+X*/(P7)R-A
’ On leave at the Laboratorium fur Physikalische Chemie, ETH-Zentrum, Universitatstrasse 22, CH-8092 Zurich, Switzerland. 2 Present address: Lumonics Inc., 105 Schneider Road, Kanata, Ontario, Canada K2K lY3.
0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
X denotes mole fraction and, (p~)~_~ and are the standard (unit pressure) relaxa(P~hLA
Z. Baalbaki et al. / Vibralional relaxation of NJ0 - Ar and CH, - Ar
tion times for pure R and for R infinitely dilute in A. In essence, using the LMR is equivalent to assuming that the different collision partners act independently, so that the total relaxation rate can be written as the sum of the rates for each collision partner considered separately. The inverse standard relaxation times on the right-hand side of eq. (I) are important quantities because they play a role in this equation which is analogous to that of the rate constants in the rate expression for, e.g., a dissociation reaction in a gas mixture. Clearly, in the study of relaxation processes in gas mixtures, it is essential to know whether the effects of different collision partners really can be added in this way and whether (for a fixed translational temperature) there do indeed exist standard relaxation times which are invariant from one mixture to another. In one respect, this question about additivity and the existence of invariant standard relaxation times has already been answered in the negative. It has in fact been known for some time on theoretical grounds [19,20] that for multistep relaxation processes in mixtures, the rates are not rigorously additive except under conditions so stringent that they are unlikely to be realised in real systems. The essential question to be answered is therefore not whether the rates are additive in principle but whether in practice. the deviations from additivity will ever be large enough to matter and, if so, under what circumstances. Of especial concern is the problem of obtaining experimental data on relaxation rates at infinite dilution in inert species, for comparison with theoretical calculations. To make such a comparison, one cannot avoid having to extrapolate rates measured at finite concentrations of the relaxing species, and then it can be crucial to know whether a linear extrapolation is valid. While this paper is primarily concerned with vibrational relaxation, similar questions [21-271 about the validity of the LMR arise for chemical reactions in which energy transfer is rate-determining, for example simple dissociation and recombination processes and unimolecular reactions at the low-pressure limit. In spite of its insecure theoretical foundations,
the validity of the LMR seems to be widely accepted, probably because in practice it generally seems to work. Experimental data on the vibrational relaxation rates of mixtures (e.g., refs. [28-301) usually do appear to give acceptable fits to eq. (1). A recent review  of diatomic dissociation and atom-atom recombination reactions has concluded that so far there is no expe~mental proof of the violation of the LMR in these cases. Theoretical studies of low-pressure unimolecular reactions  and homonuclear dissociation  suggest that deviations from the LMR are likely to be small and hard. to detect. Nevertheless the literature does contain expe~mental indications of non-additivity in several reactions 12632-351 and energy transfer processes, e.g., the vibrational relaxation  of H, following Raman excitation, and the relaxation  of N,O by H, and D,. Unfortunately, there do not seem to have been any systematic experimental investigations of the range of usefulness of the LMR. Existing evidence is therefore generally a by-product of studies made mainly for other purposes. One of the difficulties in evaluating this evidence is that little is known. about the form of deviations which should be looked for. For this reason, we recently made a theoretical investigation  of the LMR for systems with linear master equations, e.g., with VT transfer only. An example of such a system would be a relaxing molecular gas at high dilution in a mixture of two inert colliders. The advantage of studying these systems is that their master equations can be solved analytically by eigenvalue techniques. Essentially we found that mixture rule plots for relaxation are most likely to be non-linear when there are substantial differences between collision partners with respect to the patterns (e.g. the energy dependences) of their transition probabilities for energy transfer processes. Mixture rule plots, if curved, will always be convex upwards, and the curvature of such plots will tend to be especially pronounced when small amounts of a highly efficient collision partner are added to an inefficient collider; in this latter case, serious errors might arise when trying to estimate relaxation or reaction rates due to one collision partner by extrapolating data taken over a limited range of compositions. To minim&e such errors, it is neces-
2. Baalbaki et al. / Vibrational relaxation of N,O - Ar and CH, - Ar
sary to measure rates with high accuracy at widely different concentration ratios, and even then extrapolations to infinite dilution may be problematical. While the case of a molecular gas at infinite dilution in a mixture of inert species is useful for gaining an understanding of basic principles, in practice one has to deal with mixtures containing finite concentrations of molecules. Then there is another possible source of mixture rule non-linearity, namely W transfer in molecule-molecule collisions. These processes make the master equation non-linear so that in general it cannot be solved analytically. However we have obtained an analytic solution for vibrational relaxation in one particular case , namely a mixture of an inert gas with a Morse oscillator diatomic species which is undergoing W and near-Landau-Teller VT energy transfer. For this case we showed that W effects will cause measurable deviations from the LMR only if the starting distribution is non-Boltzmann. They would not be expected to cause significant deviations in shock wave experiments in which there is initially a Boltzmann distribution corresponding to the pre-shock temperature. Note, however, that this result applies only to this specific diatomic model, and that it leaves open the question whether W effects will cause non-linearities in the mixture rule for relaxation of polyatomic molecules. Recently  we have investigated the rate of vibrational relaxation of pure N,O under shock wave conditions. We found that when the Bethe-Teller relaxation time 7tit, (defined below) is measured carefully over a range of working times, it can be shown to have a significant explicit time dependence. I.e., for a given translational temperature T,,,,, the relaxation of the vibrational energy to its equilibrium value cannot be represented by a single standard relaxation time P’T,ib. Instead, depending on the conditions of the experiment, prtib may either increase or decrease as the relaxation proceeds. We showed by numerical solution of the non-linear master equation for the relaxation process that this behavior can be understood qualitatively and semiquantitatively in terms of a multi-level relaxation involving both VT and W energy transfer. While
