- Email: [email protected]
Studies of the relaxation of internal energy ofmolecular hydrogen
S T U D I E S OF THE RELAXATION OF INTERNAL ENERGY OF MOLECULAR H Y D R O G E N J. E. DOVE, D. O. JONES, AND H. TEITELBAUM
Department of Chemistry, University of Toronto, Toronto, Canada The vibrational relaxation of gaseous H2 in mixtures of H~-Kr and H2-Kr-He has been studied by the laser sehlieren method in shock waves at 1350°-3000°K. From the results, the relaxation times for pure H2, and for H2 highly dilute in Kr or He, are found to be given by log~oPr~2_E2= (33.06-4-2.48)T-~/a_ (8.39=t=0.19) log~oPrE2-K~= (36.984-5.39) T-t/a- (8.29=t=0.42) Iog~0PrH2-U~= (36.91 =t:2.75) T-1Ia- (8.74=t=0.22) where Pr is in atmos see. If the observed relaxation is interpreted as a pure vibration-translation process, then the experimental collision numbers Z1o for deexeitation of H2 by He are about 100 times smaller than the theoretical values of Shin. The measured density changes in the observed relaxation zones are too large to be accounted for by pure vibrational relaxation alone, and it is suggested that the process being observed experimentally is in reality a coupled relaxation of rotational and vibrational energy. Solutions of the master equation for the relaxation of internal energy of nonrigid rotating I~Iorse oscillator H2 molecules, for the temperature range 1500°-15000°N, support the suggestion that rotation and vibration are strongly eoupled during the observed relaxation process.
Introduction The rates of chemical reactions can be strongly dependent on the state of internal excitation, as well as the translational energy, of the reacting molecules. Under certain circumstances, therefore, observed rates of reaction can be dependent on rates of molecular energy transfer. Moreover, reaction products--especially of exothermic reactions--are often formed initially in a highly nonequitibrium energy distribution which then relaxes towards equilibrium by collisions. An understanding of molecular energy interconversion processes is thus a prerequisite to the complete elucidation of the chemistry of combustion systems. In obtaining an understanding of basic molecular processes, the hydrogen molecule and the hydrogen-helium system are of particular interest, because they often offer the best opportunity for comparison of experimental results with a priori theoretical calculations. Since there is a
real possibility of an accurate quantum mechanical calculation of vibration-translation energy transfer in H2-He, without any major approximations, the importance of obtaining reliable experimental data for this system is clear. So far, however, there do not appear to be any published measurements of vibrational relaxation in H2-He. This lack of data reflects the severe experimental difficulties with this particular mixture. Methods based on density changes are hampered by the already low density of H2-He. Sensitivity is often a problem, because the vibrational heat capacity of H2 is very small at the temperatures at which relaxation measuremerits are usually made. Shock-tube measuremerits are especially difficult because of the technical problems of driving a sufficiently strong shock wave into such a light gas. Studies of vibrational energy transfer between H2 and a number of simple molecules other than He have, however, been reported. 1,2 Kiefer and Lutz a'4 have measured vibrational relaxation times in
177
ELEMENTARY REACTIONS IN COMBUSTION
178
H2-Ar and D2-Ar mixtures in shock waves, using their highly sensitive laser schlieren technique. The stimulated Raman effect has been used 5,~ to study vibrational relaxation of pure hydrogen at room temperature. Recently, Moore 7 has used fluorescence measurements to study vibrational energy transfer between electronically excited HD (B 12~+) and He. The technical difficulties of studying H2-He in shock waves can be partly overcome by adding a heavy inert species to the test gas. The rate of relaxation of He by He can then be found by use of the mixture rule. s Accordingly, we have used the laser sehlieren technique to measure the vibrational relaxation rates of H2-Kr and H 2 - H e - K r mixtures in shock waves. In addition, we made model calculations of the relaxation of H2, based on solutions of the master equation, to study the possible role of the rotational degree of freedom in the relaxation process.
Experimental Methods The S]~ock Tube The shock tube consists of a steel test section of ll.4X8.9-cm rectangular cross section and 730 em long, which is coupled by a transition section to a driver 15 em diam and 500 cm long. The observations were made 600 cm downstream of the diaphragm position. Prescored aluminum and brass diaphragms were pressure burst by H2 to shock-heat the test gas. An Edwards 2-in. diffusion pump with DC-705 silicone fluid, located near the end plate of the shock tube, evacuated the tube to a pressure better than 10-4 torr before the test gas was introduced. The combined leak and outgassing rate was about 10-a torr/min, and the time between filling the tube and initiating the shock wave was always less than 1 rain. Gas Mixtures The gases used were: He He Kr
Matheson Prepnrified grade >99.95%. (02< 20 ppm). Canadian Helium Ltd. High Purity grade > 99.995%. Air Products and Chemicals Inc. Research grade > 99.99%. (Xe-I-Ar-I-N2-[O2< 15 ppm).
Gas mixtures were prepared manometrically in steel tanks and allowed to mix for more than 24 hours before use. The shock tube was filled di-
rectly from these tanks, and the initial pressure read with a wide-bore mercury manometer. The purity of the gases, and of stored mixtures, was checked mass spectrometrically. No additional impurities introduced in mixing or storage could be detected. Table I shows the mixtures used and the initial conditions in the experiments. Shock-Veloclty Measurement Transit times of the shock wave between four gold-film resistance gauges were measured to 4-0.1-gsee time resolution by solid-state timeinterval meters, enabling the temperature of the shock-heated gas to be calculated to =t=0.2%. The following temperatures, pressures, and densities were computed using the standard shock equations and thermoehemical tables: T~, To, Te, P,,, P~, Pc, P~, P~, P~. The subscripts denote: a, conditions immediately behind the shock front with only the translational degree of freedom excited; c, translation and rotation excited, vibration unexcited; e, thermal equilibrium, i.e., all internal degrees of freedom excited. Then AP~,, ATe, and Ap~ are the changes due to vibrational relaxation alone, e.g., Apv = Pc--Pc, and Ap~+~, AT~,+~, Ap~+~ are changes due to rotational and vibrational relaxation combined, e.g., Apv+r=p~--pa. Table I[ lists the range of postshock conditions. Schlieren Apparatus The collimated beam from a continuous wave H e - N e laser traverses the shock tube, through flush-mounted optically flat windows, at right angles to the direction of propagation of the shock wave, and falls on to a photodiode whose sensitive surface is divided into quadrants. Interconversion of translational energy of a relaxing molecular gas creates axial density gradients, thus causing a deflection of the laser beam which can be measured by monitoring the output of the appropriate quadrants of the detector. The relaxation rate of the internal energy then can be calculated from the measured gradient. The equipment can measure gradients as small as 10.9 g em -4, and has a spatial resolution of 0.01 cm. The theory of this type of system has been described by Kiefer. 4 For a linear or an exponential, or near-exponential density gradient, and for small deflections of a uniphase laser beam, the voltage output V is given by V = kD~w (dp/dy),
(1)
where k is an instrument constant, D the dis-
RELAXATION OF MOLECULAR HYDROGEN
179
TABLE I Range of initial conditions Mixture No.
Mixture composition 30.12% 49.93% 33.07% 33.04%
H2, H2, tt~, H2,
69.88% 50.07% 33.76% 33.73%
No. of exWs.
p1 (torr)
T~ (°K)
10 13 15 9
9.9-11.5 8.3-11.3 9.2-11.7 10.2-2t.6
297.0-298.2 295.4-300.3 297.7-300.4 297.6-299.6
Kr Kr IIe, 33.17% Kr He, 33.23% t(r
tanee from the center of the shock tube to the detector, 5 the specific refractivity (GladstoneDale constant) of the gas mixture, w the width of the shock tube, and dp/dy the time-dependent density gradient at the center of the laser beam.
suiting expression for dp/dy can be very well approximated by
dp/ dy = (Pc~P1~fTg) [ Cp/ ( Cp-- Cvlb ) "]/~,ov Xexp{--[-Cp/ (Cp--Cvib)](tJro)},
(2)
Method of Analysis of ExperimenLal Results
The basic model is due to Blackman2 It is assumed that: (i) the translational and rotational degrees of freedom are equilibrated immediately behind the shock front; (ii) vibrational relaxation then occurs relatively slowly; (iii) the effects of chemical reaction are negligible. Calculations confirm the correctness of (iii) under our conditions. A relationship is obtained between density gradient and vibrational energy gradient by assuming that: (iv) the relaxation process occurs at essentially eonstant pressure (el. Table I I ) ; (v) the changes in translational temperature during vibrational relaxation are small enough for the relaxation time r~, and the heat capacity, to be represented by average values (calculations showed that this was iustified, ef. Table I i ) ; (vi) the relaxation of vibrational energy follows the Bethe-Teller law. a The re-
where U is the shock velocity, Cp the heat capacity of the gas mixture, Crib the vibrational contribution to Cp. The subscript 1 refers to initial conditions, and subscript g to gas-particle time (as opposed to l = l a b time). Then ro is the Bethe-Tetler relaxation time in gas-time coordinates and (3)
d t J d h = p/pi.
Using Eqs. (1), (2), and (3), one can show that a plot of the logarithm of the output voltage against lab time should be a straight line whose slope is S = -- I - C / ( C p - - Crib )1 (P~/PI) (2.30379)-I,
and whose intercept gives the voltage V, which is directly proportional to the total density
TABLE II Range of post-shock conditions Mixture No. 3, 4
U, Ii]In/btsec Pc, torr T~, °K ATe., °K
Pe/Pl Ap~ X 108, g/ee Ap,.+~X106, g/cc
1.036-1.528 204.0-508.7 1551-2986 12-94 3.836 4.408 1.47-6.94 14.7-26.3 0.004-0.013
(4)
1.150-1.876 186.7-452.1 1350-3002 11-150 4.026-4.929 1.08-6.34 15.9-26.5 0.004-0.017
1.360-2.153 205.8-467.2 1376-2952 8-100 3.759-4.473 0.832-3.25 8.04-17.7 0.003-0.013
180
ELEMENTARY REACTIONS IN COMBUSTION
change for the relaxation process being observed, :5000
(5)
0.6
The relaxation time Pr~, reduced to 1 atmos, can be calculated from S,
0"4
0.2
Prg= - ECp/ ( G - Cvib ) -]
t600
i500
_- (a) ,,i -
=1
(pJpl) ( P J 7 6 0 ) (2.303S)-1.
