Intermolecular Transfer of Vibrational Energy BY J. D. LAMBERT, A. J. EDWARDS, D. PEMBERTON AND J. L. STRETTON Physical Chemistry Laboratory, University of Oxford Received 15th January, 1962 Measurements of ultrasonic dispersion have been made in mixtures of pairs of polyatomic gases, which have near-resonant vibrational energy levels. Single dispersion was observed for mixtures of CH30CH3 with CC12F2, CH3Cl and SFs. Double dispersion was observed for mixtures of C2F4 with CHF3, CF4 and SF6 and for a mixture of C2I& with SF6. All mixtures showed a linear dependence of reciprocal relaxation time on molar composition, extending over the whole com- position range. The results are interpreted on the hypothesis that rapid transfer of vibrational energy from molecule A to molecule B in a “ complex collision ” is followed by deactivation of B in col- lisions involving the normal transfer between vibrational and translational energy.There is little experimental evidence about the direct transfer of vibrational energy between molecules in collision in the gas phase. Measurements of vibra- tional relaxation times in pure gases by ultrasonic and other techniques have been mainly concerned with vibrational-translational energy transfer. On theoretical grounds it would appear unlikely that there is any appreciable hindrance to a reson- ance transfer of a quantum of vibrational energy from one molecule to another in a homo-molecular collision, and chemical spectroscopic evidence 1 confirms that equilibrium distribution of vibrational energy between molecules of a single species is very rapidly attained.The role of the “ complex collision ”, in which a vibrational quantum is transferred froin molecule A, plus or minus the necessary increment of translational energy to excite a different vibrational quantum of molecule B, has been investigated by ultrasonic techniques only for rare cases, where energy transfer between the different vibrational modes of a single species of polyatomic molecule involves a measurable relaxation time (CH2C12,2 and SO2 3). The complex collision can also play an important role in the overall vibrational- tianslational energy transfer process in heteromolecular collisions. The possible energy transfer processes are here : A* + A-+A+A (vib-+trans) (1) A*+B-+A+B (vib-trans) (2) A* +B-+A+B* (vibjvib) (3) B*+B-+B+B (vibjtrans) (4) B* +A-+B+A (vib-trans).(5) If A is a relaxing molecule with a low transfer probability, and B a non-relaxing molecule or a molecule with a much higher transfer probability, a rapid process (3) may be followed by a rapid process (4) or (5), giving rise to a greatly enhanced vibration-translation transfer probability for molecule A. Such a mechanism may be one of the factors responsible for the powerful “catalytic” action of certain foreign molecules in promoting energy transfer.4 Rapid vibration-vibration energy transfer would be expected to occur between molecules with near-resonant vibra- tional frequencies. This paper describes ultrasonic dispersion measurements on a number of mixtures of polyatomic gases, which fulfil this criterion to varying degrees.6162 VIBRATIONAL ENERGY TRANSFER The usual expression for the composite relaxation time 18 of a mixture of a re- laxing gas A with another gas B, where only processes (1) and (2) operate, is : where x is the mole fraction of component B, and BAA and PAB are the relaxation times corresponding to the transfer probabilities in processes (1) and (2) respectively. When complex collisions occur, the validity of this expression will depend on the relative rates of the three processes, (3). (4) and (S), iiou7 involved in the vibrational deactivation of A. If (3) or (5) is the late-controlling step, eqn. (6) will still be valid, giving a linear dependence of 1/P on x. If (4) is the rate-controlling step, this will no longer be so, since this rate is proportional to x2.Such non-linear dependence has been demonstrated for the variation of the relaxation time of 0 2 (v = 1580 