this result refers explicitly to the pure polyatomic gas, it provides an additional reason for questioning the validity of the LMR because it shows that at least one of the standard relaxation times on the right-hand side of eq. (1) is in practice not always a time-invariant quantity. In the present paper, we describe the extension of these shock wave relaxation measurements to mixtures of N,O with argon, the diluent used in most of the unimolecular reaction studies. One of the objectives of this work was, naturally, to obtain a value of (p~)~+-, Ar, and indeed we have previously published  such a value based on the assumption that the linear mixture rule was obeyed. In view of the points discussed above, particularly the finding of a time dependent p~“ib in pure N,O, it is clearly necessary to reconsider the use of the LMR to extrapolate the relaxation measurements to infinite dilution in Ar. Below we describe and analyze measured relaxation rates over the whole range of [N,O] : [Ar] ratios that is accessible with our laser schlieren measurement technique. We also describe master equation simulations of vibrational relaxation in N,O-Ar mixtures, taking into account both W and VT processes. Another simple polyatomic molecule whose unimolecular decomposition has been studied a number of times is methane [41-471. We have therefore also measured the vibrational relaxation of methane over a wide range of mixture ratios with argon. Methane has the advantage that at shock tube temperatures it has a larger vibrational specific heat than nitrous oxide and therefore gives a larger relaxation signal. This allows relaxation measurements to be made over a larger range of conditions than for N,O and has enabled us to investigate mixture rule effects more fully. However because of the much more complex vibrational level structure of CH,, we have not so far made master equation calculations for this molecule.
2. Experimental methods and analysis All experiments were carried out in incident shock waves using the laser schlieren technique
2. Baalbaki et al. / Vibrational relaxation of NJ0 - Ar and CH, - Ar
[48,49]. The pressure driven metal shock tube had a rectangular cross section of 114 x 89 mm. Density gradients due to vibrational relaxation were monitored by measuring the time-dependent deflection of a continuous-wave helium-neon gas laser beam perpendicular to the shock tube axis and directed along the 114 mm dimension. Deflection of the laser beam was detected by measuring the output of a quadrant photodiode using a bridge configuration and differential amplifier system. The amplifier output was photographed from an oscilloscope whose sweep speed was carefully calibrated using a highly stable quartz crystal controlled oscillator. The electronic rise time was 10 ns, and the effective time resolution of the optical system = 0.1 ILLS.The equipment and methods have been described more fully elsewhere [29,50]. The relaxation time measured by this technique was an energy relaxation time r based on the Bethe-Teller law dE/dt=(E,-E)/r.
The method of evaluating r used a point-by-point analysis of the relaxation signal V(t) to obtain the time dependence of the hydrodynamic variables. However values of T itself were obtained from a logarithmic plot of V(t) against t. In the analysis, it was assumed that all of the vibrational modes are strongly coupled and relax simultaneously to equilibrium so that the appropriate value of the specific heat to use was that corresponding to the total vibrational energy. This method is based on that of Kiefer and Lutz  and is described more fully in ref. ; the relation of this method to others is also discussed there. Experiments were carried out on pure nitrous oxide and on four mixtures containing nominally 1, 4, 9, and 30% of nitrous oxide in argon. The measured relaxation times constitute one set of data analyzed in this paper. The other data are measured relaxation times for pure methane and for mixtures containing nominally 0.5, 1, 2, 5, 10, 25 and 50% of methane in Ar. The actual mixture compositions, and the conditions for these two sets of measurements are given in tables 1 and 2. The nitrous oxide used for these experiments was Matheson “regular” grade > 98%, the main impurity being air. It was purified by repeated
Table 1 N,O experiments.
Shock wave temperature
6.0-17.0 10.0-20.0 6.0-34.0 2.0-75.0 0.06-56.0
850-1770 800-1460 790-2230 430-2100 390-2600
T, W) 1.00% 4.00% 9.15% 30.53% 100.00%
N,O, N,O, N,O, N,O, N,O
99.00% 96.00% 90.85% 69.47%
Table 2 CH, experiments.
Ar Ar Ar Ar
PI ([email protected]
Shock wave temperature
6.8-21.0 7.0-40.0 7.0-322 5.0-319 6.9-160 6.8-154 6.8-146 6.0-90.7
840-1700 650-1660 390-1620 387-1325 370-1420 350-1150 340-930 340-680
T2 (K) 0.50% 1.15% 2.04% 4.91% 9.91% 24.74% 48.85% 100.00%
CH,, CH,, CH,, CH,, CH,, CH,, CH,, CH,
99.50% 98.85% 97.96% 95.09% 90.09% 75.26% 51.15%
Ar Ar Ar Ar Ar Ar Ar
fractional distillation after freezing in liquid nitrogen, only the middle fraction being retained each time. The purity of the final distillate was checked by mass spectrometry and was estimated to be better than 99.99%. Methane, Matheson “pure” grade > 99.5%, and argon, Matheson “prepurified” grade > 99.998% were used as received. The main impurity in the methane was nitrogen whose efficiency for relaxation of hydrocarbons is known to be very low . 3. Experimental results and discussion 3.1. Relaxation mixtures
A Landau-Teller plot of the measured vibrational relaxation times for pure nitrous oxide and the nitrous oxide mixtures is shown in fig. 1. The 30 and 100% lines are clearly curved and were fitted by non-linear least squares to log,,pr
= a + bT-‘13 + CT-‘/~,
2. Baalbaki et al. / Vibrational relaxation of N,O - Ar and CH, - Ar
where a, b, and c are fitting parameters. Relaxation data for the 1 and 4% mixtures could be measured only over a much smaller temperature range and showed no evident curvature. Twoparameter fits were therefore made, c being omitted: log,,pr
= a + bT-“3.
