J . Chem. SOC., Faraday Trans. 1, 1984,80, 769-780 Average Vibrational Energy Transfer during a Single Collision of Excited Molecules with Heat-bath Molecules BY IZHACK OREF* Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel AND B. SEYMOUR RABINOVITCH Department of Chemistry BG10, University of Washington, Seattle, Washington 98195, U.S.A. Received 10th May, 1983 A collisional energy-transfer probability of the form B(E+ AE) B'(E') B'(E - AE) dE' 0 is assumed where B(E) is the Boltzmann distribution and A E is the incremental energy transferred and which can display negative as well as positive values. Single collisions between vibrationally excited substrate molecules with heat-bath molecules are considered. The depen- dence of the average energy per collision transferred up, down and overall on the initial energy content, on the temperature and on the size of the bath and substrate molecules is calculated and compared with experimental data in the literature.Vibrational-relaxation and energy-transfer studies are of current interest in photo- physics and molecular dynamics. At low levels of excitation energy transfer between polyatomic molecules by resonant vibrational-vibrational (V-V) energy transfer is frequently the most important mechanism. Studies have been done in the So manifold using single-mode excitation by CO, laser1-' and shock-tube heating.8qg In the S, manifold relaxation studies were made by tuned laser excitation followed by resolved S, + So fluorescence."+l* At high levels of excitation, recent photophysi~al,l~-~' multiphoton excitationl8.l9 and shock-tube studies20 support the finding from chemical- activation studies,' that vibrational energy transfer by highly vibrationally excited polyatomic molecules takes place in the gas phase on virtually every collision. For weak colliders an important mechanism is vibrational-translational, rotational (V-T, R) transfer. The small contribution of the latter processes can be judged from the small values of the collisional efficiencies of the rare gases and diatoms and weak colliders in general. The pragmatic numerical value of the average energy which is transferred depends on the nature and energetics of the collision partners as well as on the collisional transition probability model which applies to the system.In a thermal system at equilibrium, the average amount of energy gained by the substrate molecule equals the average and the equilibrium assumption forces the distribution to remain Boltzmann after any (statistical) sequence of collisional events. In a unimolecular reacting model system at high pressures, the Boltzmann distribution is essentially maintained over all energy space. At low pressures, and for the case Eo % RT, an operational definition of a strong collision is that the Boltzmann vibrational distribution is maintained up to E,. The above definitions are used in conjunction with weak-collision models to obtain 769770 VIBRATIONAL ENERGY TRANSFER the degree of ‘weakness’ of a collision, i.e. the collisional efficiency relative to strong collision. There are several empirical weak-collider energy-transfer models,21* 28+ 27 such as exponential or Poisson-analytic forms.These models are used to evaluate the average vibrational energy transferred to the substrate, (AE, ) i.e. up transition, or from the: substrate, (AE,) i.e. down transition, or the overall average, (AE). The conservation relations are maintained by imposing detailed balance. The models are helpful in the correlation of experimental data. In a non-equilibrium (frequently reacting) system the situation is more complicated. Such a system may correspond to chemical, photochemical28* 29 and l a ~ e r ~ ~ - l ~ activation of reaction. Here a molecule excited to a high vibrational level is allowed to collide with a thermal heat-bath molecule.In the process a quantity of energy is transferred. The <AE) transferred depends not only on the energy content of the reactant molecule and the temperature of the bath but on the energy-transfer probability model which is used. Detailed balance is not maintained in the non-equilibrium system, and an alternative requirement to the conservation relation is that a non-reacting system reaches