T °K 2000
O'S
v,= kb#w CoJp~)[C~/ (c~- c~b )] (ap/u,.).
X
2500
(6)
,i
o
E
"~ o.s
Thus, from each experiment, we obtain two quantities, Pr~ and Ap, where Ap (henceforth ~ labelled Apobs) should correspond to Ap~ calcu- o 0.6 lated from the shock equations, if vibrational relaxation is the process being observed.
A
~1~~, (b I)
1
I
,
I
T
I
/ /
I
0"4
Experimental
Results
Observations The experimental oscilloscope traces of the detector output showed large initial deflections caused by the shock front, followed by a decaying signal which represented the molecular relaxation process. Curvature of the shock front 1° is a limitation n on the proximity to the shock front at which meaningful relaxation measurements can be made. In our experiments, measurements could begin typically 1 to 1.5 relaxation times after shock arrival; in a few cases, interference by the initial deflection lasted for a period approaching 3T. After the initial disturbance, the plots of log V against lab time were linear, and we found no evidence of a significant deviation from an exponential variation of density gradient with time. All the results reported here are based on linear plots extending over times greater than 3r. The slopes of these log plots could generally be measured to within 1%. The time origin, which must be accurately known in order to find the intercept, was measured from a separate display of the output with a much smaller vertical amplification. At low temperatures, the relaxation signals were so small that weak hydrodynamic disturbances, probably originating in the boundary layer, interfered with the measurements. These disturbances have been reported previously.4 At high temperatures, we detected signals which Were due to dissociation of H2. The temperature range 1350~-3000°K was found suitable for our measurements.
0-2
0
/~}
-070
"080 (T °K) 13
I
"090
FIG. 1. Landau-Teller plots of experimentally measured relaxation times. (a) 49.9% H~, 50.1% Kr. (b) O 30.1% H~, 69.9% Kr; [~ 33.1% H2, 33.7% He, 33.2% Kr.
Results The equipment was tested by measuring relaxation times of a mixture of 30% H2~-70% Ar at 1612°-2608°K. Excellent agreement was found with previous work. A total of eleven experiments deviated by an average of 7% from the Pr values measured for this mixture by Kiefer and Lutz? The mixtures listed in Table I were then studied under the conditions summarized in Table II. Figure 1 shows Landau-Teller plots for the three mixture compositions. The lines fit the expressions: Mixture 1 log~0Pr = (35.584-1.98) T -u3 --
(2.3564-0.153),
(7)
Mixture 2 logioPr = (34.834-2.42) T -:]~ -
-
(2.3764-0.189),
(s)
RELAXATION OF MOLECULAR HYDROGEN
181
T °K t-C
3000
2500
2000
t600
I
I
I
I
1300
co
:x 0 . 5
0
E
0
0
0
3 b
g
o
J
-0.5
1 .070
I -075
I .0~0 (T
I -os5
I .090
°K)-113
Fro. 2. Landau-Teller plot of relaxation time of H~ very dilute in lie. Mixtures 3, 4 logioPr= (35.28-+-1.26) T -1In --
(2.490:i:0.101),
with standard deviations as shown. atmos #see; T= Te.
(9)
Pr is in
Data Reduction Standard relaxation times for relaxation of H2 by each of the collision partners, H2, Kr, and He, can be calculated from our data using the mixture equation
1/Pr= ~
(Xi/PrH2--i),
(10)
i
where Xi is the mole fraction of constituent i. First, we used the data for Mixtures 1 and 2 to find PT~t~-~2 and Pr~2_~r , the relaxation times for pure H~ and for H2 infinitely dilute in Kr, as follows: Initial estimates of PrH~-H2 and PrH2-r., were obtained by substituting values obtained from Eqs. (7) and (8) into Eq. (10). These initial values were then refined by back-substitution into the actual experimental values of Pr. In this way, 23 values were obtained for each relaxation time, PrH2-H2 and Prm--Kr, and were then least-squares fitted to give Iogl0PrH2--H2= (33.06=t=2.48 ) T -lla -
-
(2.3914-0.193),
(ii)
]OglOPrHr-Kr= (36.985d=5.39) T -1In -- (2.291-4-0.419),
(12)
for the range 1350°-3000°K. The larger standard deviation for Kr appears to be a reflection of the low efficiency of that gas. These data for relaxation by He and Kr were then used, in conjunction with Eq. (10) and the experimental points for Mixtures 3 and 4, to calculate the values of Prt~2-n~, which are shown as points in Fig. 2. The line is the leastsquares fit log10Pre2_H~= (36.91=t=2.75) T -1In -
-
(2.7425=0.221),
(13)
for the temperature range 1375°-2950°K. In summary, our results show that, in this temperature range, H2 and He are of almost equal efficiency in relaxing H~, while Kr is about one-third as efficient.
Discussion
Our values of Pr~2_~, are about 50% greater than Pr measured for H2-Ar by Kiefera; the relative magnitudes appear reasonable. For pure H2, the standardized relaxation times calculated from our measurements on H2-Kr mixtures are slightly more than twice the values of Pr~2-~2 previously found a in studies of H2-Ar. This discrepancy is perhaps not very serious, because the measurements are difficult, but it is somewhat larger than might have been expected, and it will be mentioned again below. For Pr~2_He, there appears to be no published experimental data. Recently, Shin 12 has made quantum mechani-
182
ELEMENTARY REACTIONS IN COMBUSTION T °K 3000
2500
2000
i600
i300
I
I
I
Q.M. THEORY/./ / ios _
f
J
f
f
f
/
/
/
Almost all our measured values fall above this line, implying that the extrapolated density changes are too large to be accounted for by vibrational relaxation alone; at the higher temperatures, APob~ approaches Ap~+~. tIowever, in assessing this result, we must carefully consider the location of the time origin, to which Apob~ is very sensitive. Our time origin is taken to be the measured instant at which the leading part of the shock wave intersects the center of the laser beam. Because the shock front is undoubtedly somewhat curved ~° near the tube walls, our choice of time origin is--if a n y t h i n g - slightly too early in time, so that the density changes are more likely to be overestimated than underestimated. If we make the alternative assumption that the time origin occurs when the center of the optical disturbance, due to the shock front, crosses the center of the laser beam, then the value of APob~ is reduced but still generally exceeds kp~ by 50% or more. How-
/
J
f
J f I"
o
//
N i05
_ ///
EXP'£/
to3 I
I
•070
~
I
"080
,
I
.090
( T °K )-~'/3
i.250001
FIG. 3. Comparison of theoretical (Shin, Ref. 12) and experimental (this work) collision numbers for deexeitation of H2 from v = 1 to v = 0 by He.
2500]
T °K 2QOQI
i60Q]
i~q
[3 z~ i.O
eal calculations of translation-vibration energy transfer in H2-He, using an accurate a priori interaction potential. His energy-dependent transition probabilities agree very well, except at high collision energies, with earlier values of Mies ~3 calculated by the distorted-wave approximation (DWA); the D W A breaks down at high collision energies. Figure 3 compares Shin's values of Z10, the collision number for deexeitation of H2 by I-Ie, with our experimental values. We have assumed the standard harmonic oscillator relation between Z~0 and Pr, and have used a mean collision diameter of 2.4 .&. In considering possible reasons for this large disagreement between theory and experiment, we note that the apparently anomalous vibrational relaxation behavior of a number of hydrides has previously been ascribed TM to partieipation of rotation in the relaxation process. Such participation might aIso occur in H2 itself. Rotational transitions were not included i n Shin's analysis, and they would greatly add to the complexity of an already difficult calculation. Some support for this suggestion of rotational effects is given by our measured density changes. Figure 4 shows our measured density changes, Apob~, which have been normalized by dividing by the calculated value of Apv+~. The solid line is Apv/Apv+r, i.e., the value expected if Apobs= Apr.
13 O*B
o o z~
#
2 o oo
o.e
g
#
o o
0,4
o
oo & oo
0.2
z
•070
"08 (T ° K)[~/3
~
~
"090
FIG. 4. Experimentally determined density changes Apob~ for the observed relaxation process. AOob~ has been divided by Apv+,, the calculated densiW change corresponding to relaxation of vibrational+rotational energy. The solid line corresponds to the value of Ao calculated for relaxation of vibrational energy alone. Zx 49.9% H2, 50.1% Kr; O 30.1 H2, 69.9% Kr; [] 33.1% H2, 33.7% He, 33.2% Kr.
RELAXATION
OF MOLECULAR
ever, we believe that this alternative assumption probably leads to an underestimate of Apob~. We also point out that, with our detector configuration, we can measure not only the horizontal signal but also the total light reaching the detector (this is a measure of light lost by large angle deflections) and also any vertical deflection signal due to possible transverse disturbantes. We therefore have more data available to interpret the relatively complex signals due to the shock front itself than if we had used the conventional knife-edge configuration. We conclude that, on balance, the evidence points to APob~ being significantly greater than Ap~,, but that the quantitative extent of the difference is uncertain. Our measurements in H2-Ar showed the same effect, and Kiefer and Lutz n have also noted a tendency for their observed density changes in tt2-Ar to be larger than Ap~,, but they do not state whether they considered the difference large enough to be experimentally significant. Two possible explanations of this type of density effect are: (i) In the measured relaxation zone, vibration and some of the rotational levels are relaxing towards equilibrium together. (ii) Rotation has already equilibrated before we can observe the relaxation zone, but vibrational energy transfer is faster in rotationatly excited than unexcited gas. There is thus an "induction period" for vibrational relaxation, as rotation equilibrates behind the shock front. The true time origin for the vibrational relaxation process therefore lies some distance behind the shock front, and extrapolation to the shock front will overestimate the density change. Our data would not enable us to differentiate between these two possibilities.
Master Equation Calculations For an anharmonic oscillator, or a vibrating rotor, there is no simple analytic description of the over-all relaxation behavior. In order to gain a better understanding of the way in which rotation can influence tile vibrational relaxation of II2 by an inert gas, we have therefore solved the master equation numerically, taking into account both vibrational and rotational transitions.
