J . Phys. Chem. 1988, 92,4333-4339 The slow convergence for atoms may not be as severe for molecules where the centers of fluctuating potential may be distributed over many bond orbitals. Charge-transfer correlation is not significant for HeNe. This conclusion has to be checked for the heavier systems. The relatively constant percentage error in the binding energy suggests that the terms such as the charge transfer contribute proportionately to all the rare-gas molecules. This method of analysis depends on the dominance of a single configuration in the wave function representation. Separate calculation of the van der Waals
4333
contribution is then possible. The present method also avoids BSSE errors for the calculation of the van der Waals contribution, and the SCF BSSE can be reduced to small values by using the ECP and near-saturated basis sets. Calculation of the van der Waals contribution for a correlated atom is also possible, and the energy curve for correlated rare-gas atoms can be determined by using effective potentials in limited M C S C F calculations. Registry No. He, 7440-59-7; Ne, 7440-01-9; Ar, 7440-37-1; Kr, 7439-90-9; Xe, 7440-63-3.
On the Distinction between Weak and Strong Collisions in the Theory of Unimolecular Reactions H.0. Pritchard Centre for Research in Experimental Space Science, York University, Downsview, Ontario, Canada M3J 1P3 (Received: May 7, 1987; In Final Form: February I , 1988)
Beginning with the conjecture that the principal difference between weak- and strong-collision reactions is intrinsic to the reactant molecule, rather than in the nature of the collisions between molecules, a reanalysis of the unimolecular falloff phenomenon is given. The rate at which the molecule acquires energy to react is identified with the bulk vibrational relaxation rate, and the rate at which reactive molecules lose their energy of activation is taken to be the collisional deactivation rate, as used in existing theory; certain choices must be made for the rates of intramolecular vibrational relaxation (randomization) among the reactive states, with the result that the same analytic expression for the unimolecular rate can be made to reproduce the falloff shapes for such diverse reactant molecules as nitrous oxide, methyl isocyanide, and cyclopropane, as well as their correct positions on the pressure axis.
Introduction At the present time, there are two principal flavors of unimolecular reaction theory to correspond with the principal classifications of reaction type: weak-collision and strong-collision reactions. Strong-collision theory itself comes in several versions, which differ mainly in the methods used to construct the specific rate function k(E): one begins with an attempt to formulate a transition state for the reaction in question and derives a k(E) function by the well-known RRKM prescription,’v2 whereas another derives the k(E) function as the inverse Laplace transform of the Arrhenius rate la^.^-^ The “strongness” quality of the collisions has taken a much less central position in these theories, but it turns out that the usual steady-state expression used for calculating the falloff of the rate with p r e s s ~ r e ~ *implies ~ - ’ that the collisional deactivation of reactive molecules must be a single pureexponential process;g1othis had been suspected for some time earlier”J* but not proved. It was also found that if this rate, often denoted by w , was assumed to be equal to the collision rate, then the predicted position of the falloff on the pressure axis was usually (1) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: London, 1972. (2) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973. (3) Slater, N. B. Theory of Unimolecular Reactions; Methuen: London, 1959. (4) Forst, W. J . Phys. Chem. 1972, 76, 342. (5) Pritchard, H. 0. Quantum Theory of Unimolecular Reactions; Cambridge University Press: Cambridge, England, 1984. (6) Rice, 0. K.; Ramsperger, H. C. J . Am. Chem. SOC.1927,49, 1617. (7) Kassel, L. S. Kinetics of Homogeneous Gas Reactions; The Chemical Catalog Co.: New York, 1932. (8) Nordholm, S. Chem. Phys. 1978, 29, 55. (9) Yau, A. W.; Pritchard, H. 0. Can. J . Chem. 1978, 56, 1389. (IO) Singh, S. R.; Pritchard, H. 0. Chem. Phys. Lett. 1980, 73, 191. ( I 1) Bunker, D. L. Theory ofElementary Gas Reaction Rares; Pergamon: Oxford, England, 1966. (12) Tardy, D. C.; Rabinovitch, B. S. Chem. Rev. 1977, 77, 369.
well within an order of magnitude of that observed, and that the falloff shape was in virtually perfect agreement with experiment. Unfortunately, this dual quality of strong collisions, their pure exponential character and their near-unit efficiency, has given rise to some confusion in their d e f i n i t i ~ n . ~ Moreover, we need to be careful about the use of the term “exponential decay” in collisional relaxation. One may have, in theory, the situation that all populations decay exponentially with a single time constant, whence the total energy decays exponentially, too: this is the “strong collision”, strictly definede8-12One can also have the less restrictive situation in which the total energy decays exponentially, but the populations do not? the condition for this to occur is that ( h E ) / E= constant;I4 such behavior has been observed in large molecules at low energies,IsJ6 but there is insufficient information content in the results to be able to say whether or not, strictly, these are strong collisions. In the remainder of this article, the terms single exponential, pure exponential, or strong collision will refer to the former kind of process. Likewise, there are variations in the treatment of weak-collision reactions, but the dominant feature of weak-collision theory is the collision efficiency factor &, which is the amount by which the collision rate must be scaled to cause the computed strongcollision low-pressure limiting rate constant to coincide with the observed ne.^**'^ Again, little is said about the nature of the collision process, but it is never conceived as being a pure-exponential relaxation; in fact, for triatomic moIecuIes,’5 ( A E ) / Evaries with E, and the relaxation clearly is not exponential. It is not necessary to go into further details of strong- and weak-collision unimolecular reaction theory, as they are so well-known: all we have done here is to select a few points for (13) Troe, J. J . Chem. Phys. 1977, 66,4745, 4758. (14) Forst, W.; Xu, G.-Y.; Gidiotis, G. Can. J . Chem. 1987, 65, 1639. (15) Troe, J. 2.Phys. Chem. (Neue Folge) 1987, 154, 73. (16) Wallington, T. J.; Scheer, M. D.;Braun, W. Chem. Phys. t e t t . 1987, 138, 538.
