J . Phys. Chem. 1984. 88. 4484-4487
4484
general interest and has specific applications in the separation of isotopes by gas dynamic and laser methods as well as in the alignment of Iz rotation. The supersonic flow fields for axisymmetric free jets were determined previously in method-of-characteristics calculations by Owen and T h ~ r n h i l l Wolff,28 ,~~ and Anderson.z9 Miller30 has carried out such calculations for the two-dimensional flow from a slit with a specific heat ratio y = 1.4. Our calculation procedures are essentially identical with those of the earlier work and duplicate Miller's results for y = 1.4. A perfect gas with constant specific heat and a uniform exit flow at a Mach number of 1.10 are assumed. The variation of centerline Mach number Ma with distance x from the plane of the slit, normalized by the slit width W , is shown in Figure 5 for several values of y. The data are well represented in the region x > W by the formula Ma = A(X/FV')(^I-')/~ - B(x/W)-=
(23)
Constants determined by fitting the calculated curves are listed in Table VII. Registry No. 12, 7553-56-2.
Distance, x / W Figure 5. Results of method-of-characteristicscalculations for free-jet
flow from a, slit. the reaction process can be examined.
Appendix A Inviscid Free-Jet Flow from a Slit We report here the results of method-of-characteristics calculations for inviscid free-jet flow from a slit. The system is of
(27) P. L. Owen and C. K. Thornhill, Aeronautical Research Council (Great Britain), R & M 2616, 1948. (28) W. S.Wolff, calculations (1962) reported by H. Ashkenas and F. S. Sherman in 'Rarefied Gas Dynamics, Fourth Symposium", Vol. 2, Academic Press, New York, 1966, p 84. (29) J. B. Anderson, AIAA J., 10, 127 (1972). (30) D. R. Miller, Doctoral Dissertation, Princeton University, Princeton, NJ, 1956.
State-Resolved Rotational Relaxatlon of Methyl Fluoride in the Free-Jet Expansion of Methyl Fluorfde-Helium Mixtures C. Douketis, T. E. Gough, G. Scoles,* and H. Wangt Centre of Molecular Beams and Laser Chemistry,$University of Watterloo, Waterloo, Ontario N2L 3G1, Canada (Received: March 24, 1983)
State-resolved rotational relaxation studies in the free-jet expansion of CH,F-He mixtures are carried out by optothermal molecular beam infrared Stark spectroscopy. Terminal rotational temperatures for the J and K degrees of freedom have been measured and found to be different. Under typical expansion conditions the J degree of freedom cools approximately a factor of two more efficiently than the K one. The relaxation behavior of CH3F is compared to other small molecules subjected to similar expansion conditions and the resulting correlation is discussed.
Introduction State-selected rotational relaxation studies are needed for a better understanding of the interactions between polyatomic molecules and for the modeling of several important phenomena relevant to gaseous discharge lasers, reacting gaseous mixtures, and the conduction of heat in the gas phase. The relaxation is, of course, related to some perturbation of the equilibrium state of the gas and the nature of this perturbation can vary greatly depending on the particular experimental method adopted. At one end of the scale one can use shock waves or free Permanent Address: Dalian Institute of Chemical Physics, Chinese Academy of Sciences, People's Republic of China. *An interdisciplinary center whose members participate in the GuelphWaterloo (G-W) Program for Graduate Work in Physics, the G-W Centre for Graduate Work in Chemistry, and the G-W Surface Science and Technology Group.
0022-3654/84/2088-4484$01,50/0
jet expansions to produce, very rapidly, large changes in the translational temperatures and in so doing study by how much the other degrees of freedom are left lagging behind. In the case of shock waves, the perturbation is toward higher translational temperatures while, in free jets, the expansion cools the translational temperature of the gas at rates of the order of lo9 K s-'. On the other hand, sound absorption, optoacoustic, thermal conductivity, and thermal lensing experiments produce perturbations which are normally much smaller in magnitude and make the system oscillate around room temperature and standard pressure conditions. How and why free-jet expansions can be used to study molecular relaxation has been, and still is, one of the main themes of John Fenn's work'-4 and, therefore, it seems particularly appropriate (1) S.P. Tang, R. J. Gallagher, and J. B. Fenn, Entropie, 42, 51 (1971).