For the 9% mixture, both two- and three-parameter fits were tried; the three-parameter fit is shown in fig. 1 and is used in the discussion below, but in fact the use of the two-parameter data would cause no significant change in our conclusions. Measurements of the relaxation time cannot begin until = 0.8 us (lab time) after shock heating because, while the density step at the shock front itself is in the laser beam it creates a large deflection which interferes strongly with the relaxation signal. The relaxation times in fig. 1 (and also those for methane-argon mixtures in fig. 4 below)
0 m 1 i $-a2 a 0" -I -a4
Fig. 1. Landau-Teller plot of measured vibrational relaxation times in N,O-Ar mixtures (this work): 0 pure N,O; T 30% N,O; 0 9% N,O; 0 4% N,O; A 1% N,O. The solid lines, marked with the percentage of N,O, are least-squares fits.
were measured as soon as reasonably possible after this interference had ceased and they therefore refer to a time of about 1 us after shock arrival. Consequently, depending on the particular mixture and on the temperature and pressure of the experiment, they are measured at varying multiples of rtit, after the start of the relaxation process. In the middle range of temperatures, 1200 > T > 600 K, measurements did not begin until t > 3.5rVib, and the quoted relaxation times refer to a state which is fairly close to vibrational equilibrium. At lower temperatures, where relaxation is slower, the measurements began at an earlier stage of the equilibration process. At high temperatures, particularly in undiluted N,O, we worked with lower initial gas pressures in order to compensate for the rapidity of the relaxation process. Consequently, in spite of the faster relaxation, the measurements were typically made at an earlier stage of the relaxation process than in the mid range of temperatures. At T > 1800 K, the measurement time t 2 qib. We have recently reported  that measured vibrational relaxation times in pure nitrous oxide have an intrinsic time dependence. One possible source of a time dependence of r is the fall in the translational temperature T,,,,, caused by the interconversion of translational and vibrational energy in the relaxation process itself. This type of effect on r, which is due to the fact that the probability of vibrational energy transfer depends on the velocities of the colliding molecules, is of course well known and relatively well understood. It causes a decrease in the relaxation rate (i.e. an increase in r) as the relaxation proceeds. However in N,O this particular effect is relatively small, and moreover at high temperatures the actually observed  change in r is in the wrong direction to be due to the decrease in T,,,,,. This time dependence of r is attributed to more subtle effects, particularly W transfer, on the relaxation process; these effects would occur even in the absence of a change in T,,,,. In the present work, using mixtures diluted with argon, the relaxation signal was smaller than for pure N,O and hence was more difficult to measure accurat$y. Consequently we have not so far attempted the difficult and time-consuming measurements that would be
Z. Baalbaki et aL / Vibrationat reiaxation of N&l- Ar and CH4 - Ar
needed to look for a possible intrinsic time dependence of r in the ~gon-diluted mixtures. In our experience, it is very difficult to decide whether a signal that is even only slightly noisy is a single pure exponential unless that signal has been measured over a number of time constants r. This difficulty is accentuated by the necessity of separating a possible intrinsic time dependence of r from the time dependence caused by the changes of &rlS mentioned above, even thopgh in our experiments those changes were not large enough to have a substantial effect on the relaxation rates of the more dilute mixtures. 3.2. The mixture mixtures
rule in nitrous oxide-argon
fractions of nitrous oxide. An inte~re~tion of these deviations as genuine and not due to an experimental artefact is supported by the good precision of the measurements; even in the worst case, any given point in fig. 1 is no more that 10% from the corresponding mixture line. It is also supported by the fact that the master equation computations for this system, and the experimental measurements of relaxation in methane-argon mixtures (described below), also show deviations of the same general form. We have therefore explored several possible methods of fitting the data
In order to try to extrapolate the measurements to infinite dilution in argon, we can read standard relaxation times for the different mixtures at different fixed temperatures in the range 800-2200 K from fig. 1. The resulting data are shown in fig. 2 as “mixture rule plots” of l/p7 against the mole fraction of nitrous oxide. Examination of the plots suggests fairly strongly that there are systematic deviations from the LMR, particularly at high temperatures and low mole
Fig. 2. Vibrational relaxation of NaO-Ar mixtures. Mixture rule plot of measured inverse standard refaxation time against mole fraction of NaO at temperatures of 800-2200 K (this work). The solid lines, marked with the t~perat~e, are smooth fits to the data points (see text).
