9738
J. Phys. Chem. 1996, 100, 9738-9744
Energy Transfer Rate Coefficients from Trajectory Calculations and Contributions of Supercollisions to Reactive Rate Coefficients V. Bernshtein and I. Oref* Department of Chemistry, TechnionsIsrael Institute of Technology, Haifa 32000, Israel
G. Lendvay Central Research Institute for Chemistry, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary ReceiVed: NoVember 13, 1995; In Final Form: March 13, 1996X
Quasiclassical trajectory calculations on the CS2-CO system were performed. Analytical biexponential functions were fit to the trajectory results, and energy-dependent energy transfer transition probability functions and rate coefficients were derived. They, in turn, were used in solutions of master equations. Unimolecular rate coefficients for cyclobutane fission and cyclobutene isomerization in Ar bath gas at various temperatures were obtained. Supercollisions, collisions which transfer more than 5 times the average energy transferred in a down collision, were found to contribute to the high-energy tail of the biexponential transition probability function. To assess their contribution to the unimolecular rate coefficients, their values which were obtained from trajectory-based double-exponential transition probabilities are compared with those obtained from singleexponential weak-collision transition probabilities. For cyclobutane fission an ∼5-fold increase in the value of the rate coefficient is found at 1000 K and ∼7-fold increase at 1500 K when the biexponential function is used. For cyclobutene isomerization the change ranges from an ∼7-fold increase at 500 K to an ∼9-fold increase at 1500 K. Since the magnitude of supercollisions and, thus, the magnitude of the tail of the probability distribution can vary depending on the system, a systematic study was performed on how the high-energy tail affects the values of the unimolecular rate coefficients. It was found that for weighing factors of the strongcollision contribution to the biexponential probability function of 0.5% or 10% of that of the weak-collision contribution (with an exponential parameter of 300 cm-1) the rate coefficient increases with the size of the strong-collision exponent and with temperature. When the contribution of the weak-collision exponent to the energy transfer probability function was artificially removed, it was found that in some cases, 0.5% of the strong-collision part contributes ∼70% to the overall rate coefficient while 99.5% of the weak-collision part contributes only 30%.
Introduction The value of a unimolecular rate coefficient of a substrate reacting in the gas phase depends strongly on the type of the bath gas present.1,2 The effect of the bath gas is through its energy transfer capabilities, which translates into activation and deactivation of the substrate molecule. A bath gas molecule which transfers large quantities of energy in one collision is called an efficient or a strong collider. It provides large values of the rate coefficient, below the high-pressure limit, relative to a bath gas molecule which transfers only small quantities of energy on each collision. The latter case, called a weak collider, is typical of monatomic or diatomic bath gases while the former is typical of large polyatomic molecules. The efficiency of a collider is measured by the value of the average energy transferred per collision 〈∆E〉. This is obtained by the expression
〈∆E〉 ) ∫0 ∆EP(E′,E) dE′ ∞
(1)
where P(E′,E) is the normalized energy transfer transition probability function starting at molecular energy state E and ending in state E′. P(E′,E) obeys detailed balance.
B(E′) P(E,E′) ) B(E) P(E′,E)
(2)
where B(E) is the Boltzmann distribution function of internal energies. X
Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(95)03341-7 CCC: $12.00
The functional form of P(E′,E) is not known experimentally, and therefore in the past, simple mathematical expressions were chosen either for their easy-to-grasp intuitive shape, Gaussian, or for their ease of handling, stepladder, or for their ease of obtaining analytical expressions, exponential. The stepladder model was chosen for cases of strong collisions while the exponential model was used mainly for weak colliders. Once an energy transfer transition probability function is chosen, the change in the population as function of energy can be obtained by solving a master equation:
df(E,t)/dt ) - ∫0 ωP(E′,E) f(E,t) dE′ + ∞
∫0∞ωP(E,E′) f(E′,t) dE′ - k(E) f(E,t)
(3)
where f(E,t) is the population distribution function at energy E at time t, ω is the collision frequency which is taken to be independent of the energy and k(E) is the energy-dependent RRKM theory rate coefficient. Once the population as a function of time and energy is known, the rate coefficient can be found. This will be discussed in the next section. In many cases the energy dependence of ω is not known, and therefore ω is taken to be independent of energy and ω is transported outside the integral. However, in the present work no such assumption is made and energy transfer rate coefficients which are products of probability and collision frequencies are used instead. This is discussed at great length below. © 1996 American Chemical Society