0 1984 American Chemical Society
State-Resolved Rotational Relaxation of Methyl Fluoride to choose this subject for our contribution to the present issue of The Journal of Physical Chemistry. Our own approach to the study of rotational relaxation in free-jet expansions was discussed recently in two paper^^)^ dealing with the relaxation of C O and HF. The state-sensitive detection technique adopted in our laboratory, and applied also in the present study, is based on the thermal detection, via a bolometer, of the energy deposited into the molecules when the molecular beam (formed after the free-jet expansion is completed) is crossed by the photon beam generated by an infrared laser.’ The system studied here, Le., a CH3F-He mixture, was chosen for three reasons. First, the v3 vibrational transition of CH3F (9.5 pm) is in close coincidence with the 9.4-pm band of the COz molecule so that the CH3F rovibrational transitions are accessible with a C 0 2 laser via Stark tuning. Secondly, CH3F is spectroscopically well characterized, particularly by nonlinears and molecular beam9 laser Stark spectroscopy. Finally, since CH3F is a prolate symmetric top which exhibits two rotational degrees of freedom, characterized by two moments of inertia, it is interesting to resolve their different relaxation behavior. The separation of the relaxation efficiency of two rotational degrees of freedom in the same molecule has been reported recently for the supersonic expansion of pure The results of that paper are, however, limited to relatively high temperatures with spectra which are not completely resolved and, very probably, affected by condensation phenomena which must occur at those pressure, concentration, and temperature conditions.
Experimental Section The molecular beam apparatus used in the measurements is a slightly modified version of that described in ref 5. The supersonic source is a room temperature 35-pm nozzle with stagnation conditions Pod (where Po is the stagnation pressure and d is the nozzle diameter) ranging from 0.5 to 20 torr cm. A 0.03-cm-diameter skimmer samples the expansion at a distance of about 1 cm from the nozzle where the beam has completely achieved terminal conditions. At 19 cm from the skimmer the molecular beam is crossed by radiation from a homemade C 0 2 laser whose output is chopped by a rotating mirror a t about 23 Hz. At the interaction region a Stark field was applied to bring the rovibrational transitions of CH3Fin coincidence with the C 0 2 laser lines. The Stark electrodes spacing was about 0.4 cm and a variable potential divider was included in the circuit so that the voltage could be internally calibrated with the resonance voltages reported in ref 8. Field strengths of 50 kV cm-’ could be easily obtained. The Stark field was measured to be stable and homogeneous to better than 1 part in lo3 and the experimental resolution was limited by this value and not by the laser frequency stability. At each resonance voltage the molecules in the beam are excited by the laser and take their new (modulated) energy content to a doped Si bolometer which operates at 2 K and is located 60-cm downstream of the interaction region. A typical laser-induced signal is of the order of 300 pV and is detected with a standard lock-in amplifier at the laser chopping frequency. The dc background generated by the continuous beam flux impinging on the bolometer is three orders of magnitude larger. The bolometer time constant is 0.005 s, the time constant of the signal integration 0.3 s, and the noise level with this time constant is about 2 pV. (2) R. J. Gallagher and J. B. Fenn, J . Chem. Phys., 60, 3487 (1974). (3) C. G . M. Quah, J. B. Fenn, and D. R Miller, Rarefied Gas Dyn.,ZZth, 2, 885 (1979). (4) S . P. Venkateshan, S. B. Ryali, and J. B. Fenn, J . Chem. Phys., 77, 2599 (1982). (5) D. Bassi, A. Boschetti, S. Marchetti, G . Scoles, and M. Zen, J . Chem. Phys., 74, 2221 (1981). (6) T. E. Gough and R. Miller, J . Chem. Phys., in press. (7) T. E. Gough, R. Miller, and G . Scoles, Appl. Phys. Lett., 30, 338 (1971). ,--. .
(8;s. M. Freund, G. Duxbury, M. Romheld, J. T. Tiedje, and T. Oka, J . Mol. Spectrosc., 52, 38 (1974). (9) C. Douketis and T. E. Gough, to be submitted for publication. (10) D. L. Snavely, S . D. Colson, and K. B. Wiberg, J . Chem. Phys., 74, 6975 (1981).