Fig. 3. Landau-Teller plot of the vibrational relaxation of N,O at infinite dilution in argon. 0 and upper solid line, five-point curved extrapolation of mixture rule plots (this work); A and lower solid line, five-point linear extrapolation of mixture rule plots (this work); v and dashed line, two-point linear extrapolation of mixture rule ptots (this work); , . . Simpson et al. (371; m Eucken and Niin ; + Huetx et al. ; A Kung (54]; + Rebel0 da Silva and de Vasconcelos  (Q excitation); x Rebelo da Silva and de Vasconcelos  (Ye excitation); q Hancock et al. ; * Yardley f57]; - f . . - Starr and Hancock [Sg]; - *- Stephenson and Moore . The last five sets of data are from experiments in which rs was excited (see text).
Z. Baalbaki et al. / Vibrational relaxation of NJ0 - Ar and CH, - Ar
which might allow us to extract the relaxation time of nitrous oxide at infinite dilution in argon. Our fitting attempts concentrated on the temperature range 1000-1600 K because we were able to measure relaxation times there for all five of the gas compositions that we studied. The solid lines in fig. 2 are subjective smooth curves drawn through the data points. A smooth extrapolation of these curves to XNzo = 0 gives values for PT&o_&, the relaxation times at infinite dilution in Ar, of 1.12 (1000 K), 0.93 (1200 K), and 0.71 (1600 K) atm us. These points are plotted as the open circles in fig. 3. An alternative method of analysis is simply to ignore the indications of curvature of the mixture rule plots and to fit the data points in fig. 2 directly by least squares to the LMR, eq. (1). The resulting fits tend, of course, to become worse as the temperature increases; at 1000 K the average deviation of the data points from the linear fit is = 58, and at 1600 K = 10%. By extrapolation we obtain the following pi values for relaxation of nitrous oxide infinitely dilute in argon: 1.02 atm ps at 1000 K; 0.82 at 1200 K; 0.59 at 1600 K. These values are from 9 to 17% less than obtained by the smooth extrapolations. They are shown as the lower set of open triangles in fig. 3. We have previously shown  that for systems with linear master equations, if there are any deviations from the LMR, those deviations will always necessarily be positive. Then it often happens that the mixture rule plots are fitted quite well by an equation of the form
t4) where an additional fitting parameter, the exponent n z 1, has been introduced. (There is no theoretical basis for this expression which is simply the best three-parameter equation that we could find  for correlating and extrapolating rate data for such systems. A useful feature of eq. (4) is that for n = 1 it goes over to the normal LMR.) Using the data in fig. 2, best fits were obtained with n = 1.52 at 1000 K, 2.10 at 1200 K,
and 3.16 at 1600 K. The resulting values of pr at infinite dilution in argon lie between those obtained by the linear and smooth curved extrapolations described above. However, as it happens, this fitting function for systems with linear master equations does not have quite the same shape as the mixture rule plots for the non-dilute polyatomic systems studied here, and it gives only moderately good fits to the nitrous oxide-argon data. Above 1600 K, the mixture rule plots appear to be quite strongly non-linear and, without data for mixtures with less than 9% N,O, we do not feel that it is useful to attempt an extrapolation to infinite dilution. Below 800 K, we were able to measure relaxation rates only for 30% and 100% nitrous oxide. If we assume that at these relatively low temperatures the LMR is sufficiently well obeyed to allow a conventional linear extrapolation to XN,o = 0, we obtain the values of ~~~~~~~~ which are shown as the upper set of open triangles in fig. 3. Each of these various sets of data for prN,o_Ar gives a good fit to the Landau-Teller temperature dependence [eq. (3b)] with the following parameters (atm us units): For 400-800 K (data from a two-point linear extrapolation of the mixture rule plots): a = - 1.395, b = 14.607. For 1000-1600 K (data from a five-point linear extrapolation of the mixture rule plots): a = - 1.640, b = 16.495. For 1000-1600 K (data from a five-point curved extrapolation of the mixture rule plots): a = - 1.317, b = 13.655. While these fits have slightly different temperature dependences, in fact the actual pr values predicted by all three agree quite closely, with a maximum difference of 18% over the whole temperature range 400-1600 K, and they all extrapolate to values close to the room temperature measurements (fig. 3) of Eucken and Numann  and Huetz et al. . Agreement with the shock tube measurements of Simpson et al.  is also fairly good; our p7 values are = 20% larger than theirs. It is generally considered that the thermal VT relaxation process in a molecule like nitrous oxide is dominated by the excitation and deexcitation of
Z. Baalbaki et al. / Vibrarional relaxalion of NJ0 - Ar and CH, - Ar
the lowest frequency (bending) vibrational mode. In fig. 3 we also show literature data on energy transfer related to selective excitation of vs, the asymmetric stretch vibration (energy transfer rate constants have been converted to equilibration times at 1 atm). It will be seen that these equilibration times are greater than the measured thermal relaxation times, indicating that coupling of the vibrational modes is less than complete. We discuss this coupling more fully elsewhere 1391. In summary, therefore, we propose the following expression as representing the vibrational relaxation time of nitrous oxide by argon, to within = 20%, over the whole temperature range 300-1600 K: log,,( pT/atm
us) = - 1.317 + 13.655T-1/3.