cm-1) with small additions of H20 (v2 = 1595 cm-I), which was in- vestigated by a shock-tube method.5 (Linear dependence was found for mixtures of 0 2 with D20 (v2 = 1178 cm-I), where no vibration-vibration transfer would be expected, so that only processes (1) and (2) operate.) Since the detailed dependence of relaxation time on composition is an important factor in elucidating the mechanism of energy transfer, all the systems described below have been investigated over as wide a range of composition as possible. For mixtures where both components show relaxation which is experimentally measurable, a double relaxation phenomenon would be expected 4 with two distinct relaxation times, PA and PB, corresponding to the vibrational deactivation of A and B, respectively, such that 1 1-x x + -Y P A BAA PAB _ - -- and 1 x 1-x -- --+----, PB PBB PBA (7) where PBB and PBA are relaxation times corresponding to processes (4) and (5), respectively.It is difficult to obtain experimental resolution of relaxation times differing by a small factor, or in systems where the specific heat contribution of one component heavily outweighs that of the other, and previous attempts to demonstrate this phenomenon have not been conclusive.6?7 Double relaxation is established for four of the systems described below. EXPERIMENTAL Measurements were made in two acoustic interferometers, which have already been described in essentials.8 The lower frequency interferometer was used with a quartz oscillator of frequency 200kc/sec and was operated at room temperature.The higher frequency interferometer was used with a 4 Mc/sec quartz crystal, with nodal mounting of the type described by Arnold, McCoubrey and Ubbelohde,g driven by a Clapp oscillator, and was operated at 25°C. Measurements were made at varying pressures between 0.1 and 1 atm. Gas mixtures were made by techniques described previously,lO and were intro- duced to the evacuated interferometer by a Topler pump. The compositions of mixtures were checked from time to time by mass spectrometer analysis or by a Janak gas chrom- atography apparatus. Gas-imperfection corrections for pure gases were made as described previously, using data listed for each gas below.The second virial coefficient of a mixture was calculated from the expression, B = (1 - x ) ~ B A + 2 ~ ( 1 -x)BAB + x 2 B B .J . D. LAMBERT, A . J . EDWARDS, D. PEMBERTON AND J . L. STRETTON 63 BA and BB are the second virial coefficients of the pure components. BAB, representing the heteromolecular interaction, was calculated by the methods of Hirschfelder, Curtis and Bird 11 where appropriate data were available, or by the methods of Guggenheh and McGlashan.12 Relaxation times were calculated by fitting theoretical dispersion curves to the experimental points, which were usually at least 15 to 20 in a set. The necessary specific heats were calculated from the spectroscopic data listed below, except for CH30CH3, where the vibrational assignment is uncertain, and calorimetric values 13~14 were used.The static specific heat of a mixture was taken as and the relaxing contributions calculated from the appropriate vibrational frequencies. (1 - x>c; + xc; , MATERIALS C2H4 : two different specially pure cylinder samples were used, which showed relaxa- tion times in excellent agreement. Gas imperfection, Berthelot equation. Spectroscopic data.15 SF6 was obtained from a cylinder. Mass-spectrometer analysis showed no detectable impurity. Gas imperfection.16.17 Spectroscopic data.18 C2F4 was a sample supplied by I.C.I. Mass spectrometer analysis showed no de- tectable impurity. Gas imperfection.24 Spectroscopic data.25 CHF3 was a sample used previously.19 Gas imperfection.19~ 20 Spectroscopic data.21 CF4 was a sample used previously.19 Gas imperfection.22 Spectroscopic data.23 CH30CH3 was obtained from a cylinder.Mass spectrometer analysis showed no CH3Cl was obtained from a cylinder specified to contain less than 200 p.p.m. by weight detectable impurity. Gas imperfection, Berthelot equation. Spectroscopic data.26 dimethyl ether as impurity. Gas imperfection.27~ 28 Spectroscopic data.23 CC12F2 impurity. M 4 0 - --- 8 - & 0 L? was obtained from a cylinder specified to contain less than 70 p.p.m. by volume Gas imperfection.29 Spectroscopic data.30 I I I I L 6 . 