0022-3654/88/2092-4333$01.50/0 0 1988 American Chemical Society
4334
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
I 2
Log p
4
Pritchard that both reactions should share a common low-pressure limiting rate,20 as shown by the solid lines, so long as they have the same activation energy. The data shown in Figure 1 have been rationalized by assuming the geometric isomerization to have an activation energy about 4 kcal mol-' lower 'than that of the structural isomerization,20 and unfortunately, since the two approximately equal activation energies appear to have been determined inde~endently,'~ we do not know how close they really are to each other. However, the reinvestigation of this reaction by Waage and Rabinovitch20 does present us with a clear contradiction: at the same fallen-off value of k,, the value of k, is 10%greater for a mixture of helium with 1 ,2-cyclopropane-d2than in the neat reactant itself. Since the falloff for each reaction is described by its own independent expression a certain value of
(Torr)
Figure 1. Falloff curves for173521,2-cyclopropane-d2at 718 K: ( X ) geometrical isomerization;(0)structural isomerization. The two solid curves approach the same value of ko, as required if the two reactions have the same critical energy E,, but do not interfere with each other. The dashed curve is what would be obtained in the usual case where the pair of reactions were in direct competition with each other, i.e., eq 1.
emphasis, as they have some relevance for what follows. Some Contradictory Observations (1) There are many examples of chemical activation experiments exhibiting "low-pressure turnup", which is taken to indicate that the collisional deactivation process has some cascading quality;'S2 however, many of the molecules for which this effect is observed are ones whose thermal decompositions can be well represented by strong-collision (Le., single-exponential decay) theory. Moreover, contemporary measurements of the energytransfer properties of complex molecules rarely show a loss of more than 1000 cm-I per collision, even from quite high energies.l5 (2) Consider a pair of competing unimolecular reactions having the same energy threshold, with one of them much faster than the other: the geometric (g) and structural (s) isomerizations of 1 ,2-cyclopropane-d2 could be such a pair, the activation energies being 65.1 f 0.5 and 65.4 f 1 . 1 kcal mol-', respectively, but the geometric isomerization is about 20 times as fast as the structural isomerization at infinite press~re.'~With the usual strong-collision assumption, the rate constant for each reaction i is given (in discrete form) by5918+'9
kunisdefines the relaxation rate w, and this in turn fixes k*?i,: thus, there is no room for the variation of kuni,,at fixed kUi,$within the framework of strong-collision theory. We have to conclude that if the relaxation is pure exponential, there are multiple domains of energy space above threshold, which may or may not be strongly coupled to each other; alternatively, there must be significant cascading quality in the energy-transfer process; or both. (3) Another difficulty arises when one tries to examine the rate of the collisional deactivation process required to position the falloff on the pressure axis. Data are rather sparse, but there are two cases, one more definitive than the other: in the case of cyclopropane, the collisional deactivation rateS is 5.5 X los Torr-' s-l, whereas the measured value of the vibrational relaxation rate2' (extrapolated only a short distance to the reaction temperature) is about 8 X lo3 Torr-' s-', Le., the observed relaxation rate is 80-100 times slower than the apparent rate deduced from the position of the falloff, according to strong-collision theory. Less convincing but nevertheless disturbing is the case of nitrous oxide where the vibrational relaxation ratez2 at 2000 K of 2.2 X lo3 Torr-' s-' is more than a factor of 1000 slower than the value of 2.9 X lo6 Torr-' s-I that is required to position the falloff correctly on the pressure axis if the relaxation is assumed to be pure exponentialz3(but in this case-of a weak-collision reaction-the acknowledged structure of the relaxation has been ignored). (4) If we imagine that the rate of a unimolecular reaction is to be measured by a shock-wave experiment, then it is intuitively obvious that the incubation time associated with the onset of reaction should be of the same order in magnitude as the vibrational relaxation time. However, standard strong-collision theory identifies the incubation time with the collisional deactivation rate at the high-pressure limit24*zs(and it also gives a physically unacceptable result at the low-pressure limit).z6 Unfortunately, so few measurements of incubation times have been made that it is not possible to make any extensive tests of the theory on this point.s These discrepancies signal the need to develop a more finegrained theory of unimolecular reactions; in fact, it is remarkable that single-exponential theories of unimolecular reaction have persisted so long, since it has long been known that single-exponential theory is incapable of describing the much simpler dissociation of a diatomic molecule.z7
where w is the collisional deactivation rate, (3, is the equilibrium population of grain r, and d, = Cid,,i, dr,ibeing the dissociation rate constant for grain r to the ith product. If one uses the inverse Laplace transform procedure to calculate the specific rate function , drqg= constant X d,,, for all r: thus, since d,,, dr,i [ = k i ( E ) ] then >> d,,s, both reactions must fall off over the same pressure range, they have the same value of p l l 2 , or alternatively, ko,s/kO,g = k,,,/ k,,,; RRKM calculations yield the same result.20 Qualitatively, because randomization within each grain is assumed to be complete, the branching ratio will be the same at all pressures. It seems highly unlikely that such a pair of competing reactions would maintain the same branching ratio at all pressures, and in the case of 1,2-cyclopropane-d2, the geometric isomerization falls The Aim of the Present Reformulation off significantly faster than does the structural i ~ o m e r i z a t i o n : ' ~ + ~ ~ The purpose of this investigation is to reformulate the nnalytic in Figure 1 , if the two reactions were to fall off according to eq theory of thermally induced unimolecular reactions in such a way 1 , the structural isomerization would follow the dashed curve. that However, this is rather an unusual reaction because the product (1) It helps to provide a framework within which all thermal of the geometric isomerization is still capable of undergoing reactions, from the ionization of hydrogen atoms, through diatomic structural isomerization to propylene, and therefore its falloff does not obey this equation: the two reactions can be treated as being (21) Dove, J. E.; Grant, A. Y.,personal communication. independent of each other, and so strong-collision theory predicts (22) Baalbaki, Z.; Teitelbaum, H.; Dove, J. E.; Nip, W. S. Chem. Phys.