Relaxation times methane-argon mixtures
2 y o-
However a strong caueat must be attached to this result. In making the necessary extrapolations to infinite dilution in argon, we assumed that there is not a sharp increase in the curvature of the mixture rule plots very close to XNzo = 0. However, we cannot be completely certain that this assumption is correct and indeed in the methane-argon results considered below there is apparently just such a strong curvature that prevents reliable extrapolation. If there is a similar strong curvature in the nitrous oxide-argon system which has so far escaped notice, then the true relaxation time at infinite dilution will be greater, possibly considerably greater, than predicted by eq. (5). These remarks apply not only to our own work but to other measurements of ( p7)N,0_Ar as well, since all of them entailed extrapolations to infinite dilution. 3.3.
Landau-Teller plots of the measured vibrational relaxation times for pure methane and for seven mixtures ranging from 0.5 to 50% methane in argon, are shown as the data points in fig. 4. As for nitrous oxide, relaxation measurements began at t = 1 ps, corresponding to’t/r 2 0.1 to 1.5 in the range 340 -Z T < 680 K for pure methane, and to t/r > 1.5 to 3.5 in the range 840 c T-C 1700 K for 0.5% CH, in Ar.
Fig. 4. Landau-Teller plot of measured vibrational relaxation times in CH,-Ar mixtures (this work): A pure CH,; v 50% CH,; 0 25% CH,; + 10% CH,; 0 5% CH,; n 2% CH,; x 1% CH,; A 0.5% CH,. The solid lines, marked with the percentage of CH,, are linear least-squares fits.
Unlike the case of nitrous oxide, all of the Landau-Teller plots were found to be linear to within experimental accuracy and were therefore least-squares fitted to eq. (3b). For pure methane we obtained log,,( pT/atm
us) = -2.680 + 19.235T-‘13.
No intrinsic time dependence of the relaxation times was seen, but we did not specifically investigate this point. (The measurements in this system actually antedated the measurements in pure nitrous oxide which we have reported recently .) It is therefore possible that a careful investigation of the different stages of the relaxation process would reveal an observable intrinsic time dependence in methane tao. Fig. 5 compares our measured relaxation times for pure methane (solid line) with the available
Z. Baalbaki et al. / Vibrational relaxation of NJ0 - Ar and CH, - Ar
Fig. 5. Landau-Teller plot of measured vibrational relaxation times in pure CH,. Solid line, this work (cf. fig. 4); dashed line, Richards and Sigafoos ; 0 Hill and Winter (611; 0 Eucken and Aybar ; X Jackson et al. ; A Cottrell and Matheson ; v Cottrell and Martin ; v Edmonds and Lamb  and Hill and Winter ; 0 Yardley and Moore  and Zittel and Moore [6P].
literature data. The literature results scatter about our line, but in general the agreement is quite good, our relaxation times being = 7% less than the average of other measurements in the same temperature range (340-700 K). The dashed line at higher temperatures in fig. 5 represents data of Richards and Sigafoos  based on a mixture rule extrapolation which is discussed below. 3.4. The mixture rule in methane-argon mixtures Proceeding in exactly the same way as for nitrous oxide, we have used data from fig. 4, at a number of fixed temperatures, to generate the mixture rule plots in fig. 6. It will be seen that the plots have the same general characteristics as for nitrous oxide, but that the curvature at low mole fractions of the polyatomic species is now much stronger, particularly at high temperatures. As for nitrous oxide, the precision of the measurements supports the interpretation of these deviations from linearity as being genuine; no point in fig. 4 deviates from the corresponding line used to generate fig. 6 by more than 6%. In our view, the
Fig. 6. Vibrational relaxation of CH,-Ar mixtures. Mixture rule plot of measured inverse standard relaxation time against mole fraction of CH, at temperatures of 400-1600 K (this work). The solid lines are smooth fits to the data points (see text) and are marked with the temperature.
curvature of the mixture rule plots is too great to allow us to extract a reliable value for the relaxation time at infinite dilution from our data. It is, however, clear that the relaxation of methane by argon is much slower than the self relaxation of methane. For the most dilute mixture, 0.5% methane in argon, we obtained log,,( pT/atm
us) = -2.563 + 26.992T-‘13,
for the temperature range 870-1710 K. Fig. 6 strongly suggests that it is reasonable to take this expression as a lower limiting value for the vibrational relaxation time at infinite dilution in argon, though the true value could be very much higher than this limit. Richards and Sigafoos  measured the vibrational relaxation of mixtures of 10 and 50% methane in argon. Their results are shown as the data points and solid lines in fig. 7. Their data agree fairly well, in the overlapping temperature ranges, with our own measurements (dotted lines); there appears to be a systematic difference of = 15% in the p7 values. The heavy dashed lines are their extrapolations to 100% methane and to infinite dilution, using the LMR. Since they measured only two mixtures, they had of course no
Z. BaaBaki et al. / Vibrational re[~ation
T/K 2m I”“”
’ I ’ ’ ’
of N,O - Ar and CH, - Ar
4. Master equation solutions In order to gain an insight into the effects to be expected in the vibrational relaxation of a polyatomic species in mixtures with an inert gas, we have solved the master equation governing the vibrational level, populations of the nitrous oxide molecule under simulated shock wave conditions. We emphasize that our objectives were to get a better underst~ding of the processes taking place and to see whether the predicted behaviour was qualitatively the same as we observed experimentally, and not to seek precise quantitative agreement between experiments and computations. Therefore we have simply set up a model that we believe to contain the essential physics of the relaxation process and have presented the results without trying, at this stage, to adjust the model to fit our experimental data.
