J. Phys. Chem. 1981, 85,2426-2429
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Effect of Rotational Excitation on Intramolecular Energy Transfer in C2H6 Edward R. Grant* Depc nent of Chemistry, Baker Laboratory, Cornel1 University, Ithaca, New York 14853
and
Jesus Santamaria
Depattamento de Quimica Fisica, Facultad de Quimicas, Universidad de Madrid, Madrid-3, Spain (Received: March IO, 198 1; In Final Form: May 18, 1981)
Studies of unimolecular decomposition rates in classical trajectory simulations of highly excited C2Hs,using several different initial condition sampling biases, have shown that in this system high rotational excitation accelerates the flow of energy into and out of C-H stretching modes. C-C scission reaction rates with initial energization biased toward skeletal modes behaved, on rotational excitation,in reasonable accord with statistical expectation. However, strong deviation from RRKM prediction was seen for the C-C reaction channel with initial condition sampling patterns biased in favor of C-H modes and for C-H reaction with all sampling patterns. In these cases the extent of rate enhancement with rotational excitation was significantly greater than that predicted by simple statistical theory.
Introduction The role of dynamics in regulating the flow of energy in highly excited polyatomics has been the subject of much recent theoretical and experimental interest. The field has matured significantly since the earliest indications of the extent and nature of dynamical effects provided by classical trajectory calculation^.^-^ A t present a number of experimental results are available to help define, although not without controversy, observational bounds for nonrandom b e h a ~ i o r , ~ while - ~ additional revealing details about intramolecular energy transfer have emerged from theoretical studies showing periodicity in small classical systems.lb13 The latter studies in particular indicate clearly that cases exist for which broadly localized regions of phase space can trap rather large amounts of vibrational energy. Such observations, confined so far to idealized systems, complement newer studies of detailed phonomenological reaction dynamics in larger classical trajectory systems.14-17 (1)D. L. Bunker, J. Chem. Phys., 37, 39 (1962); 40, 1946 (1964). (2) D. L. Bunker, Meth. Comput. Phys., 10, 287 (1971). (3) D. L. Bunker and W. L. Hase, J. Chem. Phys., 59,4621 (1973); 69, 4711 (1978). (4) J. D. Rynbrant and B. S. Rabinovitch, J . Chem. Phys., 54, 2275 (1971); B. S. Rabinovitch, J. F. Meagher, K. J. Chao, and J. R. Barker, ibid., 60,2932 (1974); J. F. Meagher, K. J. Chao, J. R. Barker, and B. S. Rabinovitch, J. Phys. Chem., 78, 2535 (1974). (5) Y. T. Lee, Ber. Bunsenges. Phys. Chem., 78, 135 (1974); J. M. Farrar and Y. T. Lee, J . Chem. Phys., 65, 1414 (1976). (6) J. D. McDonald, Annu. Rev. Phys. Chem., 30,29 (1979); M. Gurnick, J. Chaiken, and J. D. McDonald, J. Chem. Phys., 74, 123 (1981). (7) J. B. Hopkins, D. E. Powers, and R. E. Smalley, J . Chem. Phys., 73, 683 (1980); S. Mukamel and R. E. Smalley, ibid.,73, 4156 (1980); J. B. Hopkins, D. E. Powers, and R. E. Smalley, ibid., 74, 745 (1981). (8) K. V. Reddy and M. J. Berry, Chem. Phys. Lett., 66, 223 (1979); R. G. Bray and M. J. Berry, J . Chem. Phys., 71, 4909 (1979). (9) R. Naaman, D. M. Lubman, and R. N. Zare, J. Chem. Phys., 71, 4192 (1979). (10) K. Sture, J. Nordholm, and S. A. Rice, J . Chem. Phys., 61, 203 (1974). ' (11) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys., 71, 2864 (1979). (12) C. Cerian and W. P. Reinhardt, J . Chem. Phys., 71,1819 (1979). (13) R. J. Wolf and W. L. Hase, J . Chem. Phis., 73, 3739 (1980). (14) J. D. McDonald and R. A. Marcus, J . Chem. Phys., 65, 2180 (1976).