(17) Schlag, E. W.; Rabinovitch, B. S. J . Am. Chem. SOC.1960, 82, 5996. (18) King, K. D.; Golden, D. M.; Spokes, G. N.; Benson, S. W. In?. J . Chem. Kinet. 1971, 3, 41 1 . (19) Vatsya, S. R.; Pritchard, H. 0. Can. J . Chem. 1981, 59, 772. (20) Waage, E. V.; Rabinovitch, B. S. J . Phys. Chem. 1972, 76, 1695.
1986, 104, 107.
Pritchard, H. 0. J . Phys. Chem. 1985.89, 3970. Pritchard, H. 0.;Vatsya, S. R.Chem. Phys. 1982, 72, 447. Pritchard, H. 0. J . Phys. Chem. 1986, 90, 4471. Pritchard, H. 0. J . Phys. Chem. 1983,87, 3179. (27) Pritchard, H. 0. React. Kinet. 1975, 1, 243.
(23) (24) (25) (26)
Collisions in Unimolecular Reactions dissociation and weak-collision unimolecular reactions, right up to traditional strong-collision reactions simply comprise a broad spectrum,% with only minor differences in kind separating adjacent types of reactions; what is to be presented may go some way toward bridging the gap between weak- and strong-collision reactions, by accounting for the strict-Lindemann behavior for the dissociation of nitrous oxide, the well-known strong-collision falloff shape for the isomerization of cyclopropane, and perhaps also the rather curious falloff shape for the isomerization of methyl isocyanide, all with the same set of formulas. (2) It overcomes the inconsistency that the rate constant for collection of energy by the molecule from its surroundings may be less-sometimes a lot less-than the rate constant with which the reactive states must be created. In so doing, it introduces an element of the cascading nature of the relaxation, rather than just including one collisional relaxation rate: so far, we can treat only a biexponential decay. At the same time, it automatically associates the incubation time with the bulk vibrational relaxation time,25 which is intuitively desirable, although there is yet very little basis in experiment for so doing.5 (3) It allows the energy space above reaction threshold to be divided up into subspaces, each connected to its own product manifold or continuum; it would appear that this is logically necessary from the discussion of the 1,2-cyclopropane-d2 system given above, even though the evidence we can muster is only slim. Notice also, however, that the same problem arises in chemical activation reactions, where we have A B [AB]*
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4335 occupies a very important place, but the shape of the falloff defies accurate representation by any simple form of the theory.499332 The view has been expressed33that it is placing too much reliance on the accuracy of the results to attempt to fit the shape exactly, and were there to be only one example of such behavior, that would be very much the case; however, at least three different variants of the methyl isocyanide isomerization (cf. Figures 7.3-7.6 of ref 5) exhibit similar features, but at different pres~ures,’~ giving some basis for thinking that they may be real. It is clear that we are rapidly approaching the time when the uncertainties in the experimental measurements of unimolecular reactions are becoming a limiting factor and when more precise measurements on certain key systems are needed before the theory can advance with confidence much further.
A Minimal Theory of Thermal Unimolecular Reactions We begin by trying to construct the minimum requirements for a theory of unimolecular reactions, specifically designed to describe the falloff phenomenon of thermal reaction processes, but at the same time being aware of other phenomena usually included within the generic framework of unimolecular reactions. The unimolecular reaction problem is really one of a coupled set of many relaxation processes-although it is usual only to consider the competition between two of these, deactivation and reaction, as, for example, in RRKM theory. One common approach is to treat the problem within the framework of an “energy-resolved” master equation,” where the system is grained according to energy and, at least numerically, it is possible to allow for the cascading quality of the relaxation by allowing the amount M [AB]* M AB of energy ( A E ) transferred per collision to vary with E in some appropriate manner; for some energy-transfer patterns, approx[AB]* C D imate analytic solutions are also possible. Nevertheless, we prefer If the whole energy space a t any energy were to be completely to deal with a less convoluted form of the master equation, written mixed, then one would have to consider the direct reaction in terms of the rates of transitions between states or groups of A+B-C+D states, for several reasons: (1) we desire the flexibility for molecules at the same energy to be able to decay to one of several because the channel would be wide open; in the usual classical products, where the discrimination between the differently reacting description,’S2 it is implicit that when transcribed into quantal states may be one that is based on symmetry, on rotational energy, terms, there are separate subspaces of [AB]* that can intermix etc., and we may have to allow for different rates of relaxation on a time scale of the order of s. between rotational and vibrational states or between vibrational A Caveat about Experimental Data states of different symmetries; (2) before too long, we can anticipate the measurement of some relevant state-resolved relaxation The only inconsistency noted above that is clear and unamrates;15 (3) likewise, we can anticipate that an analytic solution biguous is that between the rate constants for vibrational relaxation can be found for a relaxation model of any degree of ~ o m p l e x i t y , ~ ~ and collisional deactivation for cyclopropane. In the 1,2-cyclowhen written in this form; if this is so, it would then be possible propane-d2 case, the measured activation energies are admitted to transcribe that solution back into the energy-resolved form, if to be too uncertain20 for us to be able to say whether Figure 1 needed. demonstrates an inconsistency of the kind proposed here or not. A minimal theory of unimolecular reactions must include, at On the other hand, it would seem that the 10% change in kuni the very least, five relaxation processes to which rate constants for the same value of kuni,sshould be real, given the descriptionZB will have to be assigned. of the care with which the experiments were done, but nevertheless, ( a ) Collisional Relaxation Processes. I . Vibrational Relaxanalyses of small amounts of products in the presence of large ation. The molecule must collect the energy it requires to reach amounts of material in huge vessels are notoriously difficult, and