Energy Transfer Rate Coefficients The common wisdom until a few years ago was that the functional form of P(E′,E) does not affect the solution of eq 3, only the value of 〈∆E〉 does. This has changed with the discovery of supercollisions. These are collisions which exchange large quantities of energy in a single event. These were found experimentally3-7 and in trajectory calculations.8-12 They are defined as collisions which transfer a ∆E which is more than 5 times the value of the average energy transferred in a down collision15 or 5 times the root mean square of the energy transferred per collision.12 For example, supercollision ∆E’s are found to be >1500 cm-1 for benzene-Ar12,15 and >2500 cm-1 for toluene-Ar.10,11 Supercollisions contribute to the tail of the energy transfer probability function and affect the value of f(E,t). In an attempt to find the contribution of supercollisions to the value of the rate coefficient, the master equation was solved13 using a double-exponential transition probability 13,14,16 for down transitions
P(E′,E)d ) (a1 exp(-∆E/R1) + a2 exp(-∆E/R2))/C(E) (4) where ∆E ) E′ - E; C(E) is the energy-dependent normalization constant which takes care of detailed balance and conservation of probability. a1 and a2 are the fractions of normal collisions (1) and strong collisions (2), respectively. The two exponential functions were assumed to describe “weak” or normal collisions and strong collisions, respectively, and R1 and R2 were assigned the values of the average energy transferred in a normal collision and in a strong collision, respectively. Since supercollisions are the high-energy end of the second exponential, the value of R2 can actually be less than the values of the energy transferred in supercollisions. By the same argument, a2 can obtain values which are larger than the fraction of supercollisions in normal collisions. One of the motivations of the present work is to explore the interrelations between the four parameters in eq 4 and how their relative values affect the values of the unimolecular rate coefficients. In model calculations of the isomerization of cyclobutene to butadiene using biexponential energy transfer probability distributions and estimated exponential parameters, it was found13 that for the 0.1% and 10 000 cm-1 for a2 and R2 respectively in 300 cm-1 normal collisions the value of 〈∆E〉 changes by a factor of 2.6 at 700 K and by a factor of ∼4 at 1500 K. Increasing the value of a2 to 0.5% increases the value of by a factor of 8.6 at 700 K and by a factor of 14.5 at 1500 K. Similar effects were seen with the values of the rate coefficient for isomerization. It increases by a factor of 4 and 11 for 0.1% and 0.5% respectively at 1500 K. Similar results were obtained for cyclobutane fission. Thus, small quantities of supercollisions have a very significant effect on the value of the rate coefficient. The value of 10 000 cm-1 in the above study13 was based on the fact that in trajectory calculations on the system of CS2 colliding with CO9 some collisions transferred as much as 10 000 cm-1. Detailed trajectory calculations10-12,15,19 indicate, however, that smaller values for supercollision are possible. In addition, fitting trajectory results to biexponential functions indicate that a2 can take much larger values.10-12,15,19 As indicated before, it seems worthwhile, therefore, to make a systematic study on the effect of the magnitude of the value of a2 and the exponent characterizing the supercollision contribution to the energy transfer probability on the rate coefficient of a unimolecular reaction. Classical trajectory calculations combined with Monte Carlo selection of initial conditions unambiguously yield the energy transfer (ET) rate coefficients R(E′,E)12,19 (instead of the probabilities P(E′,E) used in eq 3 above). An important condition that ensures that the calculations do take into account
J. Phys. Chem., Vol. 100, No. 23, 1996 9739 all trajectories that contribute significantly to energy transfer is that the maximum impact parameter bm be large enough. The proper value of bm was chosen according to the method described in refs 20 and 21. Once bm is known, a large set of trajectories is calculated at a given value of the initial energy, E, in the relaxing molecule. The initial conditions are selected so that the internal phase space of the excited molecule is sampled microcanonically and the intermolecular parameters sample a canonical ensemble.22-25 In order to obtain energy transfer rate coefficients, the resulting trajectories are binned according to the value of the energy transferred, ∆E ) E′ - E. The fraction, Ptraj(E′,E), of obtaining energy E′ when the initial energy is E is calculated by
Ptraj(E′,E) ) Ntraj(E′,E)/(Ntraj(E) δ(E))
(5)
where Ntraj(E) is the total number of trajectories with initial energy E in the excited molecule, Ntraj(E′,E) is the number of trajectories resulting in final energy corresponding to the bin around E′, and δ(E) is energy increment (bin width). The inelastic collision rate coefficient is given by
R(E′,E) ) πbm2 〈V〉Ptraj(E′,E)
(6)
No unique value, however, can be obtained for the elastic rate coefficient R(E,E) from classical trajectories: it diverges as bm is increased as elastic collisions occur at any impact parameter. However, the lack of a unique value for R(E,E) does not affect the calculations. They are insensitive to the height of the elastic “peak” (on the R(E′,E) vs E′ plots) because the elastic term does not appear in the master equation in either the presence or absence of reactions. Master equation calculations require an analytical function for R(E′,E) unlike the fluctuating functions obtained from trajectory calculations. Therefore, analytical curves are fitted to the trajectory calculated histograms. In the fitting so far, a biexponential function was applied 12,19 but other functions can probably be used as well. As long as the bin width in the histograms is fine enough to represent the shape of the rate coefficient function R(E′,E) as a function of E′, the results of the fit will be insensitive to the grain size. Early master equation calculations have used assumed energy transfer probability functions such as exponential, stepladder, or Poisson,1,2,26 which obeyed conservation and detailed balance. Those were used basically as fitting tools for experimental results. However, the meaning of probability function in trajectory calculations is different. The trajectory method gives directly energy transfer rate coefficients from which, in principle, the probability can be defined by