To generalize upon these theoretical results and especially to link theory with experiment, it is important to understand the effects of external factors that influence relaxation in real systems. For example, it has been shown that large impact parameter near-elastic collisions can alter internal energy distributions in systems probed by laserinduced f l ~ o r e s c e n c e . ~ ~ J ~ In the present work we have found by classical trajectory simulation that overall rotational energy has an effect on internal energy flow. We show, by examining the persistance of mode-specific sampling effects on trajectory lifetimes that high rotational excitation couples restricted internal degrees of freedom. The subject of our study is C2HP In earlier classical trajectory calculations on the unimolecular decomposition of this system15 we found evidence for apparent nonRRKM behavior; specific normal mode biases excitation patterns produced lifetimes and lifetime distributions that were clearly dependent on the initial energy distribution. Initial excitation biased broadly in favor of C-H modes retarded both C-C and C-H reaction channels, whereas C-C biased sampling accelerated reaction. Pure C-H asymmetric stretch sampling produced a large initial reactive transient. Pure C-H symmetric stretch sampling gave rise to an induction period of several vibrational periods over which there was no reaction. We also found more subtle, intrinsic non-RRKM behavior. Overall reaction rates in randomly sampled trajectory simulations were compared with RRKM predictions precisely drawn from the same potential energy surface. Trajectory rate constants for both C-C and C-H scission reaction channels were smaller than statistically predicted, with a greater deviation for the C-H channel. (15) E. R. Grant and D. L. Bunker, J. Chem. Phys., 68,628 (1978); J. Santamaria, D. L. Bunker, and E. R. Grant, Chem. Phys. Lett., 56, 170 (1978). (16) W. L. Hase and D.-F. Feng, J. Chem. Phys., 61,4690 (1974); ibid., 64, 651 (1976); C. S. Sloane and W. L. Hase, ibid., 66, 1523 (1977). (17) W. L. Hase, R. J. Wolf, and C. S. Sloane, J. Chem. Phys., 71,2911 (1979). (18) R. Naaman, D. M. Lubman, and R. N. Zare, Chem. Phys., 32,17 (1978). (19) J. C. Weisshar, A. P. Baronavski, A. Cabello, and C. B. Moore, J . Chem. Phys., 69, 4720 (1978).
0022-365418112085-2426$01.25/00 1981 American Chemical Society
Intramolecular Energy Transfer in C2H,
These observations led us to conclude that our classical model polyatomic is intrinsically non-RRKM. Intramolecular energy randomization in this system is restricted on the time scale of dissociation. C-H bonds are the worst offenders. We have now extended this study to examine the influence of high rotational excitation on the behavior of this system. Taking account of energetic effects we find that rotational excitation reduces the difference between observed trajectory rate constants and those predicted statistically. We draw two important conclusions from our observations: (1)rotational excitation appears to dynamically accelerate intramolecular energy transfer particularly for C-H modes, and (2) the improved correspondence between trajectory and RRKM lifetimes, which is apparently the result of enhanced intramolecular energy transfer, tends to validate our earlier conclusion that such transfer was originally restricted and our molecule intrinsically non-RRKM. Classical Trajectory and RRKM Calculations Our studies of rotational effects used the same classical trajectory program described in ref 15. Its potential surface of Morse and attenuated angular harmonic oscillators permits decomposition via C-H and C-C scission: C2HG CzHj + H (1)
The Journal of Physical Chemistry, Vol. 85, No. 16, 198 1
"2 1655 cm'l
"3 9 3 6 cm-l
2427
6 ' 1555 c m - I
'/I2 9 5 6 cm-1
"9 1165 cm-l
Flgure 1. Normal modes energized to create sampling bias labeled
C-CI. All modes excited equally.
"5
2863 cm-'
v7
(2)
2968 cm"
V8
0)
1546 cm"
Figure 2. Normal modes energized to create sampling bias labeled C-HI. All modes excited equally.