the reactive region from its surroundings. The rate constant an independent corroboration would be most reassuring. assigned to this process has to be identified with the slowest bulk Again, when we come to compare the shapes of falloff curves relaxation process for the system-usually the vibrational relaxbetween different reactant species, we may find ourselves on quite ation: we will denote this rate by Xo, directly proportional to shaky ground. For the relatively large molecules, e.g., cyclopressure p , i.e., X, = pro where ro has units of pressure-’ time-’. propane, cyclobutane, methylcyclopropane and ethylcyclopropane: 2. Vibrational Deactivation. Excited molecules are deactivated the observed shapes of the falloff are in agreement with the in competition with reaction, at a rate that we do not know with predictions of strong-collision theory, and it is usual to accept them precision but that is generally accepted, for strong-collision systems as correct. At the opposite extreme, that of very simple molecules, at least, to be within about an order of magnitude of the collision it has been speculated2gthat the falloff should conform not to the rate; we will begin by taking the deactivation rate to be the usual strong-collision shape but to the strict-Lindemann one instead: value that is used in strong-collision theory, denoted by hl = w unfortunately, there is only one good example of such behavior = pr,, where rl also has units of pressure-I time-’. To assign a among a handful of reactions, and that is the thermal decomsingle rate constant to the deactivation process is inconsistent with position of nitrous o ~ i d e . ~ ~ JFor O molecules of intermediate complexity, the thermal isomerization of methyl i ~ o c y a n i d e ~ ~ the known cascading quality, as illuminated by the chemical
+
+
-- + -+
(28) Yau, A. W.; Pritchard, H. 0. J. Phys. Chem. 1979, 83, 134, 896. (29) Yau, A. W.; Pritchard, H. 0. Chem. Phys. Lett. 1978, 60, 140. (30) Olchewski, H. A.; Troe, J.; Wagner, H. Gg. Eer. Bunsen-Ges. Phys. Chem. 1966, 70, 450. (31) Schneider, F. W.; Rabinovitch, B. S . J . Am. Chem. SOC.1962, 84, 4215.
(32) Forst, W. In Reaction Transition States; Dubois, J. E., Ed.; Gordon and Breach: London, England, 1972; p 75. (33) Rabinovitch, B. S.,personal communication, February 1984, August 1986, February 1987. (34) Pritchard, H. 0. J . Phys. Chem. 1986, 90, 3501. (35) Vatsya, S . R., personal communication.
4336 The Journal of Physical Chemistry, Vol. 92, No. 15, I988 activation experiments, but that is all that we have been able to do up to the present time; in fact, one should consider a whole spectrum of collisional relaxation processes having rates lo,XI, X2, A,, ...,whereas, for the time being, we will lump them all into XO and XI. As noted above, we expect that it will eventually be possible to solve this multirate problem, whence artifacts caused by this lumping should disappear. ( 6 ) Special Processes above Reaction Threshold. In the zeroth-order approximation, we may divide the states of a molecule above reaction threshold into two classes, reactive and unreactive, and the principal ingredient of the calculation of the specific rate constant k ( E ) for strong-collision reactions is the estimation of the relative numbers in each class at every energy E ; this is so for either the RRKM method or the inverse Laplace transform method of calculating k(E). Although the region above threshold is envisaged to be a continuum of states, in any practical calculation, the range is subdivided into small elements of width 6E, for purposes of numerical integration. It is convenient, operationally, to assume that the processes described below-often called intramolecular vibrational relaxation (IVR), or randomization-do not spread across the boundaries of these elements, or grains. The theory has not yet developed to the stage where it is possible to consider a continuum of randomization processes that spread over these discrete, and arbitrary, grains; for the present, all we can do is to vary the width of the grains and observe the extent to which the final result is altered.23 3. Mixing of Reactive States. We begin by discussing a fragmentation reaction. Typically, the zeroth-order reactive states will have widely differing lifetimes (cf. Figure 1 of ref 23), but on the other hand, they are so closely spaced that they overlap strongly, the width of many of the shorter-lived states exceeding the spacing between the reactive states by 2 or 3 orders of magnitude. Under these conditions, it appears that the interaction between the states is so strong that they can all be considered to decay at the same rate.23136 Thus, we will assume that there is one intramolecular relaxation process between reactive states, whose rate, which we will label pr,o, approaches infinity: it is not therefore necessary to know the decay rates for the individual states, only the number of such states within any energy increment and their average rate of decay. All contemporary theories adhere to this simplification. The case of isomerization is rather different, but the result is the same: an isomerization takes place via a large-amplitude oscillation, and states undergoing those motions that evolve from reactant to product and back again do not exhibit large resonance widths; however, it seems likely that in the band of excited states within a few kT above threshold, all have comparable torsional frequencies3’ Hence, for quite a different reason, it seems that the zeroth-order reactive states in an isomerization can equally be assumed all to have the same rate constant for conversion to products! 4 . Intramolecular Vibrational Relaxation or First-Order Randomization. This is the process that has been presumed to occur since the earliest days of unimolecular reaction theory, that the reactive states within the grain can mix with those that are classified unreactive. Although there have been few attempts to assign a magnitude to the rate of this process in calculating the falloff of the rate with p r e s s ~ r e ,it~is~widely , ~ ~ agreed, again from the results of chemical activation experiments,m that the rate of the analogous process, usually described in terms of classical concepts, is about 10l2s-! we will assign the label K , , ~to the rate of this process. 5 . Collisionally Induced Intramolecular Vibrational Relaxation or Second-OrderRandomization. There is, as yet, no clear experimental evidence that “intramolecular” vibrational relaxation assisted by long-range interactions with neighboring molecules
Pritchard
TABLE I pressure dependence
label Pr.0
0
!+I.