4. I. The model
( T/K Fig. 7. Landau-Teller plot of vibrational relaxation times in CH, and CH,-Ar mixtures. Shock tube data of Richards and Sigafoos ; A and upper solid line, 10% CH,. 0 and lower solid line, 50% CH,. Dashed lines, two-point linear extrapolations to pure CH, (lower line) and infinite dilution (upper line) (ref. 1601).Dotted lines, measured relaxation times for 0.5, 10, 50 and 100% CH, (this work). The two-point extrapolation from the 10 and 50% data, using the linear mixture rule, leads to a value for the relaxation time at infinite dilution that is substantially lower than the actual measured relaxation time at 0.5% CH,.
way of checking the correctness of this extrapolation. However fig. 4 shows that the extrapolation using the LMR greatly overestimates the rate of relaxation by argon. Our measured relaxation time for the 0.5% mixture is already 1.6 times as great as their value extrapolated to infinite dilution. Since the true value of pr at infinite dilution is almost certainly substantially larger than our 0.5% value, it is clear that even though the actual measurements of Richards and Sigafoos appear to be good, using them for a two-point linear extrapolation gives a result which is very seriously in error.
The model simulated vibrational relaxation in N,O/Ar mixtures, including both VT and W transitions. N,O has an asymmetric stretch vibration (1285 cm-l), a doubly degenerate bending mode (589 cm-l), and a symmetric stretch (2224 cm-‘). All of the vibrational eigenstates having energies up to 4067 cm-’ were included in a simulation of the relaxation at 750 K. (These eigenstates account for 99% of the equilibrium vibrational energy and over 99.6% of the equilibrium population at 750 K. The reason for this choice of temperature is explained below.) For the master equation solutions, these states were grouped into twenty-one “levels” of similar energy; the maximum number of states in a level was ten. The master equation therefore consisted of twenty-one ordinary differential equations governing the time dependence of the populations of these levels. Rotation was not explicitly included in the model. In all of the computations, the gas mixture was initially at equilibrium at 295 K. The computations simulated a shock wave which suddenly changed the translations temperature from 295 K to a higher temperature T2. The distribution over vibrational levels then relaxed to an equilibrium
Z. Baalbaki et al. / Vibrational relaxation of I?fl - Ar and CH, - Ar
corresponding to T,. The translational temperature was held constant after shock wave arrival, so that variations in relaxation rates resulting from changes in the translational temperature during the relaxation process were not simulated; any variations in the relaxation rates and the associated relaxation times were a consequence of the changes in the populations of the internal levels. 4.2. Transitions and transition probabilities
was assumed to be related to the rate constant for the 10 transition by an energy gap law k;,$
= kz& Xexp(
CN,O 1Acji 1)~
For W transitions:
N,O( i) + N,O( k) + N,O( j) + N,O( 1)
(Note that throughout this section, the abbreviation “10” is used to denote the transition from the first excited vibrational state (010) to the vibrational ground state (000); “01” denotes the reverse transition. Note also that we label as “VT” all processes in which only one molecule changes its vibrational energy, and reserve the term “VV” for those in which both molecules change. Some authors use the term “W” to include single molecule intermode transfers such as (020) + Ar + (100) + Ar.) The (time-dependent) vibrational relaxation time 7’ was evaluated as a Bethe-Teller energy relaxation time defined by l/7’ = (dc/dt)/(
cm - c),
where z is the instantaneous ergy
total vibrational en-
< = c NiN20+0
an energy gap law was also used for the rate constants: kj,’
where it is implied that j > i and k > 1. (The gap law expressions used here are discussed further in ref. .) In each of the above cases, rate constants for the reverse transitions were calculated by detailed balance. For the computations reported here, the following values of the parameters were used: T2 = 750 K. All c = 0.010 cm (compare the Lambert-Salter  values of c = 0.0083 for hydrogen-containing molecules and c = 0.017 for molecules containing no H-atoms). k,Ad/k,N,,’ = 0.1, k::/k 2&=100. ,
and E, is the total energy at equilibrium. In the plots, the computed relaxation times are expressed in dimensionless form by division by a hypothetical “VT relaxation time” 7 for pure N,O where 117 =
where AC denotes a difference in vibrational energies, and 589 cm-’ is the energy of the (010) level. The rate constants for VT transitions by collisions with argon were formulated similarly:
For the integration of the master equation, a dimensionless time scale x was used: x=N totalkNz0 1o,vTt,
Ntotal (k%% - k$% ).
The rate constant for a downward VT transition from level j to level i by collision with N,O
For VT transitions, only those processes were included for which I Acji I < 700 cm-‘. The effect of this is that the only allowed processes in which only one quantum number changes are those with Au = f. 1 for the bending mode, e.g., (000) + (010) or (020) --* (010). However a substantial number of more complex VT transitions, in which more than one mode changes its energy, are also allowed.