+
C2H6 2CH3 (2) We use a normal mode sampling algorithm that allows either random sampling or initial excitation biased in favor of a chosen mode or collection of modes. In our earlier work we took steps to subtract spurious angular momentum generated during sampling of vibrational coordinates. As a final step in the initialization processes we then added specific amounts of overall and internal angular momentum sampled from thermal distributions. We used this same facility to produce rotational excitation in the present work by simply specifying a high temperature for the sampled overall angular momentum distribution (the distribution sampled for CH3 internal rotation was held at 300 K). The potential energy surface was fully characterized so that it was possible to compare trajectory results with precisely drawn RRKM calculations. Vibrational analyses were carried out at intervals along each reaction coordinate. Loosened frequencies so determined were used in minimum state density searches to find self-consistent critical configurations for each energy of reaction. The RRKM program20 also allowed us to properly account for the energetic effects of overall rotational excitation on unimolecular decomposition. Such effects accompany moment of inertia changes associated with the decomposition process.21 As a molecule dissociates its moments of inertia, especially those connected with the reaction coordinate, generally increase. Depending on the nature of the fragmentation pathway, these increases, proceeding from reactant toward critical configuration (transition state), may either be small, as in the case of hydrogen atom loss, or large, as might accompany a hydrocarbon sketetal C-C scission. In all cases angular momentum will be conserved. For constant angular momentum, as a moment of inertia increases the rotational energy associated with that moment decreases. Total energy is, of course, conserved and the rotational energy +
(20) W. L. Hase and D. L. Bunker, Quantum Chemistry Program Exchange, 234 (1975). (21) See, P. J. Robinson and K. A. Holbrook, "Unimolecular Reactions", Wiley, New York, 1972.
lost appears as an increase in the internal energy of the reactant; it adds to the energy available to drive the reaction. Thus, for unimolecular decomposition accompanied by appropriate structural changes, rotational excitation accelerates reaction. For the present case, the moment of inertia changes for both C-H and C-C reaction channels were parameterized as a function of reaction coordinate displacement and included as components of respective minimum state density searches in the RRKM calculations. The appropriate correction to the critical configuration energy in each case was determined by taking the difference between the average rotational energy for a thermal distribution of reactant and the energy corresponding to the same angular momentum in the appropriate transition-state geometry. As will be discussed below trajectory results displayed effects beyond those which could be explained by accounting for the energetic effects of rotation. Results Several broadly localized sampling patterns, two from our earlier work and two new ones, were chosen to examine the dynamical effects of rotational excitation. Modes sampled to create the patterns C-CI and C-HI are illustrated respectively in Figures 1 and 2. The sampling pattern labeled C-HI1 is simply the complement of C-CI. The same logic applies to C-CII. In all cases available vibrational energy was partitioned equally among the modes of particular pattern being sampled. The phase of each mode in the pattern was independently randomized for each trajectory. To facilitate comparison with existing 210 kcal mol-l rotationally cold data for these patterns, rotationally hot trajectories were run with the same total energy. To achieve this, overall angular momentum was sampled from a thermal distribution a t 6000 K, internal angular momentum was thermalized, and the remaining available energy was apportioned to vibration by scaling. Following initialization, each trajectory was integrated for 0.4 ps or until a reaction occurred. Reactive events were counted and rate constants computed in the normal way.'5p22 Results for each channel, with each of four
2428
The Journal of Physical Chemistry, Vol. 85, No. 16, 1981
Grant and Santamaria
TABLE I: Summary of Trajectory and RRKM Theory Unimolecular Decomposition Rate Constants for Classical C,H, at a Total Energv of 210 kcal mol-' unimolecular rate constants," s-l rotational temp, K RRKM c-CI c-CII C-HI C-HI1 C-H channel 300 7.3 x 10'2 8.5 x 10" 4.3 x 101' 2.2 x 10" 3.3 x 10" 6000 5.0 X 10l2 7.8 X 10" 6.1 X 10" 2.6 X 10" 4.0 X 10" C-C channel 300 2.5 X 10" 1.5 X 10" 1.1 x 10l2 9.4x 10'0 2.5 X 10" 6000 3.9 x 10'2 1.9 x 10l2 3.9 x 10l2 3.9 x 10" 5.2 X 10"
" Statistical and trajectory for different sampling biases. TABLE 11: Parameters for RRKM Calculations Frequencies in Molecule (cm-l) 2872 1655 1555 2968 2968 1545 1545 1165 1165 2976 2976 1520 955 2862 936 1520 955 Internal rotor in molecule: reduced moment of inertia = 1.61amu A symmetry number = 3 Frequencies in C-H Critical Configuration (cm" ) 1630 2067 2966 2901 1548 1534 2972 2975 1528 834 1102 1006 991 181 1185 110 Internal rotor: reduced moment of inertia = 2.35 symmetry number = 1 Frequencies in C-C Critical Configuration (cm-' ) 2973 2973 2974 2974 1534 1534 1533 1533 2851 2851 1041 1041 411 411 379 379 Internal rotor: reduced moment of inertia = 1.61 symmetry number = 3