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4485 0.78
1.6
3.2
7.8 L
20
ELECTRIC FIELD (kY-CM’)
25
Figure 1. A laser Stark spectrum of several Q branch transitions of CH3F tuned into resonance with the P(18) line of the C 0 2 laser. The scan is linear. Nozzle stagnation conditions are indicated on the left of the figure where the product Pod (torr cm) is reported. Po is the stagnation pressure and d the nozzle diameter. The sensitivity of the detector was decreased, in the lowest trace, by a factor of 2.5 with respect to its value in the other three spectra. The Q(1,O) line is induced by the Stark field.
O0f
1
2
3
4
5
6I
7 I
LASER POWER ( W )
Figure 2. Bolometer signal, plotted as a function of the laser power, is shown for several CH,F rovibrational transitions.
Results and Discussion An example of a spectrum which illustrates a number of important features of this work appears in Figure 1. When the P( 18) line of the COz laser is used several CH3F transitions are available when the Stark field is tuned from 0 to 50 kV cm-’. Here, in a range from 20 to 25 kV cm-’, all accessible AM components of the Q(2,l) and the Q(3,3) transitions appear. Also, one component of the Stark-field induced Q( 1,O) transition appears at about 20.6 kV cm-’. The relaxation behavior of CH3F in the expansion as a function of nozzle conditions is clearly evident in the spectra of Figure 1. As the stagnation pressure is increased from top to bottom the population distribution over the rotational states shifts to lower energies so that the intensities of transitions originating from higher rotational levels decrease while the intensity of transitions from the lower levels increase. This is particularly obvious in the case of Q(3,3) where at P& = 7.8 torr cm we find negligible population in the (3,3) level of u = 0. The major experimental difficultly in the relaxation studies mentioned previously5was the measurement of the laser intensity a t the laser-molecular beam crossing. This problem is most efficiently removed if (a) the laser intensity is sufficient to saturate
4486 The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
Douketis et al.
TABLE I: Laser-Induced Bolometer Signals for Expansion of (1 i: 0.1)% Methyl Fluoride in Helium at Pod = 3.2 and 11 torr ,ma bolometer signal, yV J,K b ,Trot, K P o d = 3.2 P o d = 11 ~
~~
0 2.45 7.35 8.62 13.52 20.87 32.01 39.37 70.19 a
83.0 63.2 26.0 64.0 31.0 11.0 8.2 3.7 1.6
I I TORR-CM
410 155 29.8 275 42.0 7.25 2.8
’
Icz1
Experimental standard deviations ran e from i 5 fiV for the Initial rotational level.
larger signals to i 2 yV for the smallest.
the driven transition and (b) the electric field homogeneity is not good enough to allow coherence effects to be observed. If these two conditions are met bolometer signals are, at large laser powers, independent of laser power. Figure 2 shows saturation curves for several CH3F transitions. We operate our laser well within the saturation regime so that we do not have to calibrate the laser power and we are not sensitive to differences in transition probabilities. The excellent agreement between the signals for q(2,l) and P(2,l) recorded under saturating conditions indicates the precision with which rotational level populations can be measured in the present experiments. In Table I we present the measured amplitudes of the signals for those levels whose transitions may be tuned into resonance with CO, lasing transitions using Stark fields of less than 50 kV cm-’. All measurements were made on a (1.0 f O.l)% mixture of methyl fluoride in helium; results for two nozzle stagnation pressures are shown. Experiments were also performed on a more dilute (0.5 f 0.1)% mixture of methyl fluoride in helium. It was found that the intensities of the signals for the 0.5% mixture plotted linearly against those of the 1% mixture. At Pod = 3.2 torr cm such a plot had a slope of 0.59 and a correlation coefficient of 0.9959. At Pod = 11 torr cm the slope was 0.54, and the correlation coefficient 0.9988. We conclude that the terminal distribution of methyl fluoride between its rotational levels is, in our experiments, not a function of the concentration of methyl fluoride. Therefore, in any simulation of the expansion process (e.g., see ref 6 and 1 1 ) it should only be necessary to consider methyl fluoride-helium, and not methyl fluoride-methyl fluoride, interactions. The intensity data of Table I is cast in the form of Boltzmann plots in Figure 3. In order to produce these plots from the experimental signals one requires the rotational energy and statistical weight of the levels probed by lase Stark spectroscopy. The former in the case of CH3F is given by