a Boltzmann distribution after a sufficient number of collisions. The purpose of the present work was to evaluate (AE,), (AE,) and ( A E ) in a non-equilibrium system with the use of a proposed model for strong-collision energy-transfer cases which allows the approach to the vibrational equilibrium condition to be attained more readily. Such a model applies in a number of experimental cases (see later), and the detailed mechanism of V-V, T, R need not enter explicitly. For the purpose of gaining an understanding of how the size and temperature of the heat bath and the size and energy content of the excited molecule affect the magnitudes of the various (AE), classical densities may be used.This does not detract from the general conclusions since for this purpose the behaviour of a real molecule may be simulated by s classical oscillators. The results are compared with some existing experimental data. PROBABILITY TRANSFER RELATIONS Consider a dilute system of substrate molecules all excited to a single value of internal energy, E, in a bath gas of temperature T. The probability of transferring an amount of energy from the excited molecule to the bath P(E, AE) can be evaluated in the following manner.25 When a collision takes place between a substrate molecule and a bath molecule an amount of energy AE is exchanged in the process.The probability that any molecule will start at Ei and reach the Ei + AE level while its partner will start at Ej and end at the Ej-AE level is F(Ei, E;, AE) = B(E,) B(E, + AE) B(E;) B(E; - AE). (1) + AE is for an up transition and - AE is for a down transition; B(E) is the Boltzmann distribution and the primed quantities indicate the bath molecules. The first term in eqn (l), B(E,), denotes the probability of obtaining a molecule in state Ei, while the rest of the expression is the transition probability, P(E,, AE), from state Ei to state Et+AE.B(E) has the form B(E) = ES-lexp(-E/RT)/(s- l)!(RT)s (2) and for the bath molecule s’ replaces s in eqn (2). The use of s does not detract from the generality of the exposition. When the results of the calculations are compared with experiments, the effective value of s’ is calculated from the value of the average energy as described later.I. OREF AND B. S. RABINOVITCH 77 1 The probability that a substrate with initial state i will exchange a given AE regardless of the initial state j of the colliding bath molecule is P(Ei,AEU)dAE = B(Ei+AE)dAE B’(E;)B‘(E;-AE)dE;/Q (3) P( Ei, AED ) dAE = B( Ei - AE) d AE (4) E E for the substrate up transition, and B’(E;) B’(E; + AE) dE;/Q for the down transition. AE is the lower limit in the integral expression in eqn (3) since the bath molecule is losing energy and has to have at least energy E’ equal to the amount transferred AE.In eqn (4) the lower limit is 0 since the bath is gaining energy and therefore can have any value. The value of the normalization parameter Q’is obtained from the conservation of probability relation 00 Q-l dAE[P(Ei, AE)] = 1 ( 5 ) -00 Detailed balance follows naturally from the form of eqn (3) and (4). For example, the equality B(Ei) P(Ei, AEU) = B(Ei+AE) P(Ei, AED) (6) is obtained by making the transformation E” = E’ - AE and substituting it in eqn (3) and replacing Ei by Ei+AE (the initial state) in eqn (4). It should be stressed again that detailed balance obtains only in an equilibrium (thermalized) system. The average quantities which are sought in this work are I (AE, ) = JOm d(AE) AE P(Ei, AE,) (AE,) = J” d(AE)AEP(E,,AE,) -00 (7) where P(Ei, AEu) and P(E,, AED) obey the normalization condition of eqn (5).RESULTS The form of the collisional probability, eqn (3) and (4), for classical oscillators is given in fig. 1 for four values of the internal energy of the reactant molecule with s = s’ = 15 and T = 1000 K. The normalized curves have a regular quasisymmetrical shape. The principal features are as follows. (a) The location of the maximum of the probability curve depends on the internal energy of the reactant molecule; the lower the value of E the higher the value of AE at which the probability curve peaks. (b) The higher the value of E, the broader the peak; at E = 0 kcal mol-lt the width at half height is ca.11 kcal mol-l while at E = 60 kcal mol-1 it is ca. 21 kcal mol-l. (c) The average energy transferred up is 21.6 kcal mol-l for the E = 0 kcal mol-1 case and down is - 18.8 kcal mol-l for the E = 60 kcal mol-1 case, i.e. the value of I(AE)I has to do with the location of the maximum and not with its width. (d) The probability t 1 kcal = 4.184 kJ.772 VIBRATIONAL ENERGY TRANSFER -40 -30 -20 -10 O 10 20 30 40 AElkcal mol-’ Fig. 1. Collisional energy-transfer probability P(E, AE) plotted against AE at 1000 K for reactant and bath molecules with 15 degrees of freedom. The curves shown are for internal energies of 0, 10, 28 and 60 kcal mol-l. curve peaks at values of AE that are different from the original value of E. The same behaviour is obtained in trajectory-type calculation^^^^ 31 where a potential is assumed and a detailed study of the energy exchange is made.By contrast, weak collisions as evaluated by the simple exponential and stepladder models probe the regions close to E wherever it may be. The dependence of the average energy up (A&, ) and down (A& ) and total energy transferred in a collision ( A E ) on the internal energy is given in fig. 2 for 400, 1000 and 2000 K and s = s’ = 15. The major points which emerge are as follows. (a) At very low internal energy up collisions predominate and ( A E ) x (A&). (b) At high internal energies down collisions predominate and (AE) x (A&). (c) At internal energies around the average energy at equilibrium (sRT), up and down collisions are both significant and ( A E ) is the average of the two.( d ) The higher the temperature the larger is (A&) at lower values of E and the smaller is I(AE,)I at higher values of E. This is so because the model requires that a Boltzmann distribution be obtained in one collision for s’ = co (a few collisions when s’ # 00). As the temperature in- creases, larger up collisions are needed for a substrate with low E to obtain the value of ( E ) which increases with the temperature. By the same token, if the molecule is energy-rich it takes smaller jumps for the high-temperature system to get to ( E ) than for the lower-temperature case, all other things being equal. Note that the intercept at E = 0 is not sRT. The intercept will approach the limiting value more closely as s’ -+ 00 ; the latter case is that of gas/wall strong-collision intera~tion.~~ The effect of the number of normal modes of the bath molecules on the average energy transferred is shown explicitly for T = 1000 K in fig.3. (a) The larger the heat bath the larger the absolute value of (AE) (up and down). (b) The incremental increase in AEis not directly proportional to the value of s’ (see discussion below); an infinitely large bath molecule (s’ = co) which collides with a substrate with s = 15 transfers on the average only a little more than a molecule with s’ = 30. The size of the reactantI. OREF AND B. S. RABINOVITCH 773 0 10 20 30 40 50 60 70 80 90 Elkcal mol-’ Fig. 2. Average energy transferred per collision ( A E ) plotted against internal energy E at 400, 1000 and 2000 K for s = s’ = 15.The solid line above ( A E ) = 0 is for (AE,) and below the line is for (AE,). The barred line indicates the overall energy transferred ( A E ) . limits the amount of energy which can be transferred up and down the energy scale. (c) At lower values of internal energy up collisions predominate and at high internal energies down collisions are the most important. ( d ) At values of internal energies close to the average thermal energy up and down collisions take place at the same time and the curves all cross at E x 30 kcal mol-l (i.e. sRT). There are two effects to a change in the temperature of a real molecular system with quantum oscillators instead of the classical ones used here. At low temperature the average internal energy of the bath molecule is low and therefore the number of equivalent classical oscillators s’ as given by ( E ) = s’RTis small.Therefore, ( A E ) should be small (fig. 3). However, at low values of T, (AE) should be larger since the bath molecule is colder (fig. 2). The two effects tend to counteract each other. The effect of the number of the vibrational modes of the bath on the average energy transferred can also be seen in fig. 4(a). The average energy of the substrate with s = 15 at 1000 K is 30 kcal mol-l; if the internal energy is below that value up col- lisions will prevail; above this value down collisions are important, as can indeed be seen from fig. 4(b). The effect of s and s’ on ( A E ) is introduced by using a reduced number of degrees of freedom parameter,25 s,, which is defined as sr = 2(s - 1) (s’ - l)/[(s - 1) + (s’ - l)].