Ptraj(E′,E) ) R(E′,E)/Z
(7)
for any collision rate Z. As long as Z is the hard sphere collision rate calculated with the maximum impact parameter of the trajectory calculations, the probabilities will be identical to those obtained by binning in eq 5. This is in contrast to fitting of experimental results where the rate coefficients were defined by multiplying ad hoc probabilities by the Lennard-Jones collision number, ZLJ. The only constraint on rate coefficients is that they obey detailed balance. To distinguish between the two types of probability functions, we indicate the trajectory one obtained from eq 7 by the subscript “traj”. In the present work, we first obtain energy transfer probability functions from trajectory calculations for the relaxation of CS2 in a bath of CO at 2000 K. Biexponential functions are fit to the probability distributions, Ptraj(E′,E), histograms at various initial energies E. (This is done in order to compare the present
9740 J. Phys. Chem., Vol. 100, No. 23, 1996
Bernshtein et al.
results with previous experimental and calculational ones. It also facilitates separating the weak- and strong-collision contributions.) Unimolecular rate coefficients are obtained from solutions of master equations. The relative magnitudes of the exponential and the weight parameters are varied in a systematic way, and their contributions to the values of the unimolecular rate coefficients are assessed. Then, on the basis of these results, we vary systematically the magnitude of the exponent of the supercollision contribution and its weighing factor in the biexponential energy transfer probability function and calculate the effect these changes have on the unimolecular rate coefficient. Finally, we remove once the weak-collision exponent and once the strong-collision exponent and observe how each one of them affects the values of the unimolecular rate coefficients and the values of 〈∆E〉. Our previous work13 on the contribution of supercollisions to the rate coefficient used the experimental values of the a’s and the R’s in an assumed probability function. Trajectory results give a totally different set of parameters. The reason for the radical difference is not yet known, but any study which uses a trajectory-based probability function has to use the values of the parameters in a consistent way, that is, those obtained from trajectory results. The present work provides, in addition to other things, a comparison between experimentally based parameters and trajectory based ones. Theory This section is divided into three topics. The first deals with the solution of the master equation and the evaluation of the rate coefficients. The second deals with the derivation of Ptraj(E′,E) from trajectory calculations, and the third addresses the nontrivial problem of normalization which ensues when trajectory calculation results are converted into energy transfer probabilities and rate coefficients. Solution of Collisional Energy Transfer Master Equation. The details of the solution of the master equation, eq 3, are given in refs 17 and 18. Briefly, the master equation is cast into a matrix form
dF h /dt ) H‚F h
(8)
where F h represents the population vector and H is a n × n matrix whose elements are the transition probabilities and the RRKM theory rate coefficients which appear in equation 3. The solution of equation 8 yields a time dependent expression for the population: n
n
n
f(Ei,t) ) ∑Rijβj exp(-λjt)/∑ ∑ Rijβj exp(-λjt) j)1
(9)
i)1 j)1
The population distribution at energy state i, Ei, is determined by the eigenvectors Rij, the constants βj which are determined by the boundary conditions, and by the set of eigenvalues λj which are real numbers and determine the value of the rate coefficient. At low and moderate temperatures where steady state, df(E,t)/dt ) 0, is obtained18 the rate coefficient is equal to the eigenvalue with the smallest absolute value. Under nonsteady-state conditions the full expression for the rate coefficient is to be used.18 It is important to point out that the value of Ptraj(E,E), i.e., ∆E ) 0, does not enter into the calculations of the population distribution and rate coefficient since it is canceled out in the master equation. Energy Transfer Transition Probabilities and Rate Coefficients from Trajectory Calculations. In the present study we used trajectory calculation methods reported before20-22 to derive energy transfer rate coefficients and probability distribu-
tions. The actual system studied in the trajectory work is CS2 + CO. Various approaches were applied in the past to obtain the energy transfer probability distribution covering both the “down” and “up” wings. It is possible to fit both wings separately, but then detailed balance is not guaranteed. From the point of view of statistical methods, it is easier to get a good and stable fit to the histograms on the “down” wing because more points are available.19 In principle, both wings can be fit simultaneously by imposing the criteria of detailed balance. This procedure can be successful if one is interested in Ptraj(E′,E) at one single energy E and a simple analytical formula is used to ensure detailed balance.12 If, however, the full probability matrix is to be calculated, the fit to Ptraj(E′,E) at many initial energies is to be done simultaneously. In this case, the number of fitting parameters is so large as compared to the number of points that it is impractical to use this approach. Therefore, the entire probability matrix was created in the way described in ref 19. The down wings of the probability distributions Ptraj(E′,E) were fitted to biexponential functions at each initial energy E