Z. Baalbaki et al. / Vibrational relaxation of N,O - Ar and CH, - Ar
To keep the size of the computation within manageable limits, only those W transitions were allowed for which the energy gap, i.e. the amount of energy to be transferred into or out of translation (or rotation), is less than 700 cm-‘. However this limitation still allows very many transitions, so that the master equation contains a very large number of terms. 4.3. Results The master equation was solved, for a number of different mole fractions of nitrous oxide, using Gear’s [70,71] method for the numerical integration of stiff ordinary differential equations. In fig. 8 we show the results as “mixture rule plots” for several representative values of the dimensionless time x. It will be seen that the plots are indeed all I
Fig. 8. Simulation,of vibrational relaxation in N,O-Ar mixtures by master equation calculations. Mixture rule plots of the inverse (dimensionless) Bethe-Teller relaxation time as a function of the mole fraction of N,O, computed at different stages of the relaxation process. The lines are marked with the value of the dimensionless time variable x (see text).
curved, and moreover that their shapes change considerably as the relaxation proceeds. At short times, the mixture rule plots are almost monotonic and mainly concave upwards. At long times, when the relaxation process is approaching equilibrium, the plots become completely convex upwards and have a form very similar to that which we have previously found  for systems with linear master equations. Note that under these conditions successive curves lie one above the other so that the relaxation process is tending to speed up as the gas becomes more highly excited internally. For present purposes, the most interesting curves are those at intermediate values of x. These curves contain a point of inflection and have essentially the same shape as the experimentally measured mixture rule plots (cf. figs. 2 and 6). Note especially the curve for x = 5, which corresponds approximately to the time at which our experimental relaxation times were measured. The general shape of this curve is extremely similar to the lower temperature curves in fig. 2. However our particular choice of parameters has caused the ratio of the relaxation times in pure N,O and at infinite dilution in Ar to be greater in the computations than in the experiments. The details of the region below = 5% N,O are of special interest, because of their relevance to the problem of extrapolation to infinite dilution, and are shown considerably enlarged in fig. 9. It will be seen that at very small mole fractions of nitrous oxide, the plots can be strongly curved, especially as the relaxation process approaches equilibrium. At long times, the mixture rule plot is virtually perfectly linear for XNSo > 0.05. Yet if this linear region is extrapolated back to XNzo = 0, the relaxation time at infinite dilution is overestimated by a factor of 3.3. Problems can also arise in extrapolation from low to high mole fractions. If the steep curves close to XNzo = 0 are extrapolated to XNzo = 1, the relaxation rate in pure N,O can be overestimated by as much as a factor of 6. We conclude that the master equation computations support the experimental observations as regards both the form and the approximate magnitude of the deviations from the LMR. They also support the warning in section 3, that the relaxa-
Z. Baalbaki et al. / Vibrational relaxation of N,O - Ar and CH, - Ar 1
with the non-linear master equation governing the relaxation of nondilute diatomic molecules , and we expect polyatomic systems with non-linear master equations to show a similar effect. Unfortunately the cost of these computations increases very steeply with increase in T because we need to include many more levels and to allow transitions with larger energy gaps. 4.4. Mechanism of the dtyiations from the LMR
Fig. 9. Mixture rule plots of simulated vibrational relaxation in N,O-Ar mixtures at low mole fractions of N,O. (Detail of fig. 8.) The relaxation becomes progressively faster (increasing inverse relaxation time) over the time period shown here. Particularly at long times, extrapolation of the nearly linear right-hand-part of the graph substantially overestimates the relaxation rate at infinite dilution.
tion times extrapolated to infinite dilution must be regarded with caution. At present the potential parameters of this system are not well enough known to allow the detailed quantitative computation of the magnitude of this effect, and further experimental investigations of relaxation rates at low mole fractions of polyatomic gas are highly desirable. So far we have solved the master equation for N,O only for a temperature of 750 K where the experimentally observed deviations are still quite small, and it would be desirable to continue these solutions to higher temperatures. For systems with linear master equations, the computed deviations  tend to increase with temperature as the multilevel character of the process becomes more pronounced. We have recently seen similar behaviour
The mechanism of these deviations from the LMR is clearly a complex network effect. (For an instructive review of network effects in dissociation and recombination reactions, see Pritchard .) In spite of the complexity, one can gain at least some insight into the origin of these non-linearities by considering simple models . For a non-equilibrium chemical process, there will often be some region in the energy level diagram which strongly influences or determines the overall rate. A somewhat extreme example of this is the socalled “bottleneck” effect. Suppose that in a thought experiment we set a non-equilibrium process going by collisions with diluent 1 and that we locate its rate determining region which we call bottleneck 1. Now add a small amount of diluent 2. If bottleneck 2 is at the same place as bottleneck 1, then we will see no special effect. However, if the bottlenecks are at different places in the energy level diagram, or if they represent different kinds of process, then the addition of diluent 2 may allow bottleneck 1 to be bypassed relatively readily. Bottleneck 2 does not yet hinder the process significantly because diluent 1 is still in excess; i.e. bottleneck 2 is bypassed by the large amount of 1. The addition of small amounts of 2 therefore gives a relatively strong effect on the overall rate. However the effect of further additions of 2 is not as great, because eventually it becomes the dominant collision partner and then bottleneck 2 is rate limiting for the overall process. For the N,O-Ar system, values of the relevant transition probabilities are not yet well enough known to allow these bottlenecks or rate-determining regions to be located with certainty. Nevertheless some suggestions can be made. At very high dilution in Ar, VT excitation of the
2. Baalbaki et al. / Vibrational relaxafion of [email protected]
- Ar and CH, - Ar
bending mode is almost certainly the rate-determining step for vibrational relaxation: (000) + Ar + (010) + Ar,
(010) + Ar + (020) + Ar,
(The energy gap AC in wavenumbers, cf. table 3, is given a positive sign for processes in which energy is fed into vibration from the translational or vibrational degrees of freedom.) The other modes can become excited in processes such as (020) + Ar --, (100) + Ar,
which, however, from energy gap considerations are likely to be relatively unimportant in determining the overall rate of relaxation of vibrational energy. For relaxation by N,O-N,O collisions, it is likely that the same process as for N,O-Ar, namely VT excitation of the lowest levels of the bending mode, is rate-determining at relatively low temperatures where only a few levels become excited. However for pure (undiluted) N,O we find  strong indications of a change in mechanism as the temperature is raised. Our analysis suggests that relaxation is promoted by a sequence of a VT (020) + M --) (100) + M,
which allows an additional bending quantum to be acquired in two steps which are significantly easier than the (000) + (010) VT transition. Above 1200 K, the experiments show  that as the relaxation proceeds it actually accelerates (7’ shortens). At these temperatures, other sequences involving relatively closely spaced higher vibrational states become possible, e.g., (040) + M -+ (120) + M,
(120) + M -+ (200) + M,
(200) + M + (011) + M,
(011) + M ---)(050) + M,
This sequence enables a molecule to gain a bending quantum by a series of rapid processes. At very high dilution in Ar, this would simply allow a nitrous oxide molecule to complete its relaxation to upper levels quickly, but would not otherwise affect the bulk relaxation rate. However, when intermolecular W transitions are included, this sequence can be followed by a fast near-resonant W process such as (050) + (000) -+ (040) + (OlO),
Ar = 11.