0
Pr,2
1
A0
1
process type coupling between dissociating states, causing them to give the appearance of all having the same decay rate; only applicable to fragmentation reactions relaxation between states labeled reactive and unreactive, within a limited span of energy (grain); see ref 5 for possible causes relaxation between states labeled reactive and unreactive, within a limited span of energy (grain), caused by long-range
intermolecular forces
AI,
x2,
...
A,,
1
rate of the slowest collisional relaxation process in the system, Le., vibrational relaxation rate rates of other collisional relaxation processes in the system, between upper vibrational levels, rotations, etc.; only XI = w is considered in this paper
is an important process in the present context, although there are good theoretical reasons for believing that they s h o ~ l d ;we ~ , have ~~ assigned the label M,,~ to the rates of these processes, again with Pr.2 = Prr,2. The properties of each of these are summarized in Table I.
Method of Calculation A general method for solution of the coupled relaxation including all processes listed in Table I is not yet available, although it may well be possible: we have used a powerful device in which a relaxation matrix with eigenvalues 0, &, XI, X2, ..., is represented as a sum of strong-collision relaxation matrices with eigenvalues lO,poL (O,rlJ, {0,~21,..., in such a way that XO = KO, XI = (KO + pl), X2 = & pl p2), ...;this approach appears to have considerable potential for generalization beyond the (p,,,pl)case solved so far.2s*42 We have also solved by a different method4, the system of this system had been equations including the processes (p,,;o;c~~,~,p~,~); treated earlier by using a separable a p p r o ~ i m a t i o n which , ~ ~ gave very accurate results by assuming that the internal relaxation could be separated from the external relaxation processes {pr,l,pr,2] process po. The reasons for this had already been given by N o r d h ~ l m ,and ~ . ~by~ the same argument, we should expect the case too. separable approach to work well for the (po,pl;pr,l,p,,2) For the (po,pl}part of the relaxation, the formula describing the falloff is2s
+ +
where, as usual, P, is the equilibrium population in grain T , and are collisional rates as defined above; notice that if X1 >> Xo, as is often the case, po = X,and p l = A,. Notice also that although there is no lower limit specified for the summation in the numerator, it is effectively r = 2, because the grains r = 0 and r = 1 have zero decay rate constant ( d r ) ;the population Po represents the general population of low-lying states, and fi, represents a pool of molecules in direct connection, at a rate (po p l ) , with the reactive states, without themselves being reactive. We now examine the effective decay rate constants ( d ? )for the grains. Grain r has a total population fir, of which fila are unreactive and Prl reactive. In the general case, where states in po and p l
+
~~
(36) Mies, F. H.; Krauss, M. J . Chem. Phys. 1966, 45, 4455. (37) Clarkson, M. E.; Pritchard, H. 0. Chem. Phys. 1987, 117, 29. (38) Thiele, E.; Goodman, M. F.; Stone, J. Chem. Phys. Letr. 1980, 69, 18. (39) Pritchard, H. 0. Can. J . Chem. 1980, 58, 2236. (40) Oref, I.; Rabinovitch, B. S . Acc. Chem. Res. 1979, 12, 166.
(41) Freed, K. F. Chem. Phys. Leir. 1984, 106, 1 . (42) Vatsya, S . R.; Pritchard, H. 0. Chem. Phys. 1981, 63, 383. (43) Vatsya, S. R.; Pritchard, H. 0.Can. J. Chem. 1984,62,2879. Notice that if there is only one grain r, and ( M , , ~ M , , ~ ) is set equal to bl,the
+
( M ~ ; M , , ~ , M , , Zand } [ ~ o , ~ formulas l ]
are equivalent. (44) Nordholm, S.Chem. Phys. 1976, 15, 59.
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4331
Collisions in Unimolecular Reactions the reactive region, of equilibrium population iii,n may have different decay rates d,, the total decay rate from the grain is Ciiri,.diand C'ii,, = Prl; here and elsewhere is used to denote the summation over reactive entities only, either states or grains as appropriate. If there is an infinitely fast rate of mixing between the reactive and the unreactive states in the g r a i n - t h e usual RRKM assumption-then the rate constant specific to grain r is
x'
(4) However, if there is some restriction in the rate of mixing among the states, we may not achieve the upper limiting value in eq 4. Here, we examine the case where there are two effective scrambling processes, a first-order one with rate p,,' and a second-order one with rate pr,2as discussed above. One method of estimating the specific rate constant has been given an alternative is simply to recast eq 3 in the appropriate form:
(5)
so long as both M,,~ and pr,2 are nonzero; notice that the same population is coupled by both processes, so that the normalization in the denominator of (3) is now set to 1. In the present circumstances, where we accept that all the d, = d are the same, eq 5 reduces to dPrl Pr,2
(dr) =
Fr,l
+ pr92 + d \
/
(6)
Although this equation is not the same as was used before, it gives the same results for the shape of the methyl isocyanide falloff as does the earlier expression; it still remains to derive rigorous expressions for the effective decay rate constant ( d , ) for grain r under incomplete mixing conditions. For example, neither eq 6 nor the one used previously approach sensible limits as both F , , ~become zero, and for the case of the small-molecule limit in which we are interested, where P , , ~= pr,2 = 0, then we have to shrink the reactive grains from P, to a population of Prl and use (7) in the general case, or for consynt di = d
(dr) = d
(8)
In the following paragraphs, this method is applied to three thermal unimolecular reactions, one at the large-molecule limit (cyclopropane), one that appears to be an intermediate case (methyl isocyanide), and one that is clearly at the small-molecule limit (nitrous oxide); some guidance is needed from the parallel state-to-state simulation in performing the latter calculation.