Method of Calculation Our calculations were made for para-H2 highly dilute in Ar. Argon was chosen as col-
HYDROGEN
183
lision partner because we intended also to calculate the dissociation behavior; most experimental data on dissociation are for H2-Ar. (The dissociation studies are being reported separately.) The diatomic molecule was treated as a nonrigid rotating Morse oscillator}6 This model somewhat underestimates the number of H2 states at high energies. However the spacings and energies of the lower levels of H2 are well reproduced, typically to better than 3°-/0; these lower levels are expected to dominate the relaxation behavior in the temperature range of interest. Collisional transition probabilities between bound rotation-vibration states of the molecule were calculated from an expression of the type
Pk_ll-J(E) = (Vi-J) 2 (R k-t )2T2 (E). The translational energy transfer term, T2(E), was calculated by the DWA, 17 and vibrational matrix elements ( V i - @ were calculated for a Morse oscillator perturbed by an exponential interaction potential} s The rotational matrix elements (R~-Z)2 were calculated using Legendre polynomial coefficients obtained from fitting an anisotropic intermolecular potential generated on the assumption of pairwise additivity} 9 The total inelastic cross section for a given energy of approach (E) was calculated by summing the contributions from all partial waves, ~9 using the modified wave-number approximation of Takayanagi? ° Finally, the collisional transition probabilities at a given temperature were obtained from the energy-dependent cross sections by numerical integration over a 5{axwellian distribution of approach energies21; probabilities for the reverse process were then calculated from the principle of detailed balance. In this way, probabilities for three types of transition between the 131 bound states have been calculated : (i) Pure vibration-translation transitions (VT), where Av=-t-1, A J = 0 . (ii) Pure rotation-translation transitions (RT), where Av=0, AJ==t=2. (iii) Simultaneous vibration-rotation-translation transitions (VRT), where Av==t=l, A J = =t=2, i.e., opposed changes in v and J . Transitions from upper-bound states to the continuum were also included} 1 The transition probabilities calculated as described above were then used in the "master equation," a set of coupled differential equations which express the rates of change of the populations of individual molecular states. The master equation was integrated numerically by the
184
ELEMENTARY REACTIONS IN COMBUSTION I
ISHOI
I
[
t
I
E -
(a) •S
Results
)~total
~Ho
_
-4
_
'2 ro~
0
....... / . . . -i
'2
i
I "3
[
I
I
I '4
I "5
I" -6
[ "7
I -s
I
I
I
,
t/~s.o
~.oJ. . . . . . . . . .
I
SHO
-
(b)
"8 ,1-
- -
"6
ivib
Ts.o
.4 "t~ot:..." "2
0
0
/
"~--rot
•."
I d.
1 "2
I
"5
I
"4
I
"5
I
'6
'7
a Bethe-Teller type of expression dEA/dt= (EA ~-- Ea ) / ~ a .
I .s
t //'~SH 0
FIG. 5. Bethe-Teller relaxation times for relaxation of rotational energy, vibrational energy, and total internal energy of H2 highly dilute in Ar, calculated from our master equation solutions for 2000°K. The conditions are the same for both eases except that VRT processes have been included in (b) but not in (a). The scales have been normalized by division by rsHo, the corresponding simple harmonic oscillator relaxation time.
algorithm of Gear, 22 to give values for all the vibration-rotation level populations at suecessive instants of time. Shock heating of the gas was simulated by starting with a population distribution corresponding to a low temperature, and using transition probabilities appropriate to some high translational temperature. 21 For this nonrigid rotating Morse oscillator, the vibrational energy of a molecule depends only on v and not on J, and the rotational energy depends only on J. Knowing the population of each state, we can therefore calculate the separate contributions of vibrational and rotational energies to the total energy of the system. I n this way, our calculations have provided values for Evib, Erot, Etotal(=Evib+Erot), dEvib/dt, dErot/dt, and dEtot,Jdt at successive stages of the relaxation process. We have also followed the time dependence of the isothermal relaxation times T, ib, r~ot, and rtotal, defined by
A problem in the calculation of VT transition probabilities is that they are very sensitive to the intermolecular potential used; unfortunately no accurate a priori potential is available for H2-Ar. Our calculated VT transition prob° abilities agree closely with values obtained from Kiefer's relaxation measurements3 for H2-Ar. To achieve this agreement, the value of the exponential repulsion parameter ~ was chosen to be 4.00 X-t, rather than the values of 3.45 ~-1 or 3.02 X-1 fronl high-energy elastic scattering measurements,23 or 5.8 to 6.0 ~ - t caleulated2t from transport-property data. The DWA, and the anisotropic potential which we used, will probably both tend to overestimate the R T transition probabilities. Nevertheless, consideration of published experimental data 24-2~ on rotational relaxation (all, however, at lower temperatures) indicates that our RT transition probabilities are at least qualitatively reasonable. For our calculated VRT transition probabilities, there do not seem to be any published data which would provide a check. We have solved the master equation for a number of temperatures in the range 1500 ° . 15000°K. I n calculations with VT and RT only, vibrational and rotational energy relaxed independently at different rates, as expected [Fig. 5(a)]. However, when VRT transitions were added to the calculations, two striking features emerged. Though vibration and rotation initially relaxed independently, after a short time the time constants for rotation and vibration coalesced, and rotation and vibration then became coupled and relaxed together towards the final steady distribution. Moreover, when coupling had occurred, relaxation of total internal energy (rotation+vibration) was substantially faster than without VRT [Fig. 5 (b)]. Qualitatively, therefore, these computations support both of the suggestions which we tents-. tively made to account for our measured values of kPob.~. Vibrational relaxation is initially slow, but it speeds up as vibration and rotation become coupled, so that the time origin for the observed relaxation process will appear to be shifted away from the shock front. Moreover, vibration and rotation are indeed relaxing together, even in the later part of the relaxation zone.
Although these calculations were made for isothermal infinitely dilute conditions, we can use our calculated energy gradients to predict
RELAXATION OF MOLECULAR HYDROGEN density gradients in a shock-wave experiment. We find that the experimental APob~ is indeed predicted to be greater than Ap~. However, at 2000°K, the shift in the time origin, and the amount of relaxation in the later parts of the relaxation zone, are both small; at 2000°K, the calculated increase in APob~ is only 3.4%, much less than observed in our experiments. At higher temperatures, the effects are larger. At 6000°K, for example, APob~is predicted to be 62% larger than Ap~. When assessing the numerical results of our master equation solutions, the following should be considered. So far, there is no consensus about the intermoleeular potential parameters of H2-Ar. Therefore, at the time that we began our computations, it seemed not unreasonable to choose e,=4.00 ~ - i ; this value compares with 3.0-3.4 i - 1 from elastic scattering data and 5.8-6.0 ~ - i from transport properties, and it leads to values of P~-+ which agree with experimental relaxation data if the experiments are interpreted in terms of a pure VT process. However, if we had used a from the elastic scattering data, our p~-0 values would have been much smaller. The situation for H2-Ar may therefore be like that for H2-He, namely, the theoretical p~-0 using an accurate potential is lower than the "experimental" pI-o. This suggests that what is being measured in a high-temperature relaxation experiment may not be directly related to pl-0 at all, but may, for example, be the result of a sequence of efficient R T and VRT processes. In our solutions of the master equation, we have taken account of about 700 different transitions. Nevertheless, this already formidable number is only a small fraction of some 17,000 theoretically possible transitions. Many of the omitted transitions wilI be nearly resonant. I t seems highly probable that some additional VRT transitions ought to be included, and that these would couple the vibrational and rotational energy even more strongly. It would be very interesting to see whether addition of more VRT processes, and perhaps a reduction of the VT transition probabilities, would reproduce our experimentally observed relaxation rates and density changes, but so far we have not had the opportunity to test this. Two additional points should be mentioned. When analyzing our experimental relaxation data, we used the mixture rule in the conventional way. If there are competing VT, RT, VRT (and presmnably also VV) processes, the use of the mixture rule appears highly questionable, and could lead to errors. Of course, it is highly unlikely that major discrepancies, such as the large apparent difference between theo-
185
retieal and experimental values of Z10 for H~-He, could be mainly due to breakdown of the mixture rule. (The observation in our experiments that the H2 relaxation rate is substantially increased when part of the Kr from Mixture 1, or equal amounts of Kr and H2 from Mixture 2, are replaced by He, already shows that the efficiency of He in relaxing H2 is comparable to that of H2 itself.) However, it does seem possible that inconsistencies of a factor of 2, e.g., in values of Pru2-~2 obtained using different diiuents, could arise in this way. We are proposing to investigate this problem by making additional experiments and computations. Secondly, we find in our computations that r,ot, as defined by a BetheTeller expression, is strongly time-dependent. For example, at 1500°K, the calculated value of ~'~ot increases by a factor of 12 as Erot relaxes from zero to 95°7o of its final value. Clearly, great caution should be exercised in using the term "rotational relaxation time" for H2 when more than a very few rotational levels are excited.
Acknowledgments This work was supported by the Defence Research Board of Canada. H. T. thanks the National Research Council of Canada for the award of a postgraduate Scholarship. We are very grateful to Professor T. E. Hull, Mrs. Bryna Fellen, and Mr. Wayne Enright for advice and discussions on problems of numerical integration.
REFERENCES i. GORDOX, R. G., ~4~LEMPERER, W., AND STEINFELD, J. I.: Ann. Rev. Phys. Chem. 19, 215 (1968). 2. BURNETT, G. M. AND NORTH, A. h{. (Eds.): Transfer and Storage of Energy by Molecules. Voh If, Vibrational Energy, Wiley-Interscienee, 1969. 3. KIEFER, J. H. AND LUTZ, R. W.: J. Chem. Phys. 4~, 668 (1966). 4. KIEFER, J. ~I. AND LUTZ, R. W. : J. Chem. Phys.
4/~, 658 (1966). 5. D~;MARTIXl, F. AND D~CUINO, J.: Phys. Rev. Letters 17, 117 (1966). 6. I)UCUINO, J. AND ]~E~ARTINI, F. :J. Chim. phys. 64, 209 (1967). 7. FINK, E. H., AKINS, D. L., A~,'D MOORE, C. BRADLEY: J. Chem. Phys. 56, 900 (1972). ~. HERZFELD, tion and
K. F. AND LITOVITZ, T. A. : Absorp-
Dispersion of Ultrasonic Waves, Academic Press, 1959. 9. BLACKI~iAN, ¥.: J. Fluid Mech. 1, 61 (1956).