Ptraj(E′,E) ) a1(E) exp(-∆E/R1(E)) + a2(E) exp(-∆E/R2(E)) (10) and the up wings were calculated by detailed balance, using the densities of states of the relaxing molecule. The energy dependence of the fitting parameters of the biexponential functions was obtained by fitting them to analytical functions of the form
ai(E) ) c1i + c2iE
(11)
The fitting parameters a1(E) and a2(E) have dimensions of 1/energy. Since the objective of our study is to find the effect of supercollisions on the rate coefficients of unimolecular reactions, and since CS2 does not dissociate at the energies used in this work, we have chosen to apply the probability matrix to the well-known reactive systems of cyclobutene and cyclobutane.13 The whole probability matrix was converted into the energy transfer rate coefficient matrix of the cyclobutene and cyclobutane compounds by multiplying by πbm2〈V〉, where the relative average velocity includes the reduced mass of the appropriate substrate and bath gas. To reiterate, the outcome of the trajectory calculations is the binning of the number of trajectories as a function of ∆E. This yields, when divided by the total number of trajectories, the value of Ptraj(E′,E). The relative height of the elastic peak is a function of bm used in a particular calculation, and therefore its size is arbitrary. The scaling of the ∆E ) 0 peak, the “normalization procedure”, is complicated and leaves room for various interpretations. One procedure to normalize probability is to calculate the value of the zero energy transition Ptraj(E,E) from the value of the overall probability of obtaining inelastic collisions:
Ptraj(E,E) ) 1- ∑Ptraj(E′,E)
for all E * E′ (12)
Various other approaches were used to solve the normalization problem,12,19,20 fitting both wings of the histogram to a predetermined function such as an exponential or a doubleexponential function, adjusting the increment size until constant values are obtained for 〈∆E〉 and for the energy transfer rate coefficients, to name only few. All these methods are supposed to take care of the magnitude of the ∆E ) 0 peak and to yield the “true” shape of the energy transfer transition probability. In the present work we use energy transfer rate coefficients instead of probabilities which, in principle, avoid some of the problems
Energy Transfer Rate Coefficients mentioned before. It should be recalled that the arbitrary peak R(E,E) does not enter into the calculations of the master equation. Details of the Trajectory Calculations. The methods applied in this work have been reported elsewhere.20-22 The calculations were done using an extensively modified version of the computer code VENUS.27 The energy transfer parameters were calculated for CS2 + CO. The potential energy surface for the system is the same as described earlier.22 Ten thousand trajectories were run at each initial vibrational energy in CS2 from 1750 cm-1 to 33 230 cm-1 in 1750 cm-1 intervals. General Considerations of the Nature of Ptraj(E′,E). There are two question that must be addressed. One can the CS2/CO probability function, P(E′,E), be used for the cyclobutene/Ar and cyclobutane/Ar systems? The second is why use it at all? Why not run trajectories directly on the polyatom/Ar systems? The ultimate way of an “ab initio” calculation of the effect of energy transfer on the kinetics of unimolecular reactions would be to start with potentials, inter- and intramolecular, use trajectories to calculate R(E′,E), substitute that in a master equation, and obtain the unimolecular rate coefficients. Instead of using assumed probability function, use the real R(E′,E). This aim cannot be achieved today as a routine matter. This is because the number of trajectories needed for such calculations is very large and the time required to obtain them is extremely long. We need about 10 000 trajectories per internal energy point and we need at least 20 energy points to have a complete mapping of Ptraj(E′,E) for a single polyatom/Ar system at a single temperature. Each cyclobutene/cyclobutane trajectory takes about 1.3 min. The overall resources, time and money, needed to run the 200 000 trajectories, for a single polyatom/Ar system, are huge and way out of our reach. On the other hand, we have at our disposal a unique and extensive bank of CS2/CO trajectories which was accumulated over years of work.9,19-22,29 A realistic study of dissociative processes in CS2, however, is not possible. Besides the dissociation threshold being above the energy range of the trajectory calculations, nothing is known about its RRKM parameters or any other parameters used in other theories to calculate k(E). Therefore, only a model calculation could be performed to reveal the major effect of the calculated Ptraj(E′,E) on the kinetics of decomposition.29 On the other hand, there are two systems, the decomposition of cyclobutane and the isomerization of cyclobutene, for which all parameters needed in the RRKM treatment are known but energy transfer information is not available. Below, we show that the application of the down probability functions obtained for the CS2/CO system for studying the energy transfer in the cyclobutane/Ar and cyclobutene/Ar systems