3752
J. Phys. Chem. 1992, 96, 3752-3756
Two approaches were used to test if limited surface removal of O3could quantitatively account for the observed differences in [O3lSsresults from the two cells. The first method sought to find a fixed level of ozone absorption per unit surface area, which, when multiplied by the total area and added to the measured [O3lSs,would result in the same total ozone production for a given [T,],,. However, no fixed level of ozone absorption per unit surface area could be determined which would fit the experimental results. It is possible that with more experimental data a level of adsorption can be found which would fit experimental data. As a second approach, wall adsorption reactions were added to the gas-phase model itself. It was found that by assuming a limited adsorption of ozone to reduce the predicted initial rate of production and by assuming a significant level of TO2adsorption to decrease the ozone destruction reactions, adjustments could be made for each concentration of T2 which would produce concentrations approaching experimental results. Although a wall effect which changes with [T,], has little physical basis, it does give an indication of which effects should be considered in a more complete, heterogeneous model. Another effect of decreasing the diameter of the reaction cell is an increased loss of radiolysis energy to the walls. In the gas-phase model, wall losses were not considered. To determine if the wall loss could account for the discrepancies between experiment and model, the fraction of total radiolysis energy absorbed by the walls was calculated. For the 10-cm cell, less than 1% of the total energy was calculated to be lost to the walls, the 2.5-cm-cell loss is calculated to be about 2.5%. Therefore, the energy losses are well within the uncertainty of the radiolysis calculations and cannot account for the measured differences. It is evident that no explanation presented here completely reconciles the effect of the increased surface-to-volume ratio on the measured O3 concentration. The closest explanation is a limited adsorption of ozone on the glass.
Conclusions Experimental results confirmed the presence of ozone as an intermediate in the tritium oxidation reaction. These studies provided the first measurements of an important intermediate of the T2 + 0,reaction. The experimental results show a linear increase in the initial rate of ozone production with increasing [T2I0. As the rate of ozone depletion reactions increase, the ozone concentration reaches a steady state. This is in agreement with predictions of the comprehensive model of the homogeneous reaction. The experimental data are qualitatively, but not quantitatively reproduced by the model. The measured initial ozone production rate was one-third of that predicted. The steady-state ozone concentration for the T2 + O2experiments increased with the 0.6 power of [TJ0, whereas a 0.3-power dependence was predicted. The addition of stable hydrogen to the reaction mixture does not affectthe initial rate of ozone production, but generally reduces the [O3ISs. A clear correlation between the amount of added H2 and reduction in [O3lSswas not found. The effect of the Pyrex surface of the reaction vessel on the ozone concentration varied with [T,],, with higher tritium levels being least affected. Attempts to modify the model by single alteration of a variety of reactions failed to produce agreement with the experimental data. This implies either that a number of the kinetic factors in the model may be significantly in error or that one or more important effects are not contained in the model.
Acknowledgment. We thank Drs. Christopher Gatrousis, Jeffery Richardson, and Carl Poppe for their constant support and encouragement, Prof. Harold S. Johnston for his critical review, and Mr. Terrance Poole and Mr. Stephen Wilson for their excellent mechanical efforts. This work was performed under the auspices of the US.Department of Energy by Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48.
Choice of Gas Kinetic Rate Coefficients in the Vibrational Relaxation of Highly Excited Polyatomic Molecules Gyorgy Lendvayt and George C. Schatz* Department of Chemistry, Northwestern University, Evanston, Illinois 60208-31 I 3 (Received: October 24, 1991; In Final Form: January 14, 1992)
We examine the convergence of average energy transfer with maximum impact parameter in classical trajectory studies of CS, collisional relaxation by He, Xe, H,,CO, CS?, and CH4, SF, relaxation by He, Ar, Xe, and SFs, and SiF4 relaxation by Ar. This leads to estimates of the gas kinetic collision rate coefficient that are substantially larger (by a factor of 3.1 on average, and a maximum of 4.7) than are obtained using the traditional Lennard-Jones collision frequency.