(c) A plot of ( A E ) against A$ yields a family of straight lines, each belonging to a different value of the internal energy, E; the linear774 VIBRATIONAL ENERGY TRANSFER 30 20 10 - L o E 2 -10 - 0 3 2 W -20 -30 I I I 1 0 10 20 30 40 50 60 70 80 -40 Elkcal mol-‘ Fig. 3. The average energy transferred (AE) plotted against the internal energy E at 1000 K and s = 15 for four values of the number of degrees of freedom of the bath molecule s’ = 5, 15, 30 and a. Full lines indicate (AE,) and (AE,). The barred lines indicate {AE). correlation over such a large spread in the values of s’, and therefore of s,, is a great simplification and enables easy correlation of various combinations of reactant and bath molecules.The limiting value of s,, for s’ = m, is 2(s- l), or s, = 5.3 for the case s = 15 (see fig. 4). The dependence of the overall average energy transferred during a collision on the temperature and the internal energy is given in fig. 5 for s = s’ = 15. The slightly concave lines are almost parallel. The spacing between them decreases by a constant increment as the internal energy increases. The spacing between the lines of E = 10 and E = 20 kcal mol-l is ca. 7.4 kcal mol-l, while between the lines E = 40 and E = 50 kcal mol-l is ca. 6 kcal mol-l. Over a limited range, however, the curves can be approximated as straight lines. One may examine collisional energy transfer in another systematic way. One may increase the size of the reactant while keeping the size of the bath molecule constant.Fig. 6 shows a graph of ( A E ) against E for s = 10 and 15 and s’ = 15. The results are very interesting; (AE,) increases as s increases, while (AE,) for s = 10 is larger than that for s = 15. In the up collision less energy is needed to bring the small molecule to its average energy while in the case of the down collisions (AE,) is larger for the small molecule since it is a smaller heat bath than the larger molecule and can retain less energy. This is precisely the prediction of the statistical which saysI. OREF AND B. S. RABINOVITCH 775 I 4 I 30 Fig. 4. The average energy (AE) plotted against the square root of the reduced number of modes, s,, for various values of E (in kcal mol-1) at 1000 K and s = 15.the larger the reactant molecule the more energy it can retain and therefore the smaller (AE,,) in a collision with a constant size heat bath. Fig. 7 shows the dependence of (AE) on the reduced number of degrees of freedom s,. Here s, increases because s increases while s’ remains constant. The average energy transferred in a collision with a bath molecule (s’ = 15 and T = 1000 K) increases as the number of degrees of freedom of the reactant increases. The reasons for such behaviour stem from the following facts. At low energy content (e.g. E = 10 kcal mol-l), as s increases more energy is needed to bring the molecule to its average energy and hence (AE) increases and is positive, i.e. up collisions predominate. At high levels of excitation (e.g. 70 kcal mol-l), as s increases the absolute value of A E decreases and is negative.In this case down collisions take place to bring the molecule to its equilibrium average energy but as s increases its equilibrium average energy increases and a smaller down step is needed. The model which is presented here anticipates the limiting energy-transfer behaviour of various experimental systems and does so in a simple and a consistent fashion. COMPARISONS WITH EXPERIMENT AND OTHER THEORIES How do the results which were obtained here compare with experiment? One system available for comparison is chemical activation. There, a substrate molecule is excited by the insertion of an atom or a radical into a double bond. For example, it is possible to oWain a butyl radical with average excess energy of 43 kcal mol-l by776 30 20 10 - I - g o 3 V .Y 2 