is reasonable. There are clearly large differences between CS2 + CO and, say, cyclobutane + Ar. The main differences are that (1) CS2 is a triatomic molecule and cyclobutane is polyatomic and (2) CO has internal degrees of freedom while Ar does not. The detailed numerical values characterizing the relaxation of CS2 and cyclobutane may be quantitatively different but qualitatively the same. All available trajectory calculations on energy transfer probabilities from highly excited triatomic and polyatomic molecules with monatomic colliders resulted in probability distributions with a long tail at high values of 〈∆E〉.8-10,12,19,29 Until the discovery of supercollisions it was accepted that the shape of P(E′,E) does not affect the value of kuni obtained from the master equation; only the value of 〈∆E〉 is important. Supercollisions have changed that inasmuch as the high-energy tail plays a role in the reactive processes. From the trajectory calculations we know that the tail of the distribution has similar weight in the down wing of P(E′,E) in all systems for which trajectory data are available. It seems realistic, therefore, to
J. Phys. Chem., Vol. 100, No. 23, 1996 9741 assume that a large part of the difference between intimate details of different systems is probably averaged out. As to the nature of the collision partner, extensive studies, performed before,22 show that CO acts like a monatomic collision partner: neither its vibration nor its rotation accepts a significant portion of the energy lost by CS2. As a result, if one borrows the functional form of the CS2 + CO system for the cyclobutane + Ar or cyclobutene + Ar system, the major difference will be in the up wing of the energy transfer probability matrix. Therefore, when we calculated the up wing of the probability matrix for cyclobutene or cyclobutane from detailed balance, we used their densities of states. Similarly, the parameters used in the calculations of the collision frequencies and in the k(E)’s are those of the cyclobutane/Ar and cyclobutene/Ar systems. Density of states seems to have a minor, yet unclear, effect on the physics of energy transfer. In collisions between large polyatoms and bath gas the size of the hot molecule plays an uncertain role in the energy transfer process. For example, in collisions between bath gas molecules and members of the homologous series CnH2n-1, with almost the same internal energy E, the only effect the size has on energy transfer is the geometric increase in cross section.30 The density of states which increases dramatically with size has no effect on the value of 〈∆E〉.30 The collisional efficiency, after normalizing for variations in hard sphere cross sections, for Ar colliding with heptane or octane is practically the same. Actually this effect is used to measure the incremental increase in cross section. In the homologous series CnF2n+2 there is a 30% decline in the value of the energy transfer rate coefficient as well as there is a decrease in the values of the collisional efficiency on going from C3 to C8.31 For aromatic substrates there is an increase in the value of the collisional efficiency for the series benzene,32 toluene,33,34 and azulene.35,36 Thus, there is still room for explorations on the effect of the density of states on the energy transfer process. The density of state considerations which apply to large polyatomic molecules may not strictly be true for a molecule as small as CS2; however, animation of polyatom trajectories15 show that a local event, an out of plane mode for example, is instrumental in the energy transfer process. This leads one to believe that the mechanism of energy transfer is similar and does not depend strongly on the density of states. Application of the “experimental” R(E′,E) derived from trajectory calculations is preferred, we think, to using an assumed probability function. The latter approach is conductive to systematic studies where relative behavior is investigated.13 The model studies reported in the present work are the closest to reality as possible, and the results, we believe, truly represent the qualitative behavior of all polyatom/Ar systems. Results and Discussion There are two objectives to this work. The first is obtaining Ptraj(E′,E) and R(E′,E) from trajectory results and applying the latter to master equation calculations, and the second is to effect a systematic change in the values of ai and R2, which control the tail of the distribution which hides supercollisions. Energy transfer probabilities from trajectory calculations on collisions between CS2 and CO at 2000 K are presented in Figure 1. The probabilities obtained at each initial energy E in CS2 are fitted by biexponential functions, eq 10. The fitting parameters a1, a2, R1, and R2 depend linearly on the excitation energy E, eq 11. The parameters for the fit are given in Table 1 and in Figure 2. The weighting factor of the high-energy exponential, a2, as it relates to the weighting factor of the low-energy exponential as a function of energy is given in Figure 3. As can be seen, the ratio varies from ∼7 at low energy to ∼11 at high excitation. The fitting was performed without taking the ∆E ) 0 transition
9742 J. Phys. Chem., Vol. 100, No. 23, 1996
Bernshtein et al.
Figure 1. Ptraj(E′,E) obtained from classical trajectory calculations for CS2 + CO at 2000 K. The histograms show the probability of getting from various initial energy E (positioned at the elastic peak) to different final energies E′. Note that the probability scale is logarithmic and the distributions characterizing different initial energies are shifted upward with increasing E for presentation purposes.