I. Introduction In unimolecular and chemical activation reactions, vibrational relaxation competes with the chemical reaction. To model such processes one often uses a master equation in which the energy of the reacting molecule is divided into b i n ~ , l -with ~ transitions between bins taking place by energy transfer and loss of population of each bin arising from reaction. Solution of this master equation provides the rate coefficient for the net reactant consumption (the rate of the unimolecular reaction) corresponding to the given temperature and pressure. In this master equation, the rate of the reaction from each energy bin is generally calculated by RRKM theory,'-* while the rate of energy transfer is obtained from a dynamical model for the energy transfer probability distribution P(E,E'). This probability is converted to a rate by Present address: Central Research Institute for Chemistry, Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 17, Hungary.
0022-365419212096-3752!§03.00/0
multiplying it by the collision rate o between the reactant and the collision partner at the given pressure and temperature,1d*9 (1) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: London, 1972. (2) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1977 ., .-. (3) Pritchard, H. 0. Quantum Theory of Unimolecular Reactions; Cambridge University Press: Cambridge, UK, 1984. (4) .Quack, M.; Troe, J. In Gas Kinetics and Energy Transfer; Specialist Periodical Report; The Chemical Society: Burlington House, London, 1977; VOl. 2. (5) Tardy, D. C.; Rabinovitch, B. S.Chem. Rev. 1977, 77, 369. (6) Gilbert, R. G.; Smith, S.C. Theory of Unimolecular and Recombination Reactions; Blackwell: Oxford, UK, 1990. (7) McCluskey, R. J.; Carr, R. W., Jr. Znt. J . Chem. Kinet. 1978, 10, 171. (8) Marcus, R. A. J. Chem. Phys. 1952, 20, 359. (9) Whyte, A. R.; Lim, K. F.; Gilbert, R. G.; Hase, W. L. Chem. Phys. Lett. 1988, 152, 377. Lim, K. F.; Gilbert, R. G. J . Phys. Chem. 1990, 94, 72.
0 1992 American Chemical Society
Vibrational Relaxation of Polyatomic Molecules
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3753
For a given collision rate the simulation of a thermal unimolecular or chemical activation reaction is not very sensitive to the details of the energy-transfer probability distribution. In fact, the experimental data can be well reproduced with an appropriate choice for one moment of P(E,E?, such as the average energy transfer ( A,??).5 Any one of the general models for P(E,E?, such as the exponential, Gaussian, or stepladder, can result in essentially the same pressure-dependent rate coefficients provided that (A,??) is appropriately adjusted. The separation of the energy-transfer rates into the product of a collision rate w and an energy-transfer probability matrix P(E,E? is, however, arbitrary, and neither the experiments on unimolecular reactions nor the physical methods for “direct” determination of energy-transfer information1*14 provide a basis for separately determining w and P. These experiments do, however, determine the product of the average energy transfer ( A,??) and the collision rate w.I5 w is then estimated separately, and it has become traditional to evaluate it using the Lennard-Jones collision frequency.’-‘ One justification for this has been given by Troe16 based on an analysis of Stace and Murrell’s early trajectory study.” Recently we have been engaged in theoretical studies of energy transfer from highly excited polyatomic molecules to develop insight into the details of the One issue important in the classical trajectory method that we have used for these studies is the maximum impact parameter or “collision diameter” needed to determine the collisional energy-transfer rates. This diameter determines w. Although the classical total cross section for an infinite range potential diverges, we have substantial numerical evidence that (A,??) times the gas kinetic rate w is finite. As a result, one can define a maximum impact parameter relevant to energy-transfer information by examining the impact parameter dependence of this product. This impact parameter can then be compared with values used in the traditional method to calculate gas kinetic collision rates. In this paper we use the results of extensive trajectory simulations to determine this optimum impact parameter for several collision systems. We will show that the theoretical hard-sphere radius (or b,,,) is generally larger than the one used in the experimental studies, which implies that the collision rate traditionally used in the evaluation of the energytransfer experiments is underestimated. As a result, the average energy-transfer values are overestimated. One earlier study that considered the proper choice of b, was that of Hu and Hase,21who did a trajectory study of the Ar CH4 system. See also Hsu et aLz2 In contrast to our purely classical criterion for choosing b,,,, their choice of b,, was motivated by the idea that transitions between different quantum states are forbidden if the average energy transfer for a given impact parameter is smaller than the average spacing between states, Le., smaller than the inverse of the density of states. They concluded that this definition produces b,, values larger than the Lennard-Jones value, but they did not present detailed results. We will consider the Hu and Hase method briefly in our analysis.