Results To use eq 6 to calculate the appropriate values of ( d , ) ,we need to know d, the rate constant with which a molecule in a reactive state becomes converted into products. For isomerization processes like these, we may speculate that the transformation comes about as the result of a large-amplitude bending motion, for which we can guess the harmonic bending frequency: for the purposes of this illustration, we will use the values of 1188 and 263 cm-' for cyclopropane and methyl isocyanide, respectively; the former is a ring deformation, the latter the C-N-C bend. The rate at which these motions transform reactant to product configurations are taken to be d = 2cv s-I, Le., 7.1 X lOI3 and 1.6 X l O I 3 s-', respectively.
For the time being, it is not devastating if these estimates are incorrect: the principal effect of choosing an incorrect value of d will be to change, through eq 6, the values of p,,' and/or M,,~ below which the value of (d,) begins to fall away from its (infinite randomization) RRKM limit; through eq 4, the choice of d also defines the supposed ratio of reactive to unreactive states in the grain, Le., /3,,/Pd, which also affects the behavior of ( d , ) under conditions where randomization is severely rate-limiting. Thus, any numbers we derive will contain a dependence on our choice of d, but the general features will remain. Cyclopropane. For the isomerization of cyclopropane at 765 K, this value of d, together with an extrapolated vibrational relaxation rate2' of po/p = 8.5 X lo3 Torr-' s-l and a collisional deactivation rate5 of w / p = 5.5 X lo5 Torr-' s-l, the well-known strong-collision shape can be attained only if B,,' L loL5s-' and pr,2/p> lo8 Torr-' s-l; the latter value is entirely reasonable, but the former may be debatable. If p,,' is reduced to 1014 s-I, pLr,2 has to be increased by at least 6 orders of magnitude to maintain the proper shape of the falloff, but if it is increased to 10l6 s-l, almost any value of will do. If pr,l and pr,z are made too small, the curve lies below that which is desired in ways that have been indicated before;39on the other hand, as their values are increased, kunidoes not increase beyond bound but appreaches the usual strongallision shape because (d,) cannot rise above the infinite randomization (RRKM) limit. Given our initial conjecture, po cannot be changed, but there is a choice in respect of p l , which we have identified with the collisional deactivation rate (actually p1= w po). If pI is made too large, the falloff curve sharpens up, and once it becomes a couple of orders of magnitude too large, the shape becomes strict-Lindemann, for reasons that have been analyzed before.42 If, on the other hand, p1is reduced from its starting value, very little happens for a while, but eventually a contradiction is encountered. By examining the low-pressure limit for eq 3, we find that pl and P1 are connected by the expression
-
PI
=
k o / ~ o- C'Pr
(9)
1 - ko( -Po - ) l+ . ' PcL1 r where ko is the limiting low-pressure rate constant and C'P, is the sum of the equilibrium populations of all states above threshold. With the original choice we made for pl, P1 = 5 X lod whereas E'& = 1.2 X at 765 K; in other words, the estimated pool of subcritical molecules that are in direct communication with the reactive states at a rate equal to w = (po pl) is about 4 X 1 O3 larger than the equilibrium population of above-threshold molecules. According to eq 9, as pl is reduced from our chosen value, P1 increases until it exceeds unity, and then it becomes negative, at which point the calculated value of k,,, drops off in the region where the falloff plot is curved. Although the value of PI appears to be reasonable, it would seem unwise to ascribe too great a significance to it at the present time since its magnitude is determined, in the main, by an attempt to model the complete multiexponential relaxation with two rates ho = po and w = (lo pI); at a later stage, when it becomes possible to add more structure to the relaxation, the size of the pool of subcritical molecules in direct communication with the reactive states will become better defined. Meanwhile, there is considerable latitude in choosing the value of pI,and to associate it with the collisional deactivation rate is appealing on physical grounds. Methyl Isocyanide. For methyl isocyanide at 503 K, the conditions for the attainment of the standard strong-collisionshape are very similar. In this case, the vibrational relaxation rate is not known, but the Lambert-Salter rule45suggests that it is about 1 collision in 10, say5 8.8 X lo5 Torr-' s-', and the experimental value for the collisional deactivation rate5 is 1.2 X lo6 Torr-' s-'. With these given, H,,~ L l O I 5 s-l and ~ , , ~>/ lo* p Torr-' s-' are required to achieve the strong-collision shape, as before. However,
+
+
~
(45) Lambert, J.
~~
D. Vibrational and Rotational Relaxation in Gases;
Clarendon: Oxford, England, 1977.
4338 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
Pritchard states of the system are so widely separated, and therefore so well defined, that a reactive state will not spontaneously change into an unreactive one at approximately the same energy and vice versa, = 0; it can only do so at a rate w = (po + p l ) , which is i.e., already included in the calculation via eq 3, so that pr,zmust also be zero. Under these conditions, we do not have the usual behavior that all states above threshold may eventually decay to products it left alone long enough-now there are two classes of molecules above threshold, those that may react and those that may not, ever! Equation 3 must be rearranged slightly to give
3 I
-2
-1
D
I
Log p
2
3
1
5
(Torr)
Figure 2. Experimental and computed falloff curves for the unimolecular reactions of cycl~propane~~ at 765 K, of methyl isocyanide3' at 503 K, and of nitrous oxide30 at 2000 K; the solid lines were all calculated by using eq 3, with the constants given in Table 11. For the reactions of
methyl isocyanide and nitrous oxide, the standard5strong-collision shapes are also shown as dashed lines. Notice also that the values of pl used in this diagram are slightly different from those that would be obtained from the w values given in ref 5 for cyclopropane and for methyl isocyanide: this is because a much finer grid has been used in the present integrations to obtain k,,,.