4 4 v -10 -20 -30 VIBRATIONAL ENERGY TRANSFER 3 500 I000 1500 2000 TIK Fig.5. The average energy (AE) plotted against the temperature of the bath for s = s’ = 15 for the various values of E (in kcal mol-I) indicated on the curves. the reaction C,H, + . H C,H,. The energy spectrum of the molecule is not a delta function since the butene possesses initial thermal energy. Nevertheless, the energy spectrum is narrow and energy-transfer studies on such systems are instructive. When an excited butyl radical collides with a series of bath molecules, the larger the collider the larger The inert gases transfer ca. 2 kcal mol-l, the diatomics a little more and the polyatomics cis-butene and SF, ca.9 kcal mol-l. All the experi- mental values of (AE) which are quoted here were calculated by using the expo- nential transition probability model for the inert gases and the stepladder model for polyatomic colliders. Also, cyclopropane excited by chemical activation by lCH, addition to C2H4 has been reported to transfer ca. 4 kcal mol-l in a He bath, ca. 6 kcal mob1 in N, and ca. 10 kcal mol-l in a C,H, bath.,l Many more examples of such systems can be found in ref. (21), but the trend is clear: the larger the bath (s’), the larger is (AE). This result agrees with statistical-model r e s ~ l t s ~ ~ - ~ ~ and with our findings. Of course two effects influence collisional efficiency: one is the size of the heat bath and the other is the potential for the collisional interaction.Insofar as the former effects operate, the results in this series follow qualitative statistical expectation. However, it is possible to make a more quantitative comparison between theory and experiment for the case of the polyatomic butyl radical and SF, that function operationally in that study21 as strong colliders. Thus, at E = 43 kcal mol-l and s z 15 for the butyl radical777 E/kcal mol-’ Fig. 6. (AE) plotted against the internal energy E for s = 10 and 15, s’ = 15 and T = 1000 K. The dashed line indicates the average value of the energy transferred while the full lines indicate (AE,) and (AED). 20- 10- ‘i 0 - -z -10- - ; dq -20- A4 2 W -30 - -40 - 10 20 30 40 50 60 70 I I -50; 3 4 s t Fig. 7. (AE) plotted against the square root of the reduced number of degrees of freedom, s,.Values of initial energy E as indicated in the figure (in kcal mol-l) for s’ = 15 and T = 1000 K.778 VIBRATIONAL ENERGY TRANSFER and s’ = 3 for SF, (found by calculating the average energy of SF, and then calculating s’ from ( E ) = s’kT when T = 300 K), the value of ( A E ) calculated from eqn (7) is 6.2 kcal mol-l. This prediction is in reasonable agreement with the experimental value of ca. 9 kcal mol-l. Not many data exist for the temperature effect on the magnitude of ( A E ) in chemically activated systems. The little there is covers the low-temperature range 200-700 K and seems contradictory. Cyclopropane colliding with C2H4 shows21 an increase and then a decrease in ( A E ) on going from 300 to 700 K. For excited C,H,F,,O colliding with CH,ClF, ( A E ) remains constant at 300 and 475 K.For excited C,H,F colliding with N, there is a five-fold increase in ( A E ) on going from 315 to 560 K, an unexpectedly large temperature effect.21 The present model predicts a moderate increase in ( A E ) with temperature rise (fig. S), and reliable experimental data are clearly needed to verify this point. In recent experiments,’ azulene was photoexcited by laser and its energy-transfer behaviour was investigated by allowing it to collide with 17 bath gases. Molecules with an energy content of 17500 cm-l transferred to the bath molecules less energy than molecules excited to 30600 cm-l. The trend is similar to the one shown in fig. 2 and 3. In addition, the larger s’ the larger the value of (AE), in agreement with fig.3 and 4. In other experiments,, the energy-transfer behaviour of laser-excited cycloheptatriene was studied for a variety of bath gases, where it was found that there is practically no energy dependence of (AE), in contradiction to the results reported in ref. (41). Another type of experiment involves changing the size of the substrate (increase in the value of s) at constant s’. This type of experiment is harder to interpret