186
ELEMENTARY REACTIONS IN COMBUSTION
10. DE BOER, P. C. T.: Phys. Fluids 6, 962 (1963). 11. DAEN, J. AND DE BOER, P. C. T.: J. Chem. Phys. 36, 1222 (1962). 12. S~IN, H. K.: J. Phys. Chem. 75, 4001 (1971). 13. MIEs, F. H.: J. Chem. Phys. 42, 2709 (1965). 14. COTTRELL, T. L., DOBBIE, R. C., McLAIN, J., AND READ, A. W.: Trans. Faraday Soc. 60, 241 (1964). 15. KIEFER, J. H. AND LUTZ, R. W.: J. Chem. Phys. 45, 3888 (1966). 16. HERZBERG, G.: Spectra of Diatomie Molecules, p. 426, Van Nostrand, 1950. 17. JACKSON, J. M. AND MOTT, N. F.: Proc. Roy. Soc. (London) A137, 703 (1932). 18. MIES, F. H.: J. Chem. Phys. 40, 523 (1964). 19. ROBERTS, R. E., BERNSTEIN, R. B., AND CUR* Tiss, C. F.: J. Chem. Phys. 50, 5163 (1969).
20. TAKAYANAGI,K.: J. Phys. Soc. Japan 14, 75 (1959). 21. DOVE, J. E. hND JONES, D. G.: J. Chem. Phys. 55, 1531 (1971). 22. GEAR, C. W.: Comm. A.C.M. 14, 176 (1971); 14, 185 (1971). 23. COLGATE, S. O., JORDAN, J. E., AMDUR, I., AXD MASON, E. A.: J. Chem. Phys. 51, 968 (1969). 24. VALLEY, L. Mr. AND AMME, R. C.: J. Chem. Phys. 50, 3190 (1969). 25. JONKMAN, R. M., PRANGSMA, G. J., ERTAS, I., I4~NAAP, H. F. P., AND BEENAKKER, J. J. M.: Physica 38, 441 (1968). 26. BOITNOT% C. A. AND WARDER, R. C., Jr.: Phys. Fluids 14, 2312 (1971). 27. YODER, M. J.: J. Chem. Phys. 56, 3226 (1972).
COMMENTS R. 1. Soloukhin, University of Novosibirsk, Novosibirsk, U.S.S.R. How was it possible to detect and/or avoid a shock-diffusion separation effect in such a typical binary mixture as H2-Kr, with a large difference of the particle mass?
Authors' Reply. In a mixture such as Kr-H2 or Kr-He-H2, there could be a substantial diffusive separation of the gases across the shock front. We have made calculations, involving the Chapman-Enskog solution of the Boltzmann equation, for the region behind the shock front. These calculations show that the total spatial extent of the diffusion effect is very small. Even at the highest temperature of our experiments, 3000°K, where the effect is most pronounced, the concentrations and other state variables in the directly observed part of the relaxation zone are ahnost completely nnaffected, and the influence on measurements of the time origin is still much too small to be experimentally significant.
Percival D. McCormack, Dept. of Mathematical Physics, University College, Cork, Ireland. The authors assume that the intermolecular potential field for the hydrogen molecule is accurate at all excitation values. But this is probably not so, and can have a significant bearing on the relaxation of rotational internal energy. The possible rotational resonance interactions--which are long range--could effect the excitation into the higher rotational levels by collisions. In classical terms, the soft potential becomes ap-
preciable. A corresponding increase in the relaxation time for these higher levels would result.
Authors' Reply. Several points seem to arise from Professor McCormack's comment, such as the possible effects of long range attractive interactions on rotational transitions among the higher levels, and the dependence of the whole interaction potential (including repulsion) on the degree of internal excitation of the H~. Possibly, he is also thinking about the influence of the rotational barrier on the dynamics of energy-transfer collisions. For certain types of collision (roughly speaking, collisions where the angular momentum is large), the rotational barrier can be thought of as merging with the repulsive potential, and thus softening that potential. However, we note that the rotational barrier is essentially a convenient conceptual device for taking angular momentum conservation into account; provided the collision dynamics are treated appropriately, the unmodified intermolecular potential remains valid and can be used. A full discussion of the above points would be very lengthy, and here we can only outline a few of the relevant factors. First, we should mention that, in the temperature range of our experiments, 1500-3000°K, few of the molecules are in very highly excited rotational states, so that those states are quite unlikely to contribute appreciably to the observed relaxation behavior, e.g., at equilibrium at 2500°K, 99.5% of the molecules have J ~ 12. Naturally, this does not mean that the upper states should be entirely neglected; higher rotational levels will be in-
RELAXATION OF MOLECULAR HYDROGEN
ereasingly populated above 3000°K, and they may also play an important role in the dissociation process. Second, we emphasize that the calculations were designed to simulate the relaxation and dissociation of He by collisions with inert-gas atoms, so that H2-H2 interactions, ineluding resonant interactions, were excluded, and it is the }I2-Ar potential with which we are concerned. Third, the attractive part of the H2-Ar potential was not included in our calculations. This attractive potential, which is expected to be stronger when the He molecules are highly excited, gives rise to collision complexes ("resonances"), and it is expected to play a role in the formation of nascent He molecules in H-atom recombination at low temperatures. I However, there seem to be good theoretical reasons for assuming that the repulsive part of the intermolecular potential dominates the energy transfer in H2-Ar, under the conditions in which we are mainly interested here. Even at very high temperatures, where the highest levels become appreciably populated, our calculations suggest that transitions among these levels, involving long-range interactions, will have at most a very weak rate-determining effect on the coupled relaxation-dissociation process. Fourth, the effect of the degree of internal excitation on the intermolecular repulsive potential was, in fact, specifically allowed for in our calculations by fitting the anisotropy of the intermolecular potential for each separate (v,J) level to a Legendre polynomial, which was then used in the calculation of the respective rotational matrix element. For the reasons explained in the paper, these calculations were made for H~-Ar, for which a good quantum mechanical potential is not yet available. We therefore generated approximate potentials by assuming pairwise additivity of forces between the Ar atom and each H atom, with a single exponential repulsion parameter between H Ar pairs. We propose to extend the calculations to H2-He, for which such approximations will not be necessary. REFERENCE l. Roberts, R. E., Bernstein, R. B., and Curtiss, C. F.: J. Chem. Phys. 50, 5163 (1970).
John H. Kiefer, University of Illinois at Chicago Circle, Department of Energy Engineering, Chicago, Illinois. When we did similar experiments on H2 in 1965,~,2 we did not find any serious deviation of the integrated density change (Ap) from that expected for relaxation of just
187
the vibrational energy. It would appear that the origin of the disagreement on this issue lies in the location of the shock front (time zero) on the oscillogram. Because of the great rapidity of H2 relaxation in laboratory time, the majority of Ap appears within the first few tenths of a microsecond, and a displacement of this order in time zero will alter Ap profoundly. Thus, it comes down to which assignment is correct. A rigorous treatment of this question is very difficult; for one thing, the shape and axial extent of the shock is uncertain to this scale. I can only suggest that our choice is self-consistent. The relaxation of the hydrogen halides, 3,4 and chlorine 5 has a rapidity similar to that of H2; yet, in these examples, a consistent agreement of the measured density change with that for vibrational relaxation, alone, was always obtained. In these experiments, the shock-front assignment was the same as in the original H2 experiments. On this basis, I would conclude that our H2 measurements must be correct. I t would indeed be a remarkable coincidence for a fixed error in shock-front location to accurately compensate a rotational coupling in all these disparate examples. REFERENCES 1. J. H. Kiefer and R. W. Lutz: J. Chem. 44, 668 (1966). 2. J. H. Kiefer and R. W. Lutz: J. Chem. 45, 3888 (1966). 3. W. D. Breshears and P. F. Bird: J. Chem. 50, 333 (1969). 4. J. H. Kiefer, W. D. Breshears, and P. F. J. Chem. Phys. 50, 3641 (1969). 5. W. D. Breshears and P. F. Bird: J. Chem. 51, 3660 (1969).
Phys. Phys. Phys. Bird: Phys.
Authors' Reply. We realized, of course, quite early in our analysis of the total density changes corresponding to the observed relaxation process in H2 that, because of the shortness of the reIaxation zone in these experiments, it was very important to locate the time origin correctly. Therefore, we made a careful analysis of the interaction of the laser beam with the shockfront region, and the time origin was assigned on the basis of that analysis. For our split diode detector, the analysis correctly predicted the observed form of the horizontal deflection signal and of the total signal from the detector. We took into account the shape and axial extent of the shock front, on the basis of an analysis of the flow behavior I in our shock tube. We also considered the effects of deviations from unidimensionality of the flow immediately behind the shock front.
188
ELEMENTARY REACTIONS IN COMBUSTION
The only other published data on Ap in H2 (and D2) with which we can compare our measured density changes are in Refs. 2 and 3. When taking account of the fact that the tabulated density changes for H~ in Ref. 2 need to be multiplied by a correction factoP of 2, nearly all the measured density changes in Refs. 2 and 3 exceed Apv, and some of the positive deviations seem quite substantial to us. I n this connection, we reiterate that, even if our time origin is shifted to significantly later times, the differences between Apob~ and Ap~ in H~ persist in our work. If we use the assignment of the time origin that Prof. Kiefer has suggested to us privately, Apobs/Apv still significantly exceeds unity in our experiments in the upper half of our temperature range. To avoid any misunderstanding, we should explain that the work which Prof. Kiefer refers to as being similar to ours was on H2-Ar. As
stated in our paper, our measurements do seem to be the first study of the relaxation of H2 by He at elevated temperatures, although a study of D2-He has been recently published. 4 It should also be emphasized that the measured relaxation times are not in any way dependent on the integrated Apobs. The interest in the values of Apob~ lies in their implications for the interpretation of the observed relaxation times.
REFERENCES 1. P. C. T. De Boer: Phys. Fluids 6, 962 (1963). 2. J. H. Kiefer and R. W. Lutz: J. Chem. Phys. ~ , 668 (1966). 3. J. H. Kiefer and R. W. Lutz: J. Chem. Phys. JS, 3888 (1966). 4. P. F. Bird and W. D. Breshears: Chem. Phys. Letters 13, 529 (1972).