TABLE 1: The Parameters, ci, of the Energy Dependent Parameters of the Biexponential Function for the Down Wing Part (eq 10) ri (a) and ai (b) for the Collision System CS2 + CO at 2000 K (a) c1, R1 R2
cm-1
c2
Figure 2. Energy dependence of the parameters of the biexponential function for the down wing part (eq 10) Ri and ai for the collision system CS2 + CO at 2000 K.
9.010 × 10-3 4.079 × 10-2
97.74 9.44 (b) c1,
a1 a2
(cm-1)-1
5.65 × 10-4 5.245 × 10-5
c2, (cm-1)-2 -6.842 × 10-9 3.271 × 10-10
in consideration. As can be seen, the parameters of the biexponential function differ from those used in a previous study13 (a2 ) 0.001 and 0.005; R1 ) 300 cm-1 and R2 ) 150010 000 cm-1). R2 is smaller and a2 is larger. They are of the same order of magnitude as those obtained from a fit to trajectory results of collision between inert bath gases and benzene and hexafluorobenzene.12 This difference is the motivation for the systematic study of the effect of R2 and a2 on the unimolecular rate coefficient reported in the last part of this work. Next, we would like to explore the contribution of the highenergy tail to the probability and compare it with collisions that transfer average energies. We therefore take the ratio of the probability of a supercollision transferring a given amount of energy to the probability of transferring an average amount of energy. It is important to point out that we cannot report absolute probabilities, desirable as it might be, but only ratios of probabilities. The reason for that is explained in the previous discussion of the normalization problem of the probability function. The ∆E ) 0 peak is a function of bm, and there are always uncertainties associated with the normalization procedure. We define a supercollision as a collision that transfers more than five times the average energy transferred in a down collision.15 The average energy transferred per down collision for the CS2 + CO system was found to be about 350 cm-1. Therefore, we have calculated the ratio of the energy transfer rate coefficient of transferring 1750 cm-1 energy to the rate coefficient of transferring 350 cm-1 the average energy trans-
Figure 3. Ratio of the strong-collision parameters a2 and R2 to the weak-collision parameters a1 and R1 as a function of the energy for the CS2 + CO system in the biexponential probability expression (10). The lines are for visual aid only.
ferred per collision, and obtained 0.088 at an initial energy of 31 480 cm-1. This indicates a significant contribution of collisions which transfer large quantities of energy in a single event. Next, we used the energy transfer rate coefficients obtained as described above to solve the master equation for cyclobutane fission and cyclobutene isomerization in Ar bath gas. Cyclobutene isomerization with threshold energy for reaction, E0,
Energy Transfer Rate Coefficients
Figure 4. Pressure dependence of kuni/kuniwc for (a) cyclobutene isomerization and (b) cyclobutane fission in Ar bath gas obtained with energy-dependent strong collider and with double-exponential probability function obtained from trajectories. 500 K (-‚-), 1000 K (‚‚‚), 1500 K (s).
of 11 540 cm-1 (33 kcal/mol) was calculated at 500, 1000, and 1500 K while cyclobutane fission with E0 ) 22 040 cm-1 (63 kcal/mol) was calculated at 1000 and 1500 K. The temperatures chosen are in the range where experimental and calculational results are available. Figure 4 shows falloff curves of kuni/kuniwc at various temperatures obtained with the fitting parameters given in Table 1 and RRKM theory rate coefficient parameters given in ref 13. kuniwc is the unimolecular rate coefficient obtained from the weak-collision contribution to the biexponential probability distribution function, namely, the part containing only a1 and R1. As can be seen from the figure, adding the high-energy tail in a form of a large value exponential enhances the value of the unimolecular rate coefficient over its weak-collision part. For cyclobutene isomerization there is a ∼7-fold increase at 500 K and an 8-9-fold increase at higher temperatures. For cyclobutane fission there is a ∼5-fold increase at 1000 K and 7-fold increase at 1500 K. There is no doubt that the high-energy term in the biexponential transition probability contributes significantly to the value of the unimolecular rate coefficient.13 As was mentioned before, one aspect of the present paper is to find the effect of supercollisions on chemical reaction for any magnitude of ∆E transferred in a supercollision. To that effect, we have studied again the isomerization of cyclobutene to butadiene and the fission of cyclobutane to two ethylene molecules by solving the master equation with a doubleexponential transition probability. Unlike what was done before,13 the present calculations use weighing factors derived both from trajectory calculations and from experimental results. For each reaction the fraction of the high-energy exponential was kept constant but the magnitude of the exponential was changed in a systematic way from 300 to 10 000 cm-1. Figures 5 and 6 show the results of the calculations for a2/a1 ) 0.995/ 0.005 and a2/a1 ) 0.9/0.1 when R1 ) 300 cm-1 is kept constant. These are referred to as “0.5% strong collision” or “10% strong collision”, respectively, “in 300 cm-1 weak collision”. Sections
J. Phys. Chem., Vol. 100, No. 23, 1996 9743
Figure 5. Scaled unimolecular rate coefficients for cyclobutane fission, kuni/kuniwc, at the low-pressure limit as a function of the value of the exponent R2 and of the percent of the high-energy term in the biexponential transition probability. (a) T ) 1000 K. (b) T ) 1500 K.