since differ- ent substrate molecules have different threshold energies for decomposition, E,, and different working temperatures, usually. Statistical theory predicts that the larger the substrate the smaller is (AE), since more energy remains in the substrate The best way to obtain reliable results is to change s in a homologous series where E, and the activated complex remain unchanged.One such system is the alkyl-radical system, where excited butyl, pentyl, hexyl and octyl radicals were allowed to collide with various di- and poly-atomic gases.43 A slight increase in the magnitude of (AE) was found going from butyl to octyl. This is opposed to the prediction of the present and statistical model given in ref. (39). Cyclopropane transfers21 CQ. 10 kcal mol-l with C,H, as a bath and 9 3 kcal mol-l with n-C5Hlo.17 Dimethylcyclopropane transfers2’ 1 1.4 kcal mol-l with cis-butene as bath.* Methylcyclopropane is reported to 7 & 1 kcal mot1 in a collision with n-C,H,,, while ethylcyclopropane is reported to 7 f 4 kcal mob1 with 2-methylpentane as bath.Clearly, a definite correlation is difficult to make. 2-Pentyl and dimethyl-2-pentyl are reported2l9 45 to transfer 4.6 kcal mot1 in collisions with CF,. Again a change in the number of modes of the substrate does not appear to cause a change in (AE). More systematic and reliable experiments must be performed in order to understand the effect that increasing s has on the magnitude of (AE). calculations give good agreement with the experimental results. However, a cut-off energy in the transitional stretching mode correlating with relative translational motion along the line of centres is introduced and empirically adjusted. The empirical adjustments were done in such a way as to force the calculations for the value of (AE) for methylcyclopropane to reproduce the experimental one.The empirical values were then used in other collision-pair calculations. It is useful here to compare other statistical models. The transition-modes * Ref. (39) quotes a value of 6 kcal mol-I from ref. (44) instead of 11.4 kcal mol-’.I. OREF AND B. S. RABINOVITCH 779 The ergodic-collision theoryg6 predicts values which are generally larger than the reported experimental values. For dimethylcyclopropane colliding with cis-but-2-ene, instead of the reported value of 11.4 kcal mol-1 it predicts values in the range 27-51 kcal mol-l. For the 2-pentyl radical colliding with CF, it predicts 9.6 kcal mol-1 instead of 4.5 kcal mol-l. Generally, agreement to about a factor of two or better is obtained between theoretical and experimentally reported values.An improvement of the results calculated by the previous theory is obtained by the impulsive-collision theory.,' In this theory the collisional period is very short and therefore only kinetic energy is available for redistribution. The value of ( A E ) for the collision between dimethylcyclopropane and C,H, is now reduced to the range 14-27 kcal mol-l compared with the experimental 11.4 kcal mol-l. If one makes in the present treatment the common assumption that the classical s is half the number of modes, good agreement is obtained with the ergodic-collision theory4s as one would expect. The advantage of simplicity and lack of empiricism makes it as a useful tool in understanding the dependence of ( A E ) on the size and temperature of the collision partners.CONCLUSIONS (a) The collisional energy-transfer probability P(E, AE) as given by eqn (3) and (4) is a smooth function which obeys the conservation of probability [eqn (5)] and detailed balance [eqn (6)]. (b) The width at half of P(E, AE) is smaller at lower values of E (fig. 1). (c) At lower values of E most of the collisions are up transition and ( A E ) = ( A E , ) . At higher values of E most of the collisions are down transitions and ( A E ) = (A&) (fig. 1,2 and 5). At intermediate regions up and down collisions are operative. ( d ) Increasing the temperature of the bath effects larger (AE,) and smaller (AE,,) (fig. 2). (e) Increasing the size of the bath molecule increases the size of the average energy jump up and down (fig. 3). (f) The reduced number of degrees of freedom is a good parameter to use in order to show the effective molecular size dependence of the energy jumps, fig.4. (g) At low level of excitation ( A E ) is larger, the greater the size of the substrate, s. The collisions are up transitions (fig. 6). (h) At high levels of excitation (AE) is smaller, the larger s. The collisions are down transitions (fig. 6). (i) As s increases for a given level of excitation the value of ( A E ) increases. It is still negative at high levels of excitation and positive at lower ones. This work is supported by the United States-Israel Binational Science Foundation. 