+
11. Methods
The method we used to model the energy-transfer process is described in detail e l s e ~ h e r e . ~Some ~ * ~ specifics ~ * ~ ~ are reviewed (10) Shi, J.; Barker, J. R. J . Chem. Phys. 1988, 88, 6211, 6219, and references cited therein. (1 1) Abel, B.; Herzog, B.; Hippler, H.; Troe, J. J . Chem. Phys. 1989,91, QlUl
_I-.
(12) Hippler, H.; Troe, J. In Bimolecular Reactions; Baggott, J . E., Ashfold, M. N. R., Eds.; The Chemical Society: London, 1989. (13) Gordon, R. J. Comments At. Mol. Phys. 1988, 21, 123. (14) (a) LBhmannsrdben, H. G.; Luther, K. Chem. Phys. Leu. 1988, 144, 473. (b) Luther, K.; Reihs, K. Ber. Bunsenges. Phys. Chem. 1988, 92, 442. (15) Troe,J. J . Chem. Phys. 1982, 77, 3485. (16) Troe,J. J . Phys. Chem. 1979,83, 113. (17) Stace, A. J.; Murrell, J. N. J . Chem. Phys. 1978, 68, 3028. (18) Lendvay, G.; Schatz, G. C. J . Phys. Chem. 1990, 94, 8864. (19) Lendvay, G.; Schatz, G. C. J . Phys. Chem. 1991, 95, 8748. (20) Lendvay, G.; Schatz, G. C. J . Chem. Phys., in press. (21) Hu, X.; Hase, W. L. J . Phys. Chem. 1988, 92, 4040. (22) Hsu,K.-J.; Durant, J. L.; Kaufman, F. J. Phys. Chem. 1987 , 91, 1895.
zh
4 O y
I
0
I
I
0
0 0 0
0
0
A
0
E ”
U I
2.5
5.0
Impact Parameter (Angstrom) Figure 1. -(A,!?)* (in cm-I) vs impactgarameter 6 (in A) for relaxation of CS2 (Evib= 93 kcal/mol) in a 300 K bath of H2(triangles), CO (squares), and CH, (circles). I
here. The calculational technique we used is aimed at the determination of the average energy transfer, ( M )at , a given energy in the relaxing molecule ( A,??) is obtained as an average over a large number of collisions simulated by classical trajectory calculations. The initial intemal energy in the energy-rich molecule is kept fixed; the collision partners are selected by Monte Carlo from a thermal bath. The method uniquely determines the product of (A,??)and the collision rate. This product, or equivalently the product of (A,??)and bma:, the maximum impact parameter, is constant if b, is large enough so that no collision outside of this range gives a significant contribution to the average energy tran~fer.~.~’ Applications of this approach to several different molecule-bath combinations have yielded values of the w ( hE) product that are in good agreement with e ~ p e r i m e n t . ~ * - ~ ~ , ~ ~ J ~ There are several ways to determine the minimum value of b , necessary to converge the cross section. One is to calculate ( A,??)bW2for increasing , 6 (requiring a new set of trajectories for each b,,) and to determine where it converges.23 A faster but much less accurate and informative modification of this technique is to select a large enough b, run a statistically large sample of trajectories, and then analyze the contribution of different bins of impact parameter to the total average energy t r a n ~ f e r .More ~ precise information can be obtained if one calculates the average energy transfer (A,??)