0 6.5 7.0 7.5 loglo c f p ) (c sec-1 atm-1) FIG. 1.-Dispersion curve for mixture of CF4 with 60.2 mole % c2F4. 0, experimental values ; -- , theoretical curve for double relaxation process ; P(CF4) = 5.2 x 10-8 ; P(C2F4) = 1.42 x 10-8 sec ; - - -, theoretical curve for single relaxation process, involving total vibrational energy of both molecules.RESULTS The reciprocal of the measured relaxation time is shown plotted against molar For the mixtures involving composition for the various mixtures in fig. 2-8.64 VIBRATIONAL ENERGY TRANSFER CH30CH3, whose relaxation time is so short as to lie outside the range of experi- mental measurement, only single relaxation was observed, corresponding to the VibrationaI relaxation of the second component. For all the other mixtures, double relaxation was observed : a specimen dispersion curve for CF4+ C2F4 is shown in - 800 Y ! 0.5 1.0 mole fraction SF6 c2H4 SF6 FIG.2.-Reciprocal relaxation times for CzHd+ SFs mixtures. mole fraction C2F4 CHF3 C2F4 FIG. 3.-Reciprocal relaxation times for CHF3 + C2F4 mixtures. fig. 1. Separate relaxation times relating to each of the components were calculated 2,s described previously.3 All the mixtures show a linear plot of l/p against x, in accordance with eqn. (7) and (8), enabling values of PA*, PAB, ,!?BB and PBA to be calculated. The relaxation times for the pure gases are in general agreement with previous measurements in this and other laboratories. The value for CH3C1 differsJ . D. LAMBERT, A. J . EDWARDS, D. PEMBERTON, AND J . L STRETTON 65 from that of Fogg,lg who used a sample known to contain 2 % of CH30CH3; it is in good agreement with measurenients of Edmonds and kamb.31 Pure CH30CH3 appears to be showing incipient dispersion at the highest value of frequencylpressure attained (107.65 c sec-1 atm-1) and a minimum mole fraction C2F4 cF4 c2F4 FIG.4.-Reciprocal relaxation times for CF4+C2F4 mixtures. mole fraction C2F4 value of the 800 ,600 7 i3 U > ,400 200 FIG. 5.-Reciprocal relaxation times for SF6+ c2F4 mixtures. possible relaxation time is estimated as 2.5 x 10-9 sec. Fig. 2-5 are accompanied by energy level diagrams to show the possible vibration-vibration transfers. The upper harmonics of energy levels are included in the diagrams where appropriate as dotted lines. C66 VIBRATIONAL ENERGY TRANSFER DISCUSSION It is convenient to discuss first the mixture, C2H4 + SF6, where complex collisions are unlikely to play an important role in facilitating energy transfer.The relaxation times for pure C2H4 and SF6 do not differ greatly, and it will be seen from fig. 2 that '4 I J 0.5 I. mole fraction CH30CH3 = -t.- FIG. 6.-Reciprocal relaxation times for CClzF*+ CH30CH3 mixtures. mole fraction CH30CH3 n 400 ----- =%=Ox - 1 CH3Cl CH30CH3 FIG. 7.-Reciprocal relaxation times for CHjCl+ CH30CH3 mixtures. C2H4 + SF6 collisions are of roughly the same efficiency as C2H4 + C2H4 collisions in deactivating C2H4, while substantially more effective than SFs + SF6 collisions in deactivating SF6. The energy level diagram shows that there is an easy transferJ . D. LAMBERT, A. J . EDWARDS, D. PEMBERTON AND J . L. STRETTON 67 between the 810 cm-1 mode of C2H4 and the 775 cm-1 mode of SF6, but this could not be responsible for increasing the efficiency of deactivation of SF6 in heteromole- cular collisions above that of the self-deactivation of C2H4.In terms of the reaction scheme given in eqn. (1) to (5), however rapid process (3) may be, the overall relaxa- tion rate cannot be faster than either process (4) or process (5). The explanation of the enhanced heteromolecular collision efficiency for SF6 would seem to lie in the normal vibration-translation transfer by process (2) being facilitated by the rela- tively low mass and steep intermolecular potential of the C2H4 molecule. The observed increase in collision efficiency by a factor of 5-9 is possible on these grounds. 