E(J,K) = B ( J
+ l)J + (A- B ) p
(1)
where B = 1.226 K (ref 8) and A = 7.391 K (ref 12). We take (1) to be an equality since for this system the centrifugal distortion terms contribute less than 0.001%. Levels with K # 0 are twofold degenerate, this degeneracy corresponding to the positive and negative values of k , while levels with K = 3n ( n integer) have an additional twofold degeneracy because the total spin I = 3 / 2 as opposed to Z = 1/2 for the E levels. Levels with K # 0 exhibit a first-order Stark effect and therefore have their 2J + 1-fold M degeneracy completely removed. Levels with K = 0 show only a second-order Stark effect; all K = 0 transitions studied by us changed M according to M = 0 f l . The calibration of the intensities of such transitions against those of K # 0 levels is the subject of a separate p~b1ication.l~
-
(11) H. Rabitz and S. H. Lau, J . Chem. Phys., 63,3532 (1975). (12) C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy”, McGraw-Hill, New York, 1955.
-64
0
, Zb IO Eaor
30
40
5b
I
60
-6-
?O
0
IO
20
30 EWT
(K)
40
50
(K)
Figure 3. State population Boltzmann plots are shown under two nozzle stagnation conditions.
TABLE 11: Terminal Translational and Rotational Temperatures in Kelvin Obtained for Expansions of 1% CH,F in Helium through a 0.03-cm Nozzlea
~~
3.2 11.0
5.8 2.4
6.2 2.8
7.0 3.4
9.4
13.1
5.2
The translational temperatures were calculated via the isentropic equation, and the evaluation of the rotational temperatures is discussed in the text. a
In a Boltzmann plot states which remain in thermal equilibrium with one another, throughout the expansion process, lie on a common straight line, whose slope establishes a reciprocal temperature. The A ( K = 3n) and E ( K # 3n) modifications of methyl fluoride are not expected to remain in thermal equilibrium throughout the rapid supersonic expansion and, therefore, should not lie on a common straight line. However, if, as might be expected, each modification independently reaches the same final rotational temperature the Boltzmann plot should consist of two parallel straight lines. Inspection of Figure 3 shows that these expectations are only partially correct. The figure shows additional structure in that a separate straight line is obtained for each set of levels with a given K and varying J. These various lines are parallel to one another, within our experimental error. To interpret these observations assume, for the moment, that the A and E symmetry modifications are not interconverted in the expansion, and consider the E species K = 1 and 2 in Figure 3). The common slope of the lines for K = 1 and 2 may be used to define a temperature, TJ,which is to be interpreted, in the spirit of the sudden freeze approximation, as the terminal rotational temperature controlling the relative populations of the various J levels within a given K stack. This terminal T J is equal to the translational temperaure at that point in the expansion where rotationally inelastic collisions cease. In Figure 3 the K = 2 line lies above the K = 1 , implying an overpopulation of the K = 2 levels relative to the predictions obtained from the experimentally determined population of the (1,l) level and T P This overpopulation indicates that in some point in the expansion before the terminal TJ was achieved the K = 1 and 2 levels ceased to interconvert via inelastic collisions. After this “K-freezing” the total population of a particular K stack remains constant, so that TKmay be extracted from our experiments by measuring the ratio of the total terminal populations of the K = 1 and 2 stacks, and evaluating the rotational tem(13) A. Adam, T.E. Gough, and A. Lewin, to be submitted for publication.