Department of Chemistry, University of Toronto, Toronto, Canada The vibrational relaxation of gaseous H2 in mixtures of H~-Kr and H2-Kr-He has been studied by the laser sehlieren method in shock waves at 1350°-3000°K. From the results, the relaxation times for pure H2, and for H2 highly dilute in Kr or He, are found to be given by log~oPr~2_E2= (33.06-4-2.48)T-~/a_ (8.39=t=0.19) log~oPrE2-K~= (36.984-5.39) T-t/a- (8.29=t=0.42) Iog~0PrH2-U~= (36.91 =t:2.75) T-1Ia- (8.74=t=0.22) where Pr is in atmos see. If the observed relaxation is interpreted as a pure vibration-translation process, then the experimental collision numbers Z1o for deexeitation of H2 by He are about 100 times smaller than the theoretical values of Shin. The measured density changes in the observed relaxation zones are too large to be accounted for by pure vibrational relaxation alone, and it is suggested that the process being observed experimentally is in reality a coupled relaxation of rotational and vibrational energy. Solutions of the master equation for the relaxation of internal energy of nonrigid rotating I~Iorse oscillator H2 molecules, for the temperature range 1500°-15000°N, support the suggestion that rotation and vibration are strongly eoupled during the observed relaxation process.
Introduction The rates of chemical reactions can be strongly dependent on the state of internal excitation, as well as the translational energy, of the reacting molecules. Under certain circumstances, therefore, observed rates of reaction can be dependent on rates of molecular energy transfer. Moreover, reaction products--especially of exothermic reactions--are often formed initially in a highly nonequitibrium energy distribution which then relaxes towards equilibrium by collisions. An understanding of molecular energy interconversion processes is thus a prerequisite to the complete elucidation of the chemistry of combustion systems. In obtaining an understanding of basic molecular processes, the hydrogen molecule and the hydrogen-helium system are of particular interest, because they often offer the best opportunity for comparison of experimental results with a priori theoretical calculations. Since there is a
real possibility of an accurate quantum mechanical calculation of vibration-translation energy transfer in H2-He, without any major approximations, the importance of obtaining reliable experimental data for this system is clear. So far, however, there do not appear to be any published measurements of vibrational relaxation in H2-He. This lack of data reflects the severe experimental difficulties with this particular mixture. Methods based on density changes are hampered by the already low density of H2-He. Sensitivity is often a problem, because the vibrational heat capacity of H2 is very small at the temperatures at which relaxation measuremerits are usually made. Shock-tube measuremerits are especially difficult because of the technical problems of driving a sufficiently strong shock wave into such a light gas. Studies of vibrational energy transfer between H2 and a number of simple molecules other than He have, however, been reported. 1,2 Kiefer and Lutz a'4 have measured vibrational relaxation times in
177
ELEMENTARY REACTIONS IN COMBUSTION
178
H2-Ar and D2-Ar mixtures in shock waves, using their highly sensitive laser schlieren technique. The stimulated Raman effect has been used 5,~ to study vibrational relaxation of pure hydrogen at room temperature. Recently, Moore 7 has used fluorescence measurements to study vibrational energy transfer between electronically excited HD (B 12~+) and He. The technical difficulties of studying H2-He in shock waves can be partly overcome by adding a heavy inert species to the test gas. The rate of relaxation of He by He can then be found by use of the mixture rule. s Accordingly, we have used the laser sehlieren technique to measure the vibrational relaxation rates of H2-Kr and H 2 - H e - K r mixtures in shock waves. In addition, we made model calculations of the relaxation of H2, based on solutions of the master equation, to study the possible role of the rotational degree of freedom in the relaxation process.
Experimental Methods The S]~ock Tube The shock tube consists of a steel test section of ll.4X8.9-cm rectangular cross section and 730 em long, which is coupled by a transition section to a driver 15 em diam and 500 cm long. The observations were made 600 cm downstream of the diaphragm position. Prescored aluminum and brass diaphragms were pressure burst by H2 to shock-heat the test gas. An Edwards 2-in. diffusion pump with DC-705 silicone fluid, located near the end plate of the shock tube, evacuated the tube to a pressure better than 10-4 torr before the test gas was introduced. The combined leak and outgassing rate was about 10-a torr/min, and the time between filling the tube and initiating the shock wave was always less than 1 rain. Gas Mixtures The gases used were: He He Kr
Matheson Prepnrified grade >99.95%. (02< 20 ppm). Canadian Helium Ltd. High Purity grade > 99.995%. Air Products and Chemicals Inc. Research grade > 99.99%. (Xe-I-Ar-I-N2-[O2< 15 ppm).
Gas mixtures were prepared manometrically in steel tanks and allowed to mix for more than 24 hours before use. The shock tube was filled di-
rectly from these tanks, and the initial pressure read with a wide-bore mercury manometer. The purity of the gases, and of stored mixtures, was checked mass spectrometrically. No additional impurities introduced in mixing or storage could be detected. Table I shows the mixtures used and the initial conditions in the experiments. Shock-Veloclty Measurement Transit times of the shock wave between four gold-film resistance gauges were measured to 4-0.1-gsee time resolution by solid-state timeinterval meters, enabling the temperature of the shock-heated gas to be calculated to =t=0.2%. The following temperatures, pressures, and densities were computed using the standard shock equations and thermoehemical tables: T~, To, Te, P,,, P~, Pc, P~, P~, P~. The subscripts denote: a, conditions immediately behind the shock front with only the translational degree of freedom excited; c, translation and rotation excited, vibration unexcited; e, thermal equilibrium, i.e., all internal degrees of freedom excited. Then AP~,, ATe, and Ap~ are the changes due to vibrational relaxation alone, e.g., Apv = Pc--Pc, and Ap~+~, AT~,+~, Ap~+~ are changes due to rotational and vibrational relaxation combined, e.g., Apv+r=p~--pa. Table I[ lists the range of postshock conditions. Schlieren Apparatus The collimated beam from a continuous wave H e - N e laser traverses the shock tube, through flush-mounted optically flat windows, at right angles to the direction of propagation of the shock wave, and falls on to a photodiode whose sensitive surface is divided into quadrants. Interconversion of translational energy of a relaxing molecular gas creates axial density gradients, thus causing a deflection of the laser beam which can be measured by monitoring the output of the appropriate quadrants of the detector. The relaxation rate of the internal energy then can be calculated from the measured gradient. The equipment can measure gradients as small as 10.9 g em -4, and has a spatial resolution of 0.01 cm. The theory of this type of system has been described by Kiefer. 4 For a linear or an exponential, or near-exponential density gradient, and for small deflections of a uniphase laser beam, the voltage output V is given by V = kD~w (dp/dy),
(1)
where k is an instrument constant, D the dis-
RELAXATION OF MOLECULAR HYDROGEN
179
TABLE I Range of initial conditions Mixture No.
Mixture composition 30.12% 49.93% 33.07% 33.04%
H2, H2, tt~, H2,
69.88% 50.07% 33.76% 33.73%
No. of exWs.
p1 (torr)
T~ (°K)
10 13 15 9
9.9-11.5 8.3-11.3 9.2-11.7 10.2-2t.6
297.0-298.2 295.4-300.3 297.7-300.4 297.6-299.6
Kr Kr IIe, 33.17% Kr He, 33.23% t(r
tanee from the center of the shock tube to the detector, 5 the specific refractivity (GladstoneDale constant) of the gas mixture, w the width of the shock tube, and dp/dy the time-dependent density gradient at the center of the laser beam.
suiting expression for dp/dy can be very well approximated by
dp/ dy = (Pc~P1~fTg) [ Cp/ ( Cp-- Cvlb ) "]/~,ov Xexp{--[-Cp/ (Cp--Cvib)](tJro)},
(2)
Method of Analysis of ExperimenLal Results
The basic model is due to Blackman2 It is assumed that: (i) the translational and rotational degrees of freedom are equilibrated immediately behind the shock front; (ii) vibrational relaxation then occurs relatively slowly; (iii) the effects of chemical reaction are negligible. Calculations confirm the correctness of (iii) under our conditions. A relationship is obtained between density gradient and vibrational energy gradient by assuming that: (iv) the relaxation process occurs at essentially eonstant pressure (el. Table I I ) ; (v) the changes in translational temperature during vibrational relaxation are small enough for the relaxation time r~, and the heat capacity, to be represented by average values (calculations showed that this was iustified, ef. Table I i ) ; (vi) the relaxation of vibrational energy follows the Bethe-Teller law. a The re-
where U is the shock velocity, Cp the heat capacity of the gas mixture, Crib the vibrational contribution to Cp. The subscript 1 refers to initial conditions, and subscript g to gas-particle time (as opposed to l = l a b time). Then ro is the Bethe-Tetler relaxation time in gas-time coordinates and (3)
d t J d h = p/pi.
Using Eqs. (1), (2), and (3), one can show that a plot of the logarithm of the output voltage against lab time should be a straight line whose slope is S = -- I - C / ( C p - - Crib )1 (P~/PI) (2.30379)-I,
and whose intercept gives the voltage V, which is directly proportional to the total density
TABLE II Range of post-shock conditions Mixture No. 3, 4
U, Ii]In/btsec Pc, torr T~, °K ATe., °K
Pe/Pl Ap~ X 108, g/ee Ap,.+~X106, g/cc
1.036-1.528 204.0-508.7 1551-2986 12-94 3.836 4.408 1.47-6.94 14.7-26.3 0.004-0.013
(4)
1.150-1.876 186.7-452.1 1350-3002 11-150 4.026-4.929 1.08-6.34 15.9-26.5 0.004-0.017
1.360-2.153 205.8-467.2 1376-2952 8-100 3.759-4.473 0.832-3.25 8.04-17.7 0.003-0.013
180
ELEMENTARY REACTIONS IN COMBUSTION
change for the relaxation process being observed, :5000
(5)
0.6
The relaxation time Pr~, reduced to 1 atmos, can be calculated from S,
0"4
0.2
Prg= - ECp/ ( G - Cvib ) -]
t600
i500
_- (a) ,,i -
=1
(pJpl) ( P J 7 6 0 ) (2.303S)-1.