1.0. thanks the Berenstein Fund for the Advancement of Science for its assistance. B.S.R. thanks the Office of Naval Research and the National Science Foundation for their assistance.B. S. R. also thanks Prof. John Albery for his hospitality at the Imperial College of Science and Technology, London, while on leave there. We would like to thank a referee for illuminating and extensive comments. J. T. Yardley and C. B. Moore, J. Chem. Phys., 1966,45, 1066. R. S. Sheorey and G. W. Flynn, J . Chem. Phys., 1980,72, 1175. M. L. Mandich and G. W. Flynn, J. Chem. Phys., 1980, 73, 1265. R. Kadibelban, W. Janiesch and P. Hess, Chem. Phys., 1981, 60, 215. D. Siebert and G. Flynn, J. Chem. Phys., 1975,62, 1212. R. K. Bohn, K. H. Casleton, Y. V. C. Rao and G. W. Flynn, J. Phys. Chem., 1982,86, 736. C. J. S. M. Simpson, P. D. Gait, T. J. Price and M. G. Foster, Chem. Phys., 1982, 68, 293. C. J. S. M. Simpson, D. C. Allen and T.Scragg, Chem. Phys., 1980, 51, 279. ' T. H. Alllk and G. W. Flynn, J. Phys. Chem., 1982,86, 3673. lo S. A. Rice, Adv. Chem. Phys., 1981, 57, 231. l1 C. S. Parmenter and K. Y. Tang, Chem. Phys., 1978, 27, 127.780 VIBRATIONAL ENERGY TRANSFER l 2 G. H. Atkinson, C. S. Parmenter and K. Y. Tang, J. Chem. Phys., 1979, 71, 68. l3 D. A. Chernoff and S. A. Rice, J. Chem. Phys., 1979, 70, 2521. l4 M. Vandersall, D. A. Chernoff and S. A. Rice, J. Chem. Phys., 1981, 74,4888. l5 C. S. Parmenter, J. Phys. Chem., 1982,86, 1735. l6 K. Y. Tang and C. S. Parmenter, J. Chem. Phys., 1983,78, 3923. M. J. Rossi and J. R. Barker, Chem. Phys. Lett., 1982, 85, 21. R. Duperrex and H. Van den Bergh, J. Chem. Phys., 1979,71, 3613. R. Duperrex and H. Van den Bergh, Proc. 2nd Int. Con$ Infrared Phys., 1979, p.217. D. C. Tardy and B. S. Rabinovitch, Chem. Rev., 1977, 77, 369. 2o A. Lifshitz, A. Bar Nun, A. Burcat, A. Ofir and R. D. Levine, J. Phys. Chem., 1982,86,791. 22 I. Oref, J. Chem. Phys., 1982, 77, 5146. 23 0. Herscovitz and I. Oref, J. Phys. Chem., 1982, 86, 1495. 24 0. Herscovitz, E. Tzidoni and I. Oref, Chem. Phys., 1982, 71, 221. 25 I. Oref, 0. Herscovitz and E. Tzidoni, J. Phys. Chem., 1983, 87, 98. 26 R. E. Harnngton, B. S. Rabinovitch and M. Hoare, J. Chem. Phys., 1960,33, 744. 27 J. Troe, J. Chem. Phys., 1977,66,4745; 4758; J. Phys. Chem., 1979,83, 114. 28 H. Hippler, K. Luther and J. Troe, Faraday Discuss. Chem. SOC., 1979,67, 173. 2g T. F. Hunter, M. G. Stock and N. Webb, J. Chem. Soc., Faraday Trans. 2, 1979,75, 738. 30 I. Oref and B. S. Rabinovitch, Chem. Phys., 1977, 26, 385. 31 R. C. Bhattacharjee and W. Forst, Chem. Phys., 1978, 30, 217. 32 D. F. Kelley, T. Kasai and B. S. Rabinovitch, J. Phys. Chem., 1981, 85, 1 100. 33 I. Oref and B. S. Rabinovitch, Ace. Chem. Res., 1979, 12, 166. 34 Y. N. Lin and B. S. Rabinovitch, J. Phys. Chem., 1970, 74, 315. 35 I. Oref, Int. J. Chem. Kiner., 1977, 9, 751. 36 I. Oref, J. Phys. Chem., 1977, 81, 1967. 37 1. Oref, J. Chem. Phys., 1981, 75, 131; 1982, 77, 1253. 38 R. J. McCluskey and R. W. Carr, J. Phys. Chem., 1978, 82, 2637. 39 R. W. Carr, Chem. Phys. Lett., 1980, 74, 437. 4o H. W. Chang and D. W. Setser, J. Am. Chem. Soc., 1969,91, 648. 41 M. J. Rossi, J. R. Pladziewicz and J. R. Barker, J. Chem. Phys., 1983,78, 6695. 42 H. Hippler, J. Troe and H. J. Wendelken, 7th Int. Symp. Gas Kinet., Gottingen, Germany, 23-27 Aug. 43 D. C. Tardy and B. S. Rabinovitch, J. Chem. Phys., 1968,48, 5194. 44 J. D. Rynbrandt and B. S. Rabinovitch, J. Phys. Chem., 1970,74, 1679. 45 J. H. Georgakakos, B. S. Rabinovitch and E. J. McAlduff, J. Chem. Phys., 1970, 52, 2143. 46 S. Nordholm, B. C. Freasier and D. L. Jolly, Chem. Phys., 1977, 25,433. 47 H. W. Shranz and S. Nordholm, Int. J. Chem. Kinet., 1981, 13, 1051. 1982. (PAPER 3/74)