and we have no simple way of estimating the required values of Prl from experimental data (in the normal unimolecular calculation, we need P,, the total population in grain r, which is easily estimated from the molecular model). However, since pr,l = = 0, according to eq 8, ( d , ) = d in this expression, and thus we which would easily be deduced from only need to know
k , = dCPri
(12)
(10)
if the value of d were known: here, we appeal to the earlier state-to-state calculation, and from Figure 5 of ref 23, we estimate d = 5 X 1O1O s-l at the most probable reaction energy for the temperature 2000 K. It is convenient to choose a critical energy, which we take to be the computed value46 of E , = 66.1 kcal mol-I, to be able to identify PI as the pool of molecules below threshold. In fact, this is not strictly necessary for the implementation of eq 11, since includes PI and is the quantity actually calculated from eq 9 in this case; all we need to know are the vaues of ko and k,. With these two inputs, with the experimentalz2p o / p = 2.25 x lo3 Torr-' s-' and pl/p = 2.9 X lo6 Torr-' s-l taken from the earlier theoretical the computed curve is strict-Lindemann and follows the experimental results acceptably, as shown in Figure 2. The estimated size of the pool below threshold P1 = 1.5 X the total population above threshold E'& = 1.5 X lo4, and the total reactive population CPrl= 2.0 X 10"; these numbers all seem reasonable in relation to each ~ t h e r . ~ ' , ~ * Comment. It might be thought, when looking at the list of variables in Table 11, that this approach requires too many parameters with which to calculate a falloff curve, but this is not the case-when compared with other approaches, the present one is quite economical. We should exclude the methyl isocyanide case, since standard approaches do not examine the competition between the external and internal relaxations and its effect on the falloff shape. To begin with, we have one extra collisional relaxation rate (po) which causes the incubation time to separate from the characteristic collisional deactivation time and which other theories cannot do. Any approach to calculating the falloff curve ignoring these two refinements requires the knowledge of the density of states of the molecule and a specific rate function k ( E ) calculated according to the method of preference. If one for example, one would then would follow the method of Troe,13949 calculate Pc or p cby rather involved procedures, whereas here, we simply assume p, to be infinite or zero, respectively, and in the latter case k(E) also reduces to the constant d . Both methods require an input of a collisional deactivation rate, either ZO or p l ,
by eq 3 does not affect the computed shape nor position of the falloff, provided that the appropriate value of P1is chosen; thus, the simpler calculation is always useful, so long as it is remembered that certain properties, such as the incubation time,z4-z6cannot be contemplated within the simpler framework. Nitrous Oxide. There is a slight problem in trying to apply this approach to the reaction of a small molecule like nitrous oxide, and we will make use of the earlier state-testate calculationz3for guidance. We are going to assume that for a small molecule, the
(46) Yau, A. W.; Pritchard, H. 0. Can. J . Chem. 1979, 57, 1731. (47) Figure 5 of ref 23 shows that the computed (d,) are not quite constant with energy; however, they do not vary much, and if substituted into eq 11, together with the values of h,pl,and ko used here, the result is so close to being strict-Lindemann that the differences cannot be seen on a graph of the scale of Figure 2; the values of B1, E'@,, and are also all within 10%of those cited in the text for the empirical calculation. (48) Application of the usual theory" with these particular numbers would give a value of B, = x&/X'p, = 1.3 X lo-*. (49) Smith, I. W. M. In Modern Gas Kinetics; Pilling, M. J., Smith, I. W. M , Eds.; Blackwell. Oxford, 1987; pp 99-134.
TABLE II: Coastants for Figure 2 C3H6
E,, kcal mol-' k,, s-' ko, Torr-I s-' p o / p , Torr-' s-I
Torr-'
65.6 3.4 x 10-4 (6.6 X lo4) 8500
PI
7.5 X lo5 5.0 x 10"
d , s-' P,,', s-' ~ , , ~ / Torr-' p, s-'
7.1 x 1013 >io15 >lo8
pJp,
s-I
CH3NC
N2O 66.1 1.0 x 105 4.0 225 1 2.9 X lo6 1.5 x 10-2 1012 5 x 10'0 4.0 X 10l2 1.5 X 10l2 0 1.2 X l o r 2 4.0 X 10" 0
38.4 9.2 X lo4 4.0 x 10-5 (8.8 X lo5) 7.8 x 105 1.1 x 10-4 1.6 x 1013 5.0 x
as we have noted earlier, the shape of the falloff appears to be significantly different from the standard shape and can be matched, as shown in Figure 2, with the values of pr,' = 4 X 10lz s-I and ~ , , ~= / p1.25 X 1OIz Torr-' s-l; the former value is one that can be accepted with ease, but the latter one is not. The exact numbers we find in this case are somewhat sensitive to our choice of the decay rate d, for which we used 1.6 X 1013 s-I: a recent calculation of the torsional motion in methyl isocyanide3' suggests that the true rate may be closer to 5 X 1OIzs-l, for which the values of p , , = 10lz s-I and ~ , , ~=/ 3p X 10" Torr-' s-l are needed to reproduce the observed falloff shape; this pair of numbers is hardly more acceptable than the other pair. Again, the behavior with respect to changing pl is the same, the curve becoming strict-Lindemann if p1 is increased too much. Also, this particular combination of po and pl leads to the choice of PI = 1.1 X lo4 for the additional pool of molecules in direct collisional coupling with the reactive states, whose population amounts to 3.4 X lo-'' at this temperature; again, this seems a not unreasonable figure. It is interesting to find that the same , ~F , , ~that were falloff shape requires the same values of F ~ and p , , ~ calculation, Le., the replacement needed in the simpler ( c ~ ~ ; ,pr,z) of the standard falloff expression
xr-l/3fl
~
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4339
Collisions in Unimolecular Reactions and a further shift factor, ts, or PI, to force the calculated lowpressure limiting rate constant to coincide with ko. Thus, the number of parameters is the same for an equivalent calculation, except that the present approach does not require involved calculation of intermediate parameter^.^^