T °K 2000
O'S
v,= kb#w CoJp~)[C~/ (c~- c~b )] (ap/u,.).
X
2500
(6)
,i
o
E
"~ o.s
Thus, from each experiment, we obtain two quantities, Pr~ and Ap, where Ap (henceforth ~ labelled Apobs) should correspond to Ap~ calcu- o 0.6 lated from the shock equations, if vibrational relaxation is the process being observed.
A
~1~~, (b I)
1
I
,
I
T
I
/ /
I
0"4
Experimental
Results
Observations The experimental oscilloscope traces of the detector output showed large initial deflections caused by the shock front, followed by a decaying signal which represented the molecular relaxation process. Curvature of the shock front 1° is a limitation n on the proximity to the shock front at which meaningful relaxation measurements can be made. In our experiments, measurements could begin typically 1 to 1.5 relaxation times after shock arrival; in a few cases, interference by the initial deflection lasted for a period approaching 3T. After the initial disturbance, the plots of log V against lab time were linear, and we found no evidence of a significant deviation from an exponential variation of density gradient with time. All the results reported here are based on linear plots extending over times greater than 3r. The slopes of these log plots could generally be measured to within 1%. The time origin, which must be accurately known in order to find the intercept, was measured from a separate display of the output with a much smaller vertical amplification. At low temperatures, the relaxation signals were so small that weak hydrodynamic disturbances, probably originating in the boundary layer, interfered with the measurements. These disturbances have been reported previously.4 At high temperatures, we detected signals which Were due to dissociation of H2. The temperature range 1350~-3000°K was found suitable for our measurements.
0-2
0
/~}
-070
"080 (T °K) 13
I
"090
FIG. 1. Landau-Teller plots of experimentally measured relaxation times. (a) 49.9% H~, 50.1% Kr. (b) O 30.1% H~, 69.9% Kr; [~ 33.1% H2, 33.7% He, 33.2% Kr.
Results The equipment was tested by measuring relaxation times of a mixture of 30% H2~-70% Ar at 1612°-2608°K. Excellent agreement was found with previous work. A total of eleven experiments deviated by an average of 7% from the Pr values measured for this mixture by Kiefer and Lutz? The mixtures listed in Table I were then studied under the conditions summarized in Table II. Figure 1 shows Landau-Teller plots for the three mixture compositions. The lines fit the expressions: Mixture 1 log~0Pr = (35.584-1.98) T -u3 --
(2.3564-0.153),
(7)
Mixture 2 logioPr = (34.834-2.42) T -:]~ -
-
(2.3764-0.189),
(s)
RELAXATION OF MOLECULAR HYDROGEN
181
T °K t-C
3000
2500
2000
t600
I
I
I
I
1300
co
:x 0 . 5
0
E
0
0
0
3 b
g
o
J
-0.5
1 .070
I -075
I .0~0 (T
I -os5
I .090
°K)-113
Fro. 2. Landau-Teller plot of relaxation time of H~ very dilute in lie. Mixtures 3, 4 logioPr= (35.28-+-1.26) T -1In --
(2.490:i:0.101),
with standard deviations as shown. atmos #see; T= Te.
(9)
Pr is in
Data Reduction Standard relaxation times for relaxation of H2 by each of the collision partners, H2, Kr, and He, can be calculated from our data using the mixture equation
1/Pr= ~
(Xi/PrH2--i),
(10)
i
where Xi is the mole fraction of constituent i. First, we used the data for Mixtures 1 and 2 to find PT~t~-~2 and Pr~2_~r , the relaxation times for pure H~ and for H2 infinitely dilute in Kr, as follows: Initial estimates of PrH~-H2 and PrH2-r., were obtained by substituting values obtained from Eqs. (7) and (8) into Eq. (10). These initial values were then refined by back-substitution into the actual experimental values of Pr. In this way, 23 values were obtained for each relaxation time, PrH2-H2 and Prm--Kr, and were then least-squares fitted to give Iogl0PrH2--H2= (33.06=t=2.48 ) T -lla -
-
(2.3914-0.193),
(ii)
]OglOPrHr-Kr= (36.985d=5.39) T -1In -- (2.291-4-0.419),
(12)
for the range 1350°-3000°K. The larger standard deviation for Kr appears to be a reflection of the low efficiency of that gas. These data for relaxation by He and Kr were then used, in conjunction with Eq. (10) and the experimental points for Mixtures 3 and 4, to calculate the values of Prt~2-n~, which are shown as points in Fig. 2. The line is the leastsquares fit log10Pre2_H~= (36.91=t=2.75) T -1In -
-
(2.7425=0.221),
(13)
for the temperature range 1375°-2950°K. In summary, our results show that, in this temperature range, H2 and He are of almost equal efficiency in relaxing H~, while Kr is about one-third as efficient.
Discussion
Our values of Pr~2_~, are about 50% greater than Pr measured for H2-Ar by Kiefera; the relative magnitudes appear reasonable. For pure H2, the standardized relaxation times calculated from our measurements on H2-Kr mixtures are slightly more than twice the values of Pr~2-~2 previously found a in studies of H2-Ar. This discrepancy is perhaps not very serious, because the measurements are difficult, but it is somewhat larger than might have been expected, and it will be mentioned again below. For Pr~2_He, there appears to be no published experimental data. Recently, Shin 12 has made quantum mechani-
182
ELEMENTARY REACTIONS IN COMBUSTION T °K 3000
2500
2000
i600
i300
I
I
I
Q.M. THEORY/./ / ios _
f
J
f
f
f
/
/
/
Almost all our measured values fall above this line, implying that the extrapolated density changes are too large to be accounted for by vibrational relaxation alone; at the higher temperatures, APob~ approaches Ap~+~. tIowever, in assessing this result, we must carefully consider the location of the time origin, to which Apob~ is very sensitive. Our time origin is taken to be the measured instant at which the leading part of the shock wave intersects the center of the laser beam. Because the shock front is undoubtedly somewhat curved ~° near the tube walls, our choice of time origin is--if a n y t h i n g - slightly too early in time, so that the density changes are more likely to be overestimated than underestimated. If we make the alternative assumption that the time origin occurs when the center of the optical disturbance, due to the shock front, crosses the center of the laser beam, then the value of APob~ is reduced but still generally exceeds kp~ by 50% or more. How-
/
J
f
J f I"
o
//
N i05
_ ///
EXP'£/
to3 I
I
•070
~
I
"080
,
I
.090
( T °K )-~'/3
i.250001
FIG. 3. Comparison of theoretical (Shin, Ref. 12) and experimental (this work) collision numbers for deexeitation of H2 from v = 1 to v = 0 by He.
2500]
T °K 2QOQI
i60Q]
i~q
[3 z~ i.O
eal calculations of translation-vibration energy transfer in H2-He, using an accurate a priori interaction potential. His energy-dependent transition probabilities agree very well, except at high collision energies, with earlier values of Mies ~3 calculated by the distorted-wave approximation (DWA); the D W A breaks down at high collision energies. Figure 3 compares Shin's values of Z10, the collision number for deexeitation of H2 by I-Ie, with our experimental values. We have assumed the standard harmonic oscillator relation between Z~0 and Pr, and have used a mean collision diameter of 2.4 .&. In considering possible reasons for this large disagreement between theory and experiment, we note that the apparently anomalous vibrational relaxation behavior of a number of hydrides has previously been ascribed TM to partieipation of rotation in the relaxation process. Such participation might aIso occur in H2 itself. Rotational transitions were not included i n Shin's analysis, and they would greatly add to the complexity of an already difficult calculation. Some support for this suggestion of rotational effects is given by our measured density changes. Figure 4 shows our measured density changes, Apob~, which have been normalized by dividing by the calculated value of Apv+~. The solid line is Apv/Apv+r, i.e., the value expected if Apobs= Apr.
13 O*B
o o z~
#
2 o oo
o.e
g
#
o o
0,4
o
oo & oo
0.2
z
•070
"08 (T ° K)[~/3
~
~
"090
FIG. 4. Experimentally determined density changes Apob~ for the observed relaxation process. AOob~ has been divided by Apv+,, the calculated densiW change corresponding to relaxation of vibrational+rotational energy. The solid line corresponds to the value of Ao calculated for relaxation of vibrational energy alone. Zx 49.9% H2, 50.1% Kr; O 30.1 H2, 69.9% Kr; [] 33.1% H2, 33.7% He, 33.2% Kr.
RELAXATION
OF MOLECULAR
ever, we believe that this alternative assumption probably leads to an underestimate of Apob~. We also point out that, with our detector configuration, we can measure not only the horizontal signal but also the total light reaching the detector (this is a measure of light lost by large angle deflections) and also any vertical deflection signal due to possible transverse disturbantes. We therefore have more data available to interpret the relatively complex signals due to the shock front itself than if we had used the conventional knife-edge configuration. We conclude that, on balance, the evidence points to APob~ being significantly greater than Ap~,, but that the quantitative extent of the difference is uncertain. Our measurements in H2-Ar showed the same effect, and Kiefer and Lutz n have also noted a tendency for their observed density changes in tt2-Ar to be larger than Ap~,, but they do not state whether they considered the difference large enough to be experimentally significant. Two possible explanations of this type of density effect are: (i) In the measured relaxation zone, vibration and some of the rotational levels are relaxing towards equilibrium together. (ii) Rotation has already equilibrated before we can observe the relaxation zone, but vibrational energy transfer is faster in rotationatly excited than unexcited gas. There is thus an "induction period" for vibrational relaxation, as rotation equilibrates behind the shock front. The true time origin for the vibrational relaxation process therefore lies some distance behind the shock front, and extrapolation to the shock front will overestimate the density change. Our data would not enable us to differentiate between these two possibilities.
Master Equation Calculations For an anharmonic oscillator, or a vibrating rotor, there is no simple analytic description of the over-all relaxation behavior. In order to gain a better understanding of the way in which rotation can influence tile vibrational relaxation of II2 by an inert gas, we have therefore solved the master equation numerically, taking into account both vibrational and rotational transitions.