sampling patterns at low and high rotational temperatures, are given in Table I. Sample sizes for these computations range from 522 for C-CII sampling at 6000 K to 646 for C-HI1 sampling at 300 K. Estimated twothirds error bounds in all cases are no greater than f20%.22 Corresponding RRKM rate constant calculations were carried out by using the following parameters derived from the trajectory potential energy surface. Minimum state density searches along reaction pathways for each channel produced effective thresholds of 70.0 kcal mol-l for C-C and 93.4 kcal mot1 for C-H scission. Critical configuration frequencies are tabulated in Table 11. The moment of inertia changes proceeding through the transition state are from 6.46, 25.50, 25.50 amu A2to 11.64, 28.14, 33.31 amu A2 for the C-H channel and to 6.46,65.42,65.42 amu A2 for the C-C channel. With these parameters the average rotational energy of the C2H6molecule at 300 K, 0.89 kcal mol-', decreases to 0.63 kcal mol-l in the C-H exit channel and 0.46 kcal mol-l in the C-C exit channel. For a rotationally hot molecule, with a larger share (on average 17.9 kcal mol-' combined (22) For most of these sampling patterns excitation biases produced lifetime distributions that were not simple single-exponentialdecays, but rather two-component curves characterized by an initial transient of higher or lower reactivity.l6 For the purpose of uniformly comparing the effects of rotation, initial transients were disregarded in computing trajectory rate constants. Reactive events in the exponential long-time region were counted to yield a rate constant by 1 At
k=-ln-
N N - NR
where N is the total number of trajectories of which NR are reactive in the time interval At. Error is governed by the statistics of NR. For example, the total reactivity with C-CI sampling at 6000 K consisted of 66 reactive events for 189 post-transient trajectories over the interval 1.4 X s for an overall rate constant of (3.0 f 0.6) X lo1*,partitioned to C-C and C-H channels by the branching ratio.
Stat isti ca I Prediction
-in E 0
RRKM
Trajectory
C-C
I
C-C
II
Results
C-H
I
C-H
II
I
C-H
II
0 c 0
a
C-C
RRKM
C-C
I
C-C
II
Reaction
C-H
Flgure 3. Effect of rotational energy on rates of unimolecular reaction as predicted by RRKM theory and observed in classical trajectories: open bars, TR = 300 K; shaded bars, TR = 6000 K.
energy) in rotation, the corresponding energies in critical configurations are 12.7 kcal mol-I in the C-H exit channel and 8.8 kcal mol-l in the C-C exit channel. Thus, to preserve the same total energy compared with normal trajectories, our rotationally hot molecules are constrained to begin with less vibrational energy. However, as they reach critical bond extensions they regain a certain fraction of this deficit. These factors have a competing effect on the calculated RRKM rate constants. The lower initial vibrational energy tends to reduce the reaction rate, while energy release in the critical configuration promotes decomposition. The net result is a calculated increase in the RRKM C-C reaction rate, where rotational energy release is large, and a decrease in the C-H rate, where the smaller moment of inertia change does not fully compensate for the lower vibrational energy. These statistical rate constants are compared with the corresponding trajectory results in Table I. All results are presented graphically in Figure 3.
Discussion As shown in Figure 3, RRKM calculation, corrected for the energetic effect of rotation, predicts that the C-H channel reaction rate will be retarded by putting available energy into rotation. Examination of the trajectory results, however, shows the opposite behavior. In all cases but one rotational excitation enhances the rate of C-H reaction. With all sampling patterns rotation is more effective than predicted in promoting C-H scission. When we turn to the C-C channel, where enhancement is predicted, we find that, for C-C sampling, enhancements in accord with expectations are obtained. However, for C-H biased sampling, where the C-C channel is severely retarded at low rotation, we find that rotational excitation
J. Phys. Chem. 1981, 85,2429-2430
produces significantly greater enhancements. The effects observed here form a systematic pattern when considered in the light of earlier results. We have evidence that the C-H vibrational degrees of freedom for our classical model of C2Hstransfer energy on a time scale that is slow compared with reaction at the energies studied. This causes retarded reaction rates for C-H scission with all but the most localized initial excitation pattern, and makes C-H biased excitation less effective in producing reaction. In the present study we see rotational excitation (taken away from vibrational) accelerating C-H reaction when it should be retarding it. Furthermore, and perhaps more significantly, we see substantial rotational enhancement of C-C reactivity for a highly unfavorable C-H sampling pattern. This can be explained if we conclude that rotational excitation accelerates intramolecular energy transfer into and out of the C-H modes in our model ethane. Enhanced energy transfer out of these modes increases the C-C scission rate for C-H biased excitation.