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4487
State-Resolved Rotational Relaxation of Methyl Fluoride
#
3.2 TORR-CM
/
TROT (K)
O-l 0
5 IO ROTATIONAL CONSTANT
15
20
(CM-I)
Figure 4. The reduced rotational temperature is plotted as a function of rotational constant for several molecules. All points correspond to dilute mixtures in helium at Pod = 3.2 torr cm. The translational temperature shown is calculated from the isentropic equation. The data for CO are obtained from ref 5 and that of HF and C02 from ref 6 and 14.
perature at which this ratio obtains. In Table I1 we summarize the information on Tj, TK, and the terminal translational temperature calculated from standard isentropic equations. With the straight lines drawn in Figure 3 as extrapolation guides, it is possible to measure the ratio of populations of the A and E symmetry modifications of methyl fluoride. For a bulk gas at room temperature this ratio is unity, while in the molecular beam we find a 5% overpopulation of the E species. This deviation from unity is not, in our view, outside the limit of experimental error. Therefore, we conclude that, not surprisingly, the A and E modifications of methyl fluoride do not interact throughout a supersonic expansion in helium. In an attempt to compare the results obtained here with the results of previous studies on other molecules under similar expansion conditions we show in Figure 4 a plot of the rotational temperatures in the beam as a function of rotational constants. In the case of CH3F we take B to be the rotational constant corresponding to the J temperature and A - B for the K temperature. To a good approximation the behavior in the figure is linear so that a scaling law can be obtained. This is
T,,,/T0 = aB
+ 0.017
( B in cm-')
(2)
where a = 5.6 X cm, with all data taken at Pod = 3.2 torr cm. It is remarkable that both CH3F temperatures fall right on the line. In order to test our results further rotational relaxation data for molecules such as HCl, DC1, NH3, H2S, and CH4 in He expansions would be useful since they have moments of inertia that would fill in the large gap between CH3F and HF in Figure
4. Rotational relaxation efficiency has been quantified in a number of ways. Most commonly workers have used the concept of a rotational relaxation number, Zrot,defined as the product of overall (14) R.Miller, Ph.D. Thesis, University of Waterloo, Waterloo, Canada 1980.
collision frequency and relaxation time. Equivalently Z,, can be expressed as a ratio of cross sections: G o t = uref/brot (3) where urOtis the rotationally inelastic cross section and urefan average cross section that determines the collision frequency. The temperature dependence and the rotational state dependence of uIotcan be, for simpler molecules, calculated and taken into account.5 However, here we will be concerned only with a ZIotthat is averaged over all T and (J,K) states. The drawback in obtaining Z,,, from (3) has to do with the somewhat arbitrary choice of a reference cross section, urep However, in the case of the CH3F system we can quote a ratio of relaxation numbers for the two rotational degrees of freedom which removes the influence of the reference cross section completely. In terms of rotational temperature this ratio can be expressed as
~ r o t v r o ~ 9 -=1
(To - TK)(TO - TJ)-'(~J)'''y-"(TK)'-l TJ"5T K
(4)
for To>> TJ,TKand y = 1.66. Therefore, at Pod = 3.2 torr cm we have the result that cr,,tJ(CTrotX)-l
= 2.1
While a t Pod = 11 torr cm the same ratio is 1.9. Calculations of these ratios are feasible with present days infinite order sudden and close coupling programs for rotationally inelastic scattering and the results should not be too sensitive to errors in the assumed potenital energy surface provided that all important components of the potential are included.
Conclusions Having assumed Boltzmann behavior, two terminal rotational temperatures in the free-jet expansion of CH3F in He have been found for the J and K distributions. The former is about a factor of 2 more efficiently cooled than the latter. These temperatures and TI,, for simple linear molecules, under similar expansion conditions and analyzed with the same assumptions, were found to correlate linearly when expressed in terms of rotational constants. Therefore, in this relaxation environment for a molecule diluted in H e the rotational cooling is largely governed by the rotational constant and it is not strongly influenced by the details of the intermolecular potential. In fact, in the work of ref 5, where a modeling of the relaxation was performed for CO in He, terminal rotational temperatures were found to be relatively insensitive to the choice of the interaction potential. Indeed differences of a factor of 3 in the well depths produced only relatively modest terminal temperature changes. A similar modeling of the relaxation for CH3Fin He is currently under consideration but it is by no means trivial. Finally, let us express the hope that there is more truth in our results than there is in the rumor of John Fenn's retirement which we will have to see with our own eyes before we will be led to believe in it. Registry No. CH3F, 593-53-3; He, 7440-59-7.