Method of Calculation Our calculations were made for para-H2 highly dilute in Ar. Argon was chosen as col-
HYDROGEN
183
lision partner because we intended also to calculate the dissociation behavior; most experimental data on dissociation are for H2-Ar. (The dissociation studies are being reported separately.) The diatomic molecule was treated as a nonrigid rotating Morse oscillator}6 This model somewhat underestimates the number of H2 states at high energies. However the spacings and energies of the lower levels of H2 are well reproduced, typically to better than 3°-/0; these lower levels are expected to dominate the relaxation behavior in the temperature range of interest. Collisional transition probabilities between bound rotation-vibration states of the molecule were calculated from an expression of the type
Pk_ll-J(E) = (Vi-J) 2 (R k-t )2T2 (E). The translational energy transfer term, T2(E), was calculated by the DWA, 17 and vibrational matrix elements ( V i - @ were calculated for a Morse oscillator perturbed by an exponential interaction potential} s The rotational matrix elements (R~-Z)2 were calculated using Legendre polynomial coefficients obtained from fitting an anisotropic intermolecular potential generated on the assumption of pairwise additivity} 9 The total inelastic cross section for a given energy of approach (E) was calculated by summing the contributions from all partial waves, ~9 using the modified wave-number approximation of Takayanagi? ° Finally, the collisional transition probabilities at a given temperature were obtained from the energy-dependent cross sections by numerical integration over a 5{axwellian distribution of approach energies21; probabilities for the reverse process were then calculated from the principle of detailed balance. In this way, probabilities for three types of transition between the 131 bound states have been calculated : (i) Pure vibration-translation transitions (VT), where Av=-t-1, A J = 0 . (ii) Pure rotation-translation transitions (RT), where Av=0, AJ==t=2. (iii) Simultaneous vibration-rotation-translation transitions (VRT), where Av==t=l, A J = =t=2, i.e., opposed changes in v and J . Transitions from upper-bound states to the continuum were also included} 1 The transition probabilities calculated as described above were then used in the "master equation," a set of coupled differential equations which express the rates of change of the populations of individual molecular states. The master equation was integrated numerically by the
184
ELEMENTARY REACTIONS IN COMBUSTION I
ISHOI
I
[
t
I
E -
(a) •S
Results
)~total
~Ho
_
-4
_
'2 ro~
0
....... / . . . -i
'2
i
I "3
[
I
I
I '4
I "5
I" -6
[ "7
I -s
I
I
I
,
t/~s.o
~.oJ. . . . . . . . . .
I
SHO
-
(b)
"8 ,1-
- -
"6
ivib
Ts.o
.4 "t~ot:..." "2
0
0
/
"~--rot
•."
I d.
1 "2
I
"5
I
"4
I
"5
I
'6
'7
a Bethe-Teller type of expression dEA/dt= (EA ~-- Ea ) / ~ a .
I .s
t //'~SH 0
FIG. 5. Bethe-Teller relaxation times for relaxation of rotational energy, vibrational energy, and total internal energy of H2 highly dilute in Ar, calculated from our master equation solutions for 2000°K. The conditions are the same for both eases except that VRT processes have been included in (b) but not in (a). The scales have been normalized by division by rsHo, the corresponding simple harmonic oscillator relaxation time.
algorithm of Gear, 22 to give values for all the vibration-rotation level populations at suecessive instants of time. Shock heating of the gas was simulated by starting with a population distribution corresponding to a low temperature, and using transition probabilities appropriate to some high translational temperature. 21 For this nonrigid rotating Morse oscillator, the vibrational energy of a molecule depends only on v and not on J, and the rotational energy depends only on J. Knowing the population of each state, we can therefore calculate the separate contributions of vibrational and rotational energies to the total energy of the system. I n this way, our calculations have provided values for Evib, Erot, Etotal(=Evib+Erot), dEvib/dt, dErot/dt, and dEtot,Jdt at successive stages of the relaxation process. We have also followed the time dependence of the isothermal relaxation times T, ib, r~ot, and rtotal, defined by
A problem in the calculation of VT transition probabilities is that they are very sensitive to the intermolecular potential used; unfortunately no accurate a priori potential is available for H2-Ar. Our calculated VT transition prob° abilities agree closely with values obtained from Kiefer's relaxation measurements3 for H2-Ar. To achieve this agreement, the value of the exponential repulsion parameter ~ was chosen to be 4.00 X-t, rather than the values of 3.45 ~-1 or 3.02 X-1 fronl high-energy elastic scattering measurements,23 or 5.8 to 6.0 ~ - t caleulated2t from transport-property data. The DWA, and the anisotropic potential which we used, will probably both tend to overestimate the R T transition probabilities. Nevertheless, consideration of published experimental data 24-2~ on rotational relaxation (all, however, at lower temperatures) indicates that our RT transition probabilities are at least qualitatively reasonable. For our calculated VRT transition probabilities, there do not seem to be any published data which would provide a check. We have solved the master equation for a number of temperatures in the range 1500 ° . 15000°K. I n calculations with VT and RT only, vibrational and rotational energy relaxed independently at different rates, as expected [Fig. 5(a)]. However, when VRT transitions were added to the calculations, two striking features emerged. Though vibration and rotation initially relaxed independently, after a short time the time constants for rotation and vibration coalesced, and rotation and vibration then became coupled and relaxed together towards the final steady distribution. Moreover, when coupling had occurred, relaxation of total internal energy (rotation+vibration) was substantially faster than without VRT [Fig. 5 (b)]. Qualitatively, therefore, these computations support both of the suggestions which we tents-. tively made to account for our measured values of kPob.~. Vibrational relaxation is initially slow, but it speeds up as vibration and rotation become coupled, so that the time origin for the observed relaxation process will appear to be shifted away from the shock front. Moreover, vibration and rotation are indeed relaxing together, even in the later part of the relaxation zone.
Although these calculations were made for isothermal infinitely dilute conditions, we can use our calculated energy gradients to predict
RELAXATION OF MOLECULAR HYDROGEN density gradients in a shock-wave experiment. We find that the experimental APob~ is indeed predicted to be greater than Ap~. However, at 2000°K, the shift in the time origin, and the amount of relaxation in the later parts of the relaxation zone, are both small; at 2000°K, the calculated increase in APob~ is only 3.4%, much less than observed in our experiments. At higher temperatures, the effects are larger. At 6000°K, for example, APob~is predicted to be 62% larger than Ap~. When assessing the numerical results of our master equation solutions, the following should be considered. So far, there is no consensus about the intermoleeular potential parameters of H2-Ar. Therefore, at the time that we began our computations, it seemed not unreasonable to choose e,=4.00 ~ - i ; this value compares with 3.0-3.4 i - 1 from elastic scattering data and 5.8-6.0 ~ - i from transport properties, and it leads to values of P~-+ which agree with experimental relaxation data if the experiments are interpreted in terms of a pure VT process. However, if we had used a from the elastic scattering data, our p~-0 values would have been much smaller. The situation for H2-Ar may therefore be like that for H2-He, namely, the theoretical p~-0 using an accurate potential is lower than the "experimental" pI-o. This suggests that what is being measured in a high-temperature relaxation experiment may not be directly related to pl-0 at all, but may, for example, be the result of a sequence of efficient R T and VRT processes. In our solutions of the master equation, we have taken account of about 700 different transitions. Nevertheless, this already formidable number is only a small fraction of some 17,000 theoretically possible transitions. Many of the omitted transitions wilI be nearly resonant. I t seems highly probable that some additional VRT transitions ought to be included, and that these would couple the vibrational and rotational energy even more strongly. It would be very interesting to see whether addition of more VRT processes, and perhaps a reduction of the VT transition probabilities, would reproduce our experimentally observed relaxation rates and density changes, but so far we have not had the opportunity to test this. Two additional points should be mentioned. When analyzing our experimental relaxation data, we used the mixture rule in the conventional way. If there are competing VT, RT, VRT (and presmnably also VV) processes, the use of the mixture rule appears highly questionable, and could lead to errors. Of course, it is highly unlikely that major discrepancies, such as the large apparent difference between theo-
185
retieal and experimental values of Z10 for H~-He, could be mainly due to breakdown of the mixture rule. (The observation in our experiments that the H2 relaxation rate is substantially increased when part of the Kr from Mixture 1, or equal amounts of Kr and H2 from Mixture 2, are replaced by He, already shows that the efficiency of He in relaxing H2 is comparable to that of H2 itself.) However, it does seem possible that inconsistencies of a factor of 2, e.g., in values of Pru2-~2 obtained using different diiuents, could arise in this way. We are proposing to investigate this problem by making additional experiments and computations. Secondly, we find in our computations that r,ot, as defined by a BetheTeller expression, is strongly time-dependent. For example, at 1500°K, the calculated value of ~'~ot increases by a factor of 12 as Erot relaxes from zero to 95°7o of its final value. Clearly, great caution should be exercised in using the term "rotational relaxation time" for H2 when more than a very few rotational levels are excited.
Acknowledgments This work was supported by the Defence Research Board of Canada. H. T. thanks the National Research Council of Canada for the award of a postgraduate Scholarship. We are very grateful to Professor T. E. Hull, Mrs. Bryna Fellen, and Mr. Wayne Enright for advice and discussions on problems of numerical integration.
REFERENCES i. GORDOX, R. G., ~4~LEMPERER, W., AND STEINFELD, J. I.: Ann. Rev. Phys. Chem. 19, 215 (1968). 2. BURNETT, G. M. AND NORTH, A. h{. (Eds.): Transfer and Storage of Energy by Molecules. Voh If, Vibrational Energy, Wiley-Interscienee, 1969. 3. KIEFER, J. H. AND LUTZ, R. W.: J. Chem. Phys. 4~, 668 (1966). 4. KIEFER, J. ~I. AND LUTZ, R. W. : J. Chem. Phys.
4/~, 658 (1966). 5. D~;MARTIXl, F. AND D~CUINO, J.: Phys. Rev. Letters 17, 117 (1966). 6. I)UCUINO, J. AND ]~E~ARTINI, F. :J. Chim. phys. 64, 209 (1967). 7. FINK, E. H., AKINS, D. L., A~,'D MOORE, C. BRADLEY: J. Chem. Phys. 56, 900 (1972). ~. HERZFELD, tion and
K. F. AND LITOVITZ, T. A. : Absorp-
Dispersion of Ultrasonic Waves, Academic Press, 1959. 9. BLACKI~iAN, ¥.: J. Fluid Mech. 1, 61 (1956).