0- 5 mole fraction CH3OCH3 FIG. 8.-Reciprocal relaxation times for SF6+ CH30CH3 mixtures.Similar theoretical considerations predict that the efficiency of deactivation of C2H4 in collisions with SF6 would be lower than in self-collision. The fact that the experimentally measured efficiency shows little change may be interpreted in terms of a complex collision involving very rapid energy transfer from the 810 cm-1 mode of C2H4 to the 775 cm-1 mode of SF6, followed by deactivation of the SF6 molecule by a combination of processes (4) and (5). This might be expected to lead to a non- linear plot in fig. 2, as the more efficient heteromolecular collisions (process (5)) would play a larger role in the mixtures richer in C2H4. Unfortunately, the close- ness of the two relaxation times makes it impossible to resolve them for the mixtures richer in SF6 and to decide whether the observed line is, in fact, curved.The little change observed in the heteromolecular deactivation efficiency of C2H4 is, however, in accord with this closeness. All the remaining mixtures show a much larger difference between the relaxation times of the two pure components. If A is the gas with the longer relaxation time (lower efficiency of deactivation), the values of PAB are consistently smaller than those of PAA to an extent which is unlikely to be accounted for by difference in molecular mass or repulsion potential. Indeed, for two mixtures, CHF3+C2F4 and CF4+ C2F4, the effect of mass would operate in the reverse direction. The energy level diagrams show that, in all cases, easy vibration-vibration transfers are available, so that a mechanism involving complex collisions seems likely to be responsible for the enhanced efficiency of heteromolecular collisions in deactivating A.For the three68 VIBRATIONAL ENERGY TRANSFER mixtures where the relaxation time of the other component B can be observed, the values of ~ B A fall a little more than those of PBB. (The appearance of fig. 3, 4 and 5 is misleading in this respect, as the ratio of the two l/p intercepts, which is the relevant factor here, is nut proportional to the slope of the line, and the variation in relaxation time of component B appears exaggerated in comparison with that of A.) This means that heteromolecular collisions are slightly less efficient than self- collisions in deactivating B ; the effect is of a size which can probably be accounted for by differences in intermolecular repulsion potential and/or mass.There is no case for invoking complex collisions to account for a decrease in efficiency of deactivation, and a simple vibration-translation transfer (process (5)) is an ap- propriate mechanism for the deactivation of B. The values of PYA given in table 1 are calculated on the assumption that the whole of the vibrational energy of B relaxes through the lowest mode by this mechanism : p7A = (C7/Cr)pBA, see below. It now remains to discuss the details of the complex collision mechanism re- sponsible for the heteromolecular deactivation of A. The three mixtures involving C 2 F 4 as the B component (fig. 3, 4 and 5) are the easier to interpret, and may be TABLE 1 RELAXATION DATA - 35 CHF3 C2F4 0.519 0.164 0.119 2.49x 10-7 1.61 X 10-9 2.30X 10-9 1.84X 10-9 1.4 1-25 0 CF4 C2F4 0.584 0.164 0.104 5.14X 10-7 1.61 X 10-9 2.46X 10-9 2.97X 10-9 1-5 0.53 - 5 SF,j CzF4 0.310 0.164 0.070 1.96x 10-7 1 6 1 X 10-9 3.20X 10-9 10.3X 10-9 2.0 0.31 +17 - 1.79X 10-8 - c2H4 SF6 0.277 0.310 - 6.37X 10-8 1.96X 10-7 - CCIzF2 CH30CH3 0.189 0.249 0.1 14 1.69 X 10-8 < 6.3 X 10-10 4.26X 10-10 - >0*7 - - 1 1 CH3CI CH30CH3 0438 0.249 0.204 9.41 x 10-8 < 6-3 x 10-10 1-57X 10-9 - >2*5 - + 18 SF6 CH3OCH3 0.310 0.249 0.086 1.96x 10-7 < 63X 10-10 1.13X 10-9 - >1.8 - - 35 considered together.For the C H F 3 + C 2 F 4 mixture, there is a resonance vibration- vibration transfer between two 507 cm-1 modes, while