+,at different fmed impact parameters, b. The desired b, is then the smallest value where the integral J(A,??),,b db converges. The energy in the hot molecule is usually set to a high value when the impact parameter dependence of (A,??)b is determined, but we have also studied whether the maximum impact parameter obtained in this manner is appropriate at lower energies. The calculations were performed with a custom-designed version of the classical trajectory program VENUS.” The inter- and intramolecular potentials (parameters for which are given in refs 18-20 and 24) were designed to describe relaxation of CS2(initial energy 93 kcal/mol) by He, Xe, H2, CO, CS2, and CH4, relaxation of SF6 (100 kcal/mol) by He, Ar, Xe, and SF,, and relaxation of SiF4 (100 kcal/mol) by Ar. The number of trajectories run at each impact parameter at a specific temperature varies from 200 to 500 depending on the average energy transfer: if the latter is small, generally a larger number of trajectories are needed to (23) Bruehl, M.; Schatz, G. C. J . Chem. Phys. 1988,89, 770. (24) Bruehl, M.; Schatz, G. C. J . Phys. Chem. 1988, 92, 7223. (25) Hase, W. L.; Duchovic, R. J.; Lu, D.-H.; Swamy, K. N.; Vande Linde, S.R.; Wolf, R. J. VENUS. A General Monte Carlo Classical Trajectory Computer Program.
3754 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
k h
1
0 0 0
0
Lendvay and Schatz
0 0
7501
w
*
0
I
0
0 0
0
-
L
k Q,
m
0
‘
I
.
2I
.
4I
*
46
a
$
A
T 10 ’
250-
(d
k
A 0
Impact Parameter (Angstrom)
get meaningful statistical accuracy. 111. Results
A. Impact Parameter Dejmdence of (AE),. Figure 1 shows the average vibrational energy transferred from vibrationally hot CS2 molecules (initial Eyiv= 93 kcal/mol) in collisions with H2, CO, and CH4 at 300 K as a function of impact parameter b. The initial rotational energy of CS2, the translational energy, and the internal energy in the bath molecules correspond to the Boltzmann distribution at this temperature. The plots show the shape one expects on the basis of the pioneering studies of Stace and Murrell” and on common sense. The largest ( h E ) b values are found for low-impact parameter collisions; then the curves decrease rapidly to zero as b increases. Another type of impact parameter dependence is displayed in Figure 2 for the s F 6 rare gas collision system at 300 K,taking the initial Evibin SF6 to be 100 kcal/mol. Here one finds a small dip in ( AE)bat low impact parameters. A relatively broad b range in which there is more or less constant energy transfer can be observed in the SF6 SF6 system (see ref 26). The maximum impact parameter needed to define a converged set of average energy transfers is determined by considering the integrand of the integral
+
+
(26) Lendvay, G.; Schatz, G. C., to be published. (27) Hippler, H.; Troe,J.; Wendelken, H. J. J . Chem. Phys. 1983, 78, hlW?
(28) Troe, J. J . Chem. Phys. 1977, 66, 4145, 4758. (29) Dove, J. E.;Hippler: H.; Troe. J. J. Chem. Phys. 1985, 82, 1907. (30) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids;McGraw-Hill: New York, 1977.
0 O
9
I
A.
A
I
a
o
-
,A
Impact Parameter (Angstrom)
I
Figure 3. b(AE)b(triangles), J b p b ( h E ) bdb (squares), and (hE) from eq 1 (circles) vs b for SF, (100 kcal/mol) in He at 1000 K.