Figure 6. Scaled unimolecular rate coefficients for cyclobutene isomerization, kuni/kuniwc, at the low-pressure limit as a function of the value of the exponent R2 and of the percent of the high-energy term in the biexponential transition probability. (a) T ) 500 K. b) T ) 1000 K. (c) T ) 1500 K.
a and b of Figure 5 show the fission of cyclobutane at 500 and 1000 K and sections a, b, and c of Figure 6 show the isomerization of cyclobutene at 500, 1000, and 1500 K. As can be seen, the unimolecular rate coefficient increases as the magnitude of the strong-collision exponent R2 increases and as the temperature increases. The increase is almost linear
9744 J. Phys. Chem., Vol. 100, No. 23, 1996
Bernshtein et al.
TABLE 2: Values of the Average Energy Transferred in a Down Collision 〈∆E〉d, Obtained at High Values of Internal Energy, as a Function of the Value of the Strong-Collision Exponent, r2a R2, cm-1 〈∆E〉d, cm-1
1500 728
3000 1721
5000 3350
10000 7937
a The weak-collision exponent is R ) 300 cm-1; the weight factors 1 are a1 ) 0.9 and a2 ) 0.1.
with the value of the strong-collision exponent. Obviously the contribution of “10% strong collision” is much larger than that of “0.5% strong collision”. The increase in the value of the unimolecular rate coefficient is dramatic. For cyclobutane and “0.5% strong collision” it increases by a factor of up to ∼3.5 at 1000 K and a factor of up to ∼9 at 1500 K at R2 ) 10 000 cm-1. For cyclobutene the increase is of similar magnitudes. The obvious conclusion is that the larger the fraction of strong collisions the larger the overall value of kuni. As can be seen from the figures the ratio kuni/kuniwc is not unique but can be obtained for more than one set of a2 and R2. To estimate the relative contribution of the two terms in the biexponential function to the value of the unimolecular rate coefficient, we have solved the master equation twice, once using only the low-energy term and once using only the high energy term. We find that for the case of R2 ) 10 000 cm-1 and a2 ) 0.5% the high-energy term contributes ∼70% and the 99.5% of the weak-collision term contributes only ∼30% to the value of the unimolecular rate coefficient. The ratio of the two contributions varies with the values of ai and Ri.28 Similar observations were made when the relative importance of the two exponential contributions was studied in the determination of the relaxation of ensembles19 in the absence of reaction and in a model study of the pressure dependence of unimolecular rate coefficients.29 It should be borne in mind that the highenergy term contributes to the transfer of low values of 〈∆E〉 as well. A systematic study with a new probability distribution function which will assess the pure contribution of supercollisions to the energy transfer probability function and unimolecular rate coefficient will be reported elsewhere.28 Not only is the value of the unimolecular rate coefficient influenced by the contribution of the high-energy tail. The value of the average energy transferred and the collisional efficiency are affected as well.13 Table 2 shows what happens to the asymptotic value of 〈∆E〉d for a mixture of a1 ) 0.9 weak collision and a2 ) 0.1 supercollision when the value of the highenergy term is increased systematically.13 As can be seen, it increases significantly as the value of R2 increases. It should be pointed out that 〈∆E〉d is an explicit function of the internal energy13,22 as is also clear from the energy-dependent shapes of the probability distributions. Therefore, the values reported in Table 2 represent the high limit values of 〈∆E〉d, which are obtained when the internal energy is much higher than the values of Ri. Conclusion The present work uses results of trajectory calculations to obtain analytical expressions for energy transfer rate coefficients and probability functions. The rate coefficients are applied to master equation calculations which yield values for unimolecular rate coefficients for the isomerization of cyclobutene to butadiene and the fission of cyclobutane to ethylene. The results are compared with results of numerical calculations reported previously. The comparison calls for a systematic study on the dependence of the unimolecular rate coefficient on the values