Ar = 110
and a W process (100) + (000) + (020) + (OlO),
Table 3 Some energy levels of N,O a) Level
Energy gap (cm-‘)
(000) (010) (020)
1 2 3
(100) (030) (110)
1 4 2
(001) (040) (120)
1 5 3
(200) (011) (050)
1 2 6
0 589 1175 1285 1759 1878 2224 2340 2469 2564 2800 2918
589 586 110 474 119 346 116 129 95 236 118
‘) The table lists the first twelve energy levels. The “energy gap” is to the next higher energy level in the table.
Then the overall effect is to allow an unexcited (000) molecule to gain a bending quantum by a sequence of rapid processes with small energy gaps. A (040) molecule is regenerated at the same time, so that the cycle can be repeated over and over again to excite other (000) molecules. Therefore, once a sufficient number of nitrous oxide molecules become excited, the overall relaxation is speeded up “autocatalytically”. The relatively slow (000) --) (010) VT process is bypassed, and the relaxation bottleneck is now in a different part of the energy level diagram, presumably at the slowest step(s) of the autocatalytic cycle(s). Thus the condition suggested for deviations from mixture rule linearity, namely that the bottlenecks for N,O-Ar and N,O-N,O relaxation be at different places, will be fulfilled. Incidentally, these relatively closely spaced upper levels are not inaccessible at high temperatures. For example, at 1700 K over 40% of the equilibrium population is in levels above (050).
2. Baalbaki et al. / Vibrational relaxation of N$ - Ar and CH,-
In experimental measurements of the vibrational relaxation of N,O-Ar mixtures we have observed moderate deviations from the linear mixture rule, particularly at low mole fractions of N,O and temperatures above 1000 K. These deviations cause uncertainties in extrapolating the measured relaxation rates to infinite dilution in argon. Nevertheless by comparing linear and nonlinear extrapolations, and by considering the data of other workers, we suggest log,,( pT/atm
= - 1.317 + 13.655 T-1’3,. to represent the relaxation time to within = 20% over the temperature range 300-1600 K. However this result should be used with caution because of the possibility that there is a strong curvature of the mixture rule plots at very small mole fractions of N,O, a region which has not yet been studied experimentally. If there is such curvature, the true rate of relaxation by Ar, particularly at high temperatures, could be substantially less than indicated by the above expression. Simulations of the vibrational relaxation in this system at 750 K, using master equation techniques with inclusion of both VT and intermolecular W transfer, predict deviations from the linear mixture rule whose form is very similar to that observed experimentally. These simulations also permit a tentative identification of the mechanistic features which are responsible for the deviations. We have also measured relaxation times in pure methane and in CH,-Ar mixtures. For pure CH, we obtain log,,( pT/atm
ps) = -2.680 + 19.253 T-1/3.
However in mixtures with argon, strong deviations from the linear mixture rule were observed. The curvature of the mixture rule plots at small mole fractions of CH, was so great that, even though we made measurements with as little as 0.5% CH,, it was was not feasible to extrapolate reliably to infinite dilution, in order to obtain a value of (PGH,-Ar. It appears possible that similar deviations from the linear mixture rule occur in the vibrational
relaxation of other polyatomic species, particularly at high temperatures. Such deviations could cause significant errors in extrapolating energy transfer measurements to obtain rates at infinite dilution. Relaxation rates by inert diluents might then be substantially overestimated, possibly giving rise to wrong conclusions in comparisons between theory and.experiment. Since the extent of the deviations will probably vary from one diluent to another, it might even happen that experiments yield a sequence of apparent collision efficiencies which differs from the true sequence. To obtain a fuller understanding of this problem, further experimental and theoretical studies of the concentration dependence of relaxation rates, especially at high dilution, are needed.
Acknowledgement This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Swiss National Sscience Foundation. We thank Dr. John H. Kiefer for a helpful discussion of relaxation mechanisms at high temperatures.
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