TABLE I: ~ ~ Q ‘ / ~ ( 2 Products ,2) and b,, Values Obtained from the Trajectory Calculations in the A + CS2, A + SFr and A + SiF, Systems
system He + CS2
Xe
+ CS,
H2
+ CS2
co + cs2 cs2+ cs2 CH,
+ CS2
temp/K 300 1000 2000 300 1000 2000 300 1000 2000 300 1000 2000 3 00 1000 2000 300
1000
He + SF,
+ SF6 Xe + SF6 SF, + SF, Ar
This integrand is plotted as a function of b in Figure 3 (together with the accumulated integral and ( A E ) ) for the system He + SF6 at loo0 K. One can see that the product b(AE)bhas a large peak at b = 3 A and then drops to zero for b 1 8 A. As a result, the integral in the numerator of eq 1 increases monotonically until it converges at 6 , = 8 A. In addition, ( AE) decreases as bmaL2 for b , > 8 A. The optimal b , is the value where the integral is within some threshold of its converged value. The maximum impact parameters obtained by setting this threshold to 3% are presented in Table I for 29 different choices of collision system and temperature. Also included are the products of the Lennard-Jones u parameter (taken from refs 27-30) and the reduced collision integral calculated using the formula of Troe,2Bi.e., the ‘traditional” estimate of bma. The trajectory results are always
-
0
A
A 0
k
@
Figure 2. - ( A E ) b vs b as in Figure 1 but for SF, (100 kcal/mol) in a 300 K bath of He (triangles), Ar (squares), and Xe (circles).
0
A
6k 4 5
0
W
Ar
+ SiF,
2000 300 1000 300 1000 300 1000 300 1000 2000 300 1000
oAQ1/*(2,2)/A
b,,,/A
3.49 3.1 1 2.94 5.44 4.40 4.08 4.15 3.51 3.29 4.86 4.04 3.71 6.13 4.95 4.46 5.08 4.17 3.88 3.66 3.29 4.85 4.10 5.48 4.54 6.12 5.08 4.74 4.61 3.92
4.9 4.7 4.5 6.3 6.1 5.8 6.6 5.9 5.5 7.0 6.6 6.3 10.2 9.0 8.1 8.1 1.3 6.2 7.5 5.9 9.3 8.7
10.0 9.8 12.3 10.9 9.9 8.7 7.9
larger than the Lennard-Jones results, with the ratio being larger than a factor of 2 in several cases. The traditional collisional radii also decrease much faster with increasing temperature than the theoretical ones. The impact parameter dependence of the energy transfer and the maximum impact parameter needed for converged calculations may, in principle, depend on the energy content of the relaxing molecule. Our calculations on the CO CS2system in the energy range 5-93 kcal/mol show that the shape of the ( M ) vs , b curves is independent of energy within statistical uncertainty at 300 K. This means that the impact parameters determined at 93 kcal/mol also are appropriate at lower energies. Similar observations were made for the He + CS2 and Ar + SiF4 system. In the Introduction we mentioned that an alternative definition of b,,, was proposed by Hu and Hase2’ based on the density of
+
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3155
Vibrational Relaxation of Polyatomic Molecules
A
1
A
ai
$ -7001
I
I
1
2.0
4.0
6.0
Impact Parameter (Angstrom)
I
Figure 4. (AE),(triangles) and its rotational counterpart (squares) (in cm-l) vs 6 for CS2 (93 kcal/mol) in CH4 at 1000 K.