of the weighing factors, ai, and exponents, Ri, in the biexponential energy transfer probability function. It is found that a small contribution of the high energy component to the probability function increases the value of the unimolecular rate coefficient in a significant way. Solution of the master equation with the high-energy component alone shows that under the conditions studied it is the major contributor to the value of the unimolecular rate coefficient as well as being a major contributor to the value of the average energy transferred per collision. Thus, supercollisions are an important vehicle in effecting chemical change. Acknowledgment. This work is supported by the IsraeliHungarian Intergovernmental Science and Technology Program (Grant Number 04), by the Hungarian National Scientific Research Fund (to G.L., Grant Number T7428), by the Ministry of Science and the Arts (to V.B.), and by the Technion Fund for Research (to I.O.). References and Notes (1) Tardy, D. C.; Rabinovitch, B. S. Chem. ReV. 1977, 77, 369. (2) Oref, I.; Tardy, D. C. Chem. ReV. 1990, 90, 1407. (3) Pashutzky, A.; Oref, I. J. Phys. Chem. 1988, 92, 178. (4) Hassoon, S.; Oref, I.; Steel, C. J. Chem. Phys. 1988, 89,1743. (5) Margulis, I. M; Asperse, S. S.; Steel, C.; Oref, I. J. Chem. Phys. 1989, 90, 923. (6) Mullin, A. S.; Park, J.; Chou, J. Z.; Flynn, G. W.; Weston, R. E. Chem. Phys. 1993, 53, 175. (7) Mullin, A. S; Michaels, C. A.; Flynn, G. W. J. Chem. Phys. 1995, 102, 6032. (8) Clarke, D. L.; Thompson, K. G.; Gilbert, R. G. Chem. Phys. Lett. 1991, 182, 357. (9) Lendvay, G.; Schatz, G. C. J. Phys. Chem. 1990, 94, 8864. (10) Bernshtein, V.; Lim, K. F.; Oref, I. J. Phys. Chem. 1995, 99, 4531. (11) Bernshtein, V.; Oref, I. Chem. Phys. Lett. 1995, 173, 233. (12) Lenzer, T.; Luther, K.; Troe, J.; Gilbert, R. G.; Lim, K. F. J. Chem Phys. 1995, 103, 626. (13) Bernshtein, V.; Oref, I. J. Phys. Chem. 1993, 97, 12811. (14) Troe, J. J. Chem. Phys. 1992, 97, 288. (15) Clary, D. C.; Gilbert, R. G.; Bernshtein, V.; Oref, I. Faraday Discuss. 1995, Dec. (16) Bernshtein, V.; Oref, I. J. Phys. Chem. 1994, 98, 3782. (17) Tzidoni, E.; Oref, I. Chem. Phys. 1984, 84, 403. (18) Bernshtein, V.; Oref, I. J. Phys. Chem. 1993, 97, 6830. (19) Lendvay, G.; Schatz, G. C. J. Phys. Chem. 1994, 98, 6530. (20) Lendvay, G.; Schatz, G. C. J. Phys. Chem. 1992, 96, 3752. (21) Lendvay, G.; Schatz, G. C. In Vibrational Energy Transfer InVolVing Large and Small Molecules; Barker, J. R., Ed.; JAI Press: Greenwich, CT, 1995. (22) Lendvay, G.; Schatz, G. C. J. Chem. Phys. 1992, 96, 4356. (23) Lim, K. F.; Gilbert, R. G. J. Phys. Chem. 1990, 94, 77. (24) Truhlar, D. G.; Muckerman, J. T. In Atom-Molecule Collision Theory, A Guide to the Experimentalist; Bernstein, R. B., Ed.; Plenum Press: New York, 1979. (25) Raff, L. M.; Thompson, D. L. In Theory of Chemical reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. III. (26) Tardy, D. C; Rabinovitch, B. S., J. Chem. Phys. 1972, 48, 1282. (27) 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, 1988. (28) Bernshtein, V.; Lendvay, G.; Oref, I. manuscript in preparation. (29) Lendvay, G. Presented at the Thirteenth. International Symposium on Gas Kinetics, Dublin, Ireland, Sept 11-16, 1994; manuscript in preparation. (30) Tardy, D. C.; Rabinovitch, B. S. J. Chem. Phys. 1968, 48, 5194. (31) Tardy, D. C.; Song, B. H. J. Phys. Chem. 1993, 97, 5628. (32) Yerram, M. L.; Brenner, J. D.; King, K. D.; Barker, J. R. J. Phys. Chem. 1990, 94, 6341. (33) Toselli, B.; Brenner, J. D.; Yerram, M. L.; Chin, W. E.; King, K. D.; Barker, J. R. J. Chem. Phys. 1990, 94, 6341. (34) Hippler, H.; Troe, J.; Wendelken, H. J. J. Chem. Phys. 1983, 78, 5351, 6718. (35) Barker, J. R. J. Chem. Phys. 1990, 94, 6341. (36) Hippler, H.; Lindemann, L.; Troe, J. J. Chem. Phys. 1985, 83, 3906.
JP953341L