for the S F 6 + C 2 F 4 and C F 4 f C 2 F 4 mixtures the values of Av for vibration-vibration transfer are only 17 and 5 cm-1 respectively.Process (3) would therefore be expected to be extremely rapid in all three cases, so that the rate-controlling step would be the faster of (4) or (5). Predominance of (4) might be expected to give rise to a non-linear plot of 1/p against x, but the measured rates of (4) and (5) differ so little that the observed linear dependence seems justified if both processes are operating. (This is in sharp contrast to the 0 2 + H 2 0 mixture mentioned above,s where the efficiency of vibra- tional deactivation of H 2 0 in self-collisions is some 103 times that in H 2 0 + 0 2 collisions.) If the mechanism is as suggested, the measured relaxation time PAB corresponds to relaxation of the whole of the vibrational specific heat contribution of both molecules (C$+ Cs), via the lowest mode of molecule B, whose contribution is C?.It has been shown that for self-collision of polyatomic molecules, where the whole of the vibrational energy of the molecule relaxes through the lowest mode, the effective relaxation time of this mode is given by provided it is assumed that energy transfer between modes is very rapid.32 ByJ . D. LAMBERT, A. J. EDWARDS, D. PEMBERTON AND J. L. STRETTON 69 analogy the effective relaxation time for energy transfer via the lowest mode of B in the heteromolecular process should be given by and the value of pfB would then be expected to lie between byB and pBA, since processes (4) and (5) control the rate. Values of fltB calculated by eqn.(9) are shown in table 1, together with values of p7B and p:A and the ratios of fltB to these. It will be seen that for the CF4fC2F4 and SF6fC2F4 mixtures the value of fl+B lies in between PyB and fl?* in accordance with expectation. For the CHF3+C*F4 mixture, fl?B lies slightly above both flyB and flyA ; in view of the complexity of the mechanism and the many assumptions involved, this is not unsatisfactory agreement. Interpretation of the remaining three mixtures (fig. 6, 7 and 8) is less satisfactory, as the full dispersion zone of the B component CH30CH3 is not accessible to the present apparatus, and only the relaxation of the A (less efficient) component can be observed in the mixtures. The observed values of f l ~ ~ are again much smaller than PAA, to an extent which is unlikely to be due to the simple effect of mass or inter- molecular potential.The energy-level diagrams again show easy vibration- vibration transfers, and the simplest hypothesis is that the same mechanism applies as for the C2F4 mixtures discussed above. Table 1 shows values of PtB calculated on the same basis, and their ratios to the minimum likely value of pBF. It will be seen that the ratios again approximate to unity, and, since the value of p”,” is only approximate and it is unlikely that PTA will be much larger or smaller, this can be regarded as satisfactory agreement. It is, of course, possible that the self-deactiva- tion of CH30CH3 (process (4)) is faster than the complex transfer, so that the latter is rate-determining. But, in view of the fact that the lowest fundamental of CH30CH3, 164 cm-1, is considerably larger than Av for any of the complex col- lisions involved, this seems unlikely, and the mechanism proposed above has the merit of giving a self-consistent interpretation of the results for both sets of mixtures.An attempted interpretation on the assumption that the complex collision is rate- determining leads to mutually contradictory results for some of the mixtures, and there is no obvious physical reason for applying it in some cases and not in others. The authors are grateful to Dr. C. J. Danby for making mass spectrometer analyses of mixtures, to Dr. J. L. Stewart for help on constructional details of the 4 Mclsec interferometer and to I.C.I. Plastics Division for generously providing the tetrafluoroethylene used.A. J. E. and D. 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