vibrational states of the hot molecule. This density is difficult to calculate accurately, but for CS, at 93 kcal/mol, several types of anharmonic oscillator models give values around 10 states/cm-’. Unfortunately, comparing the inverse of this estimate with ( h E ) b in Figure 1 to derive a value for b,,, is problematic because our statistical uncertainty in calculating ( h E ) b is a factor of 10 or more larger than the inverse density of states for b,, values comparable to those listed in Table I (obtained using our criterion). Because the inverse of the density of states is so small, we would have to increase the number of trajectories at least 100-fold in order to determine precise 6,, values so that we could apply the Hase and Hu method. For some of the SF6collision systems listed in Table I it is possible to make somewhat better estimates of b,, and in all cases, the Hu and Hase values are larger than what we estimate. This indicates that the H u and Hase criterion requires convergence of ( M ) b to a smaller value than does ours. Another difference between the H u and Hase method and ours concerns the energy dependence of b,,,. In the H u and Hase method, b , goes to zero at low energy because the inverse density of states eventually becomes larger than ( M ) b for any 6. For CO CS2 at 2000 K we find that b,,, = 0 at 5 kcal/mol according to the Hu and Hase method, while our method predicts b,,, = 6.3 A. B. Rotational Energy Transfer. The detailed trajectory studies also show how rotation of the excited molecule participates in the energy transfer. Figure 4 shows the rotational and vibrational energy (( U R ) b and ( A E ) b ) transferred from CS2 with an initial vibrational energy of 93 kcal/mol in collisions with methane. The translational temperature is 1000 K, and the initial rotational energy of CS, and internal energy of CHI are selected from Boltzmann distributions corresponding to this temperature. The b dependence of ( h E ) b is typical of what we have seen in Figures 1 and 2. The absolute value of ( A E R ) b is generally smaller than that of ( m ) b (as makes sense given that the initial rotational energy is sampled from a Boltzmann distribution while the initial vibrational energy is much hotter), but ( A & ) b does not necessarily decrease monotonically with increasing impact parameter. In fact, the (AE ) b is found to be positive at low impact parameters then decreases, turns negative, and has a minimum (1.0-2.5 at medium impact parameters (4-5 A). The location of this negative minimum coincides with the center of mass distance a t the minimum of the intermolecular potential in all the systems we studied (He, Xe, H,, CO, CS2 + CSz and He, Ar, Xe, CO SF6). At larger impact parameters ( A E R ) b decreases to zero but often (CO + CSz, CS, + CS,) slower than (A,!?)b. As a consequence, if one calculates sequences of successive collisions
+
a),
+
where rotational energy transfer plays a role in determining evolution of the sequence, the correct maximum impact parameter will be larger than the one obtained from the impact parameter dependence of the vibrational energy transfer. One explanation for the b dependence of ( h E R ) b in Figure 4 is as follows. For small b, the collisions sample the repulsive inner wall of the intermolecular potential, resulting in an impulsive interaction which rotationally excites the molecule. As 6 increases to the potential minimum, attractive forces become dominant, leading to a locking of the relative orientation of the molecules during collision. Because of the larger moment of inertia of this combined collision system, the rotational energy of this system and of the molecules into which it fragments is smaller than the initial rotational energy. At still larger impact parameters the molecules do not interact, and rotational energy transfer is small. C. Energy-TrrursferDistrWion Functions, The most important consequence of the fact that the collision diameters used in the traditional evaluation of the energy-transfer experiments are underestimated is that the average energy transfer is overestimated. Let us now determine by how much it is overestimated. When experiments are modeled and the parameters of the energy-transfer probability function are fit, the only quantity that is in reality obtained is the rate of energy transfer from initial energy E to E’after collision, k(E’,E).6*9The quasielastic energy-transfer rate k(E,E) is not determined by any model study because it does not contribute to the net energy transfer. As usual, we may decompose k(E’,E) into the product of the collision rate, o,and the probability of the energy transfer, P(E’,E), i.e.