2429
These results tend to validate our earlier findings on the importance of dynamical effects in the unimolecular decomposition of highly excited molecules. We concluded from a comparison of trajectory rates with RRKM calculations that, under 300 K thermal rotational conditions, our model C2H6 was intrinsically non-RRKM; overall trajectory rates were consistently smaller than expected. With high rotational excitation, trajectory results uniformly move closer to RRKM prediction, and apparent non-RRKM behavior associated with localized initial excitation diminishes. This elasticity in trajectory decomposition rates both clearly indicates the role of weak perturbations in promoting intramolecular energy transfer and provides additional, rather compelling, evidence for intrinsic non-RRKM behavior in isolated classical polyatomics.
Acknowledgment. We are grateful to the National Science Foundation for support of this work (CHE7920858).
COMMENTS where F,, the standard chemical potential of this species, is a function of T and p only. Let yr be the fraction of micelles with aggregation number N,. Obviously
Thermodynamics and Statistical Mechanics of Nonionic Micelles
Sir: We published some years ago a general thermodynamic account of micellization in solutions of nonionic surfactants. A recent paper by Wulf’ contains several erroneous statements concerning our work,z and we feel it will be useful to correct these errors and to comment on some related aspects of the interpretation of experimental data. Our first point concerns the statement that we assume micellar monodispersity. For a single-component surfactant solution at equilibrium in which micelles and monomers behave ideally, our eq 16-49 may be written as 1.1’
= F(T,p,lY)
+ kT In x,
1.1’
where N is the mean micelle aggregation number and x, is the mole fraction of micelles. Also
NI.1 (2) where p is the chemical potential of surfactant monomer. Equations similar to eq 1 have been used by Hill3 and it is clear both from his work and from our paper that monodispersity is not implied. If it were, then N would not be a continuous variable. To demonstrate this more clearly we consider a series of micellar species in equilibrium both with each other and with monomer. Let the rth micellar species have an aggregation number N,. Assuming ideality, the chemical potential of this species, p r , may be written as p, = F,(T,p) + k T In x , = N r p (3) I.1. =
(1) A. Wulf, J.Phys. Chem., 82, 804 (1978). (2) D. G. Hall and B. A. Pethica in “Nonionic Surfactants”, M. J. Schick, Ed., Marcel Dekker, New York, 1967. (3) T. L. Hill, “Thermodynamics of Small Systems”, Part 2, W. A. Benjamin, New York, 1963, Equations 7-198, 10-199, 10-257. 0022-3654/8 1/2085-2429$01.25/0
(44
Yr = X r / X m
(4b)
N = CYJ, r
(44
= Cyrpr = m1.1
(4d)
r
From eq 3 and 4 we obtain p* = C y T r k T C y , In yr + k T In x ,
+
r
(1)
Cxr r
x, =
r
(5)
which when compared with eq 1 shows that
F = CYJr = k T C y r In Yr r
(6)
r
Noting that Crdyr = 0 we readily deduce from eq 3-6 that at constant T and p d F = C ( F r k T In y,) dy, + k T In xmCdyr r
+
r
= C p r dyr = PCLcNr dyr = P r
r
(7)
This justifies writing F = F(T,p,IY) in eq 1 and shows clearly that all deductions based on eq 1 refer to polydisperse micelles. A similar procedure has been outlined by Hill4 and has been used to discuss both single-component2 and multicomponent micelle^.^ In some respects eq 3 is a better starting point for discussing polydisperse micelles than eq 1. Our second point concerns the approximation that changes in solvent composition are negligible. We do not (4) Reference 3, Chapter 10, p 131. (5) D. G. Hall, Trans. Faraday Soc., 66, 1351, 1359 (1970).
0 1981 American Chemical Society