186
ELEMENTARY REACTIONS IN COMBUSTION
10. DE BOER, P. C. T.: Phys. Fluids 6, 962 (1963). 11. DAEN, J. AND DE BOER, P. C. T.: J. Chem. Phys. 36, 1222 (1962). 12. S~IN, H. K.: J. Phys. Chem. 75, 4001 (1971). 13. MIEs, F. H.: J. Chem. Phys. 42, 2709 (1965). 14. COTTRELL, T. L., DOBBIE, R. C., McLAIN, J., AND READ, A. W.: Trans. Faraday Soc. 60, 241 (1964). 15. KIEFER, J. H. AND LUTZ, R. W.: J. Chem. Phys. 45, 3888 (1966). 16. HERZBERG, G.: Spectra of Diatomie Molecules, p. 426, Van Nostrand, 1950. 17. JACKSON, J. M. AND MOTT, N. F.: Proc. Roy. Soc. (London) A137, 703 (1932). 18. MIES, F. H.: J. Chem. Phys. 40, 523 (1964). 19. ROBERTS, R. E., BERNSTEIN, R. B., AND CUR* Tiss, C. F.: J. Chem. Phys. 50, 5163 (1969).
20. TAKAYANAGI,K.: J. Phys. Soc. Japan 14, 75 (1959). 21. DOVE, J. E. hND JONES, D. G.: J. Chem. Phys. 55, 1531 (1971). 22. GEAR, C. W.: Comm. A.C.M. 14, 176 (1971); 14, 185 (1971). 23. COLGATE, S. O., JORDAN, J. E., AMDUR, I., AXD MASON, E. A.: J. Chem. Phys. 51, 968 (1969). 24. VALLEY, L. Mr. AND AMME, R. C.: J. Chem. Phys. 50, 3190 (1969). 25. JONKMAN, R. M., PRANGSMA, G. J., ERTAS, I., I4~NAAP, H. F. P., AND BEENAKKER, J. J. M.: Physica 38, 441 (1968). 26. BOITNOT% C. A. AND WARDER, R. C., Jr.: Phys. Fluids 14, 2312 (1971). 27. YODER, M. J.: J. Chem. Phys. 56, 3226 (1972).
COMMENTS R. 1. Soloukhin, University of Novosibirsk, Novosibirsk, U.S.S.R. How was it possible to detect and/or avoid a shock-diffusion separation effect in such a typical binary mixture as H2-Kr, with a large difference of the particle mass?
Authors' Reply. In a mixture such as Kr-H2 or Kr-He-H2, there could be a substantial diffusive separation of the gases across the shock front. We have made calculations, involving the Chapman-Enskog solution of the Boltzmann equation, for the region behind the shock front. These calculations show that the total spatial extent of the diffusion effect is very small. Even at the highest temperature of our experiments, 3000°K, where the effect is most pronounced, the concentrations and other state variables in the directly observed part of the relaxation zone are ahnost completely nnaffected, and the influence on measurements of the time origin is still much too small to be experimentally significant.
Percival D. McCormack, Dept. of Mathematical Physics, University College, Cork, Ireland. The authors assume that the intermolecular potential field for the hydrogen molecule is accurate at all excitation values. But this is probably not so, and can have a significant bearing on the relaxation of rotational internal energy. The possible rotational resonance interactions--which are long range--could effect the excitation into the higher rotational levels by collisions. In classical terms, the soft potential becomes ap-
preciable. A corresponding increase in the relaxation time for these higher levels would result.
Authors' Reply. Several points seem to arise from Professor McCormack's comment, such as the possible effects of long range attractive interactions on rotational transitions among the higher levels, and the dependence of the whole interaction potential (including repulsion) on the degree of internal excitation of the H~. Possibly, he is also thinking about the influence of the rotational barrier on the dynamics of energy-transfer collisions. For certain types of collision (roughly speaking, collisions where the angular momentum is large), the rotational barrier can be thought of as merging with the repulsive potential, and thus softening that potential. However, we note that the rotational barrier is essentially a convenient conceptual device for taking angular momentum conservation into account; provided the collision dynamics are treated appropriately, the unmodified intermolecular potential remains valid and can be used. A full discussion of the above points would be very lengthy, and here we can only outline a few of the relevant factors. First, we should mention that, in the temperature range of our experiments, 1500-3000°K, few of the molecules are in very highly excited rotational states, so that those states are quite unlikely to contribute appreciably to the observed relaxation behavior, e.g., at equilibrium at 2500°K, 99.5% of the molecules have J ~ 12. Naturally, this does not mean that the upper states should be entirely neglected; higher rotational levels will be in-
RELAXATION OF MOLECULAR HYDROGEN
ereasingly populated above 3000°K, and they may also play an important role in the dissociation process. Second, we emphasize that the calculations were designed to simulate the relaxation and dissociation of He by collisions with inert-gas atoms, so that H2-H2 interactions, ineluding resonant interactions, were excluded, and it is the }I2-Ar potential with which we are concerned. Third, the attractive part of the H2-Ar potential was not included in our calculations. This attractive potential, which is expected to be stronger when the He molecules are highly excited, gives rise to collision complexes ("resonances"), and it is expected to play a role in the formation of nascent He molecules in H-atom recombination at low temperatures. I However, there seem to be good theoretical reasons for assuming that the repulsive part of the intermolecular potential dominates the energy transfer in H2-Ar, under the conditions in which we are mainly interested here. Even at very high temperatures, where the highest levels become appreciably populated, our calculations suggest that transitions among these levels, involving long-range interactions, will have at most a very weak rate-determining effect on the coupled relaxation-dissociation process. Fourth, the effect of the degree of internal excitation on the intermolecular repulsive potential was, in fact, specifically allowed for in our calculations by fitting the anisotropy of the intermolecular potential for each separate (v,J) level to a Legendre polynomial, which was then used in the calculation of the respective rotational matrix element. For the reasons explained in the paper, these calculations were made for H~-Ar, for which a good quantum mechanical potential is not yet available. We therefore generated approximate potentials by assuming pairwise additivity of forces between the Ar atom and each H atom, with a single exponential repulsion parameter between H Ar pairs. We propose to extend the calculations to H2-He, for which such approximations will not be necessary. REFERENCE l. Roberts, R. E., Bernstein, R. B., and Curtiss, C. F.: J. Chem. Phys. 50, 5163 (1970).
John H. Kiefer, University of Illinois at Chicago Circle, Department of Energy Engineering, Chicago, Illinois. When we did similar experiments on H2 in 1965,~,2 we did not find any serious deviation of the integrated density change (Ap) from that expected for relaxation of just
187
the vibrational energy. It would appear that the origin of the disagreement on this issue lies in the location of the shock front (time zero) on the oscillogram. Because of the great rapidity of H2 relaxation in laboratory time, the majority of Ap appears within the first few tenths of a microsecond, and a displacement of this order in time zero will alter Ap profoundly. Thus, it comes down to which assignment is correct. A rigorous treatment of this question is very difficult; for one thing, the shape and axial extent of the shock is uncertain to this scale. I can only suggest that our choice is self-consistent. The relaxation of the hydrogen halides, 3,4 and chlorine 5 has a rapidity similar to that of H2; yet, in these examples, a consistent agreement of the measured density change with that for vibrational relaxation, alone, was always obtained. In these experiments, the shock-front assignment was the same as in the original H2 experiments. On this basis, I would conclude that our H2 measurements must be correct. I t would indeed be a remarkable coincidence for a fixed error in shock-front location to accurately compensate a rotational coupling in all these disparate examples. REFERENCES 1. J. H. Kiefer and R. W. Lutz: J. Chem. 44, 668 (1966). 2. J. H. Kiefer and R. W. Lutz: J. Chem. 45, 3888 (1966). 3. W. D. Breshears and P. F. Bird: J. Chem. 50, 333 (1969). 4. J. H. Kiefer, W. D. Breshears, and P. F. J. Chem. Phys. 50, 3641 (1969). 5. W. D. Breshears and P. F. Bird: J. Chem. 51, 3660 (1969).
Phys. Phys. Phys. Bird: Phys.
Authors' Reply. We realized, of course, quite early in our analysis of the total density changes corresponding to the observed relaxation process in H2 that, because of the shortness of the reIaxation zone in these experiments, it was very important to locate the time origin correctly. Therefore, we made a careful analysis of the interaction of the laser beam with the shockfront region, and the time origin was assigned on the basis of that analysis. For our split diode detector, the analysis correctly predicted the observed form of the horizontal deflection signal and of the total signal from the detector. We took into account the shape and axial extent of the shock front, on the basis of an analysis of the flow behavior I in our shock tube. We also considered the effects of deviations from unidimensionality of the flow immediately behind the shock front.
188
ELEMENTARY REACTIONS IN COMBUSTION
The only other published data on Ap in H2 (and D2) with which we can compare our measured density changes are in Refs. 2 and 3. When taking account of the fact that the tabulated density changes for H~ in Ref. 2 need to be multiplied by a correction factoP of 2, nearly all the measured density changes in Refs. 2 and 3 exceed Apv, and some of the positive deviations seem quite substantial to us. I n this connection, we reiterate that, even if our time origin is shifted to significantly later times, the differences between Apob~ and Ap~ in H~ persist in our work. If we use the assignment of the time origin that Prof. Kiefer has suggested to us privately, Apobs/Apv still significantly exceeds unity in our experiments in the upper half of our temperature range. To avoid any misunderstanding, we should explain that the work which Prof. Kiefer refers to as being similar to ours was on H2-Ar. As
stated in our paper, our measurements do seem to be the first study of the relaxation of H2 by He at elevated temperatures, although a study of D2-He has been recently published. 4 It should also be emphasized that the measured relaxation times are not in any way dependent on the integrated Apobs. The interest in the values of Apob~ lies in their implications for the interpretation of the observed relaxation times.
REFERENCES 1. P. C. T. De Boer: Phys. Fluids 6, 962 (1963). 2. J. H. Kiefer and R. W. Lutz: J. Chem. Phys. ~ , 668 (1966). 3. J. H. Kiefer and R. W. Lutz: J. Chem. Phys. JS, 3888 (1966). 4. P. F. Bird and W. D. Breshears: Chem. Phys. Letters 13, 529 (1972).