P(E’,E) = k(E’,E)/o The energy-transfer probability is normalized
(2)
1
J - P ( E ’ , E ) dE’ = 1
(3)
and it determines the average energy transfer
(AE)= J ( E ’ - E)P(E’,E) dE’
(4)
If, however, we choose a larger collision diameter resulting in a larger collision rate, w , , the same energy-transfer rate will decompose into a different product: PI(E’,E) = k ( E ’ , E ) / w ,
(5)
The new energy-transfer probability matrix must also be normalized JP,(E’,E) dE’ = 1
(6)
but it results in a different number for the energy transfer
(AE),= J ( E ’ - E)P,(E’,E) dE’
(7)
At first glance, one would think the correlation between the old and the new energy-transfer probabilities is just scaling by the ratio of collision rates PI(E’,E) = ( w / w , ) P ( E ’ , E )
(8)
But then, if P is normalized to unity, P I is not. This problem can be resolved only if there is another difference between the energy-transfer probability distributions. If both w and oIcorrespond to collision diameters that are large enough, there is negligible contribution to the energy transfer from collisions taking place between these two collision diameters. The larger diameter means a larger number of quasielastic collisions, and PI(E,E)must be relatively larger than the quasielastic peak P(E,E). If one takes a certain number N of collisions corresponding to the collision rate w , the number of collisions transferring the molecule to the level E’is NP(E’,E). If one runs Mmore collisions but all with impact parameters which are between b and b,, the number of nonelastic collisions will not change but that of the quasielastic ones increases by M. In order that the proper sampling of the impact parameter be maintained, the ratio of the number of collisions must satisfy N / ( N + M) = b2/b12= w/wI. Then the
3756 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 number of collisions leading to the level E’is NP(E‘,E)
- E), and the new probabilities are
+ M6(E‘
+
Pl(E’,E) = N / ( N M)P(E’,E) + M / ( N + IU) &(E’- E) = (w/uI)P(E’,E) + (1 - w / u I ) 6(E’- E) (9) Now the distribution function is correctly normalized: l P l ( E ’ , E ) dE‘= w / u l
+ (1 - w / w I )
1
(10)
If one substitutes eq 9 into eq 4, one finds that the average energy transfer is scaled down by a factor of w / w , . This last result is known from earlier s t ~ d i e s . ~ , ~The * ~ new ~ * ~information * here is that the probability distribution function is governed by eq 9. This equation indicates that the inelastic probability distribution is scaled by w / w , just like the average energy transfer. The scaling of the average energy transfer by w/wl means that if one underestimates the collision diameter by a factor of a,the average energy transfers will be overestimated by a factor of l/az, The absolute value of the average energy transfer is subject to this kind of uncertainty in both the physical (spectroscopic) and chemical (chemical or photochemical activation) experiments devoted to determing the average energy transfer. In studies of thermal unimolecular reactions, when information on energy transfer is extracted, it is customary to fit the pressuredependent rates in terms of a collisional efficiency factor 8. This factor is multiplied by w to yield an “effective collision freq~ency”,~’ and (31) Oref, I.; Tardy, D.C. Chem. Reu. 1990, 90, 1407.
Lendvay and Schatz it is the & product that is determined from the experimental data. As a result, scaling of the collision diameter by a results in 8 scaled as l/(u2. fl is related to (AE), but the functional relationship is nonlinear, so in this kind of evaluation ( AE) does not scale according to l/az. In the range 0.03 < 8 < 0.3, according to Troe’s exponential model formula%the scaling power law for ( M )varies as a-2.2- a-3.2 .
IV. Conclusions Using classical trajectory calculations, we studied the impact parameter dependence of the energy transfer in the collisional relaxation of vibrationally highly excited molecules. We found in a large number of hot moleculmllision partner combinations that the maximum impact parameters needed to obtain converged energy-transfer moments are significantly larger than the product of the Lennard-Jones u parameter and the square root of the reduced collision integral O* (2,2) (the quantity generally used in the calculation of vibrationally inelastic collision rates). As a result, the experimental and model theoretical studies relying upon the use of the Lennard-Jones collision frequency underestimate the collision rate and overestimate the average energy transfer. The size of the error in collision rate ranges from a factor of 1.3 to a factor of 4.7 in the examples in Table I, with an average value of 3.1. Acknowledgment. This research was supported by NSF Grant CHE-9016490. Registry No. CS2, 75-15-0; SF6, 2551-62-4; He, 7440-59-7; Xe, 7440-63-3; H2, 1333-74-0; CO, 630-08-0; Ar, 7440-37-1; CHI, 74-82-8; SiF4, 7783-61-1.