Discussion The thermal isomerization of cyclopropane is the archetypical strong collision reaction, and the thermal decomposition of nitrous oxide is equally the archetypical weak-collision reaction. In each case, we find the same collisional pattern, that the vibrational relaxation rate is more than an order of magnitude slower than our estimate for the collisional deactivation rate. On the other hand, there do exist reactions, e.g., of ethyl chloride, cyclobutane, methyl isocyanide, etc., where the two rates appear to be of similar magnitude, although in each of these, we have no experimental value for the vibrational relaxation rate, only an estimate based on the Lambert-Salter rule; two of these are strong-collision reactions, but the latter may be an intermediate case. Thus, we see a marked failure to classify these reactions into strong- and weak-collision types on the basis of the known collisional properties of the molecules, and perhaps we should search for some other basis upon which to make the distinction. An obvious alternative is to make an association with the large- and small-molecule behavior, in more-or-less the same sense as this is understood by spectroscopists: in the small molecules, the states are unique and cannot evolve into other states without the intervention of external forces, whereas in the large molecules, the opposite is true. To explore this conjecture, it is necessary to develop a computational scheme within which we can explore the coupling of these intramolecular relaxation processes into the more familiar reaction versus collisional deactivation competition. The present exploration is only a step toward a full multiexponentiaI treatment of the unimolecular falloff, in which a wide spectrum of collisional relaxation rates coupling different degrees of freedom within the reacting molecule can be included. It forces us to make certain choices about the fraction of molecules pl, which must be connected to the reactive states with collisional rate w = (po + fil). but the values we have to choose are not unphysicaLM As a corollary, it is clear that we can accommodate within the states above threshold various domains that can lead to competing reactions, whose falloff rates are not inevitably constrained to bear a simple relationship to each other-as were the two incorrect shapes shown in Figure 1 for the special case of equal-energy thresholds. However, we must beware of artifacts arising from our attempt to model an obviously multiexponential relaxation process as a biexponential one: for example, in traditional theory, changing the value of the collisional deactivation rate shifts the falloff along the pressure axis; however, here, changing the equivalent quantity fil alters the sharpness of the falloff without moving it along the pressure axis, and increasing pl makes it tend toward the strict-Lindemann form. The values of pl that cause the falloff to have the correct curvature are roughly those values of the collisional deactivation rate w needed to position the falloff correctly in the more simple theory (strictly, pl = w - p0). However, this is probably an artifact: examples where a change from two-state (single-exponential) to threestate (biexponential) theory ~
(50) It should be possible to find a correlation between fll and the values deduced from various deactivation experiments.
~~
~
cause a switchover in the parametric dependence have been observed before,27 and, like other deficiencies noted above, this ambiguity too will probably be resolved when a full multiexponential theory of the falloff can be developed. We have also accepted, on reasonable theoretical ground^,^^^^ that the internal and external relaxations can be treated as separable and are then forced to make some assumptions about randomization rates: first, that all overlapping states connected to the same product continuum are completely mixed (Le., at infinite rate), which is not unreasonable. Nor is it unreasonable to suppose that for very small molecules, states not connected to a dissociation continuum are so widely spaced that they cannot mix spontaneously at any rate, but only under the influence of external perturbations. For the large, traditional strong-collision molecules like cyclopropane, we are forced to choose a value of fir,’ of at least lOI5 s-’, which is perhaps surprising but may not be impossible; although the mechanisms for such mixing are fairly o b v i o ~ si.e., ,~ broadening associated with infrared emission and black-body radiation, little has been done to define a rate that can be used in the context of a competition between this mixing and other kinetic processes. On the other hand, the conclusion that the collisional scrambling of states at reaction energy must be greater than lo8 Torr-’ s-’ cannot give much cause for argument. For methyl isocyanide, which appears to exhibit a curious falloff shape, specific values of 10l2s-’ and 3 X 10” Torr-’ s-’, respectively, had to be chosen; the former is entirely acceptable, but these very high collisional rates do present a problem. In elastic scattering, it is usual to consider a maximum value of the cross section, given by the simple formula Q 8 [ C / h ~ for ] ~a/ sixth ~ power longrange potential;51 for the molecules considered here, with intermolecular interaction parameted of the order of c / k ii: 500 K and u = 5 A, we have C = 4.3 X erg cm6 and a collision diameter of about 50 A, whereas the value of p r z just quoted corres onds to an effective diameter of about 800 A. The figure of 50 corresponds, however, to the maximum impact parameter that will produce an observable deflection in the collision, having regard for the constraints set by the uncertainty principle. Perhaps, we are not asking here that there shall be an observable deflection--only that as the result of a very long-range encounter, the reactant molecule will be left at the same energy, but in a different set of states. At these large distances, e.g., 400 A, the 6-12 intermolecular potential is still about lod7cm-I, more than an order of magnitude greater than the spacing between the states5-so it is not clear that such a collisional “tickling” cannot occur. If it cannot, then one will have to anticipate that near inflections, if they really do occur in experimental falloff curves,34 must be the result of some unusual energy-transfer pattern, which may be elucidated once a full multiexpnential theory is developed; however, no assumed pattern of coIlisionu1transition probabilities has ever yielded other than a monotonic (log-log) falloff curve,52 but modern experiments that measure relaxation rates as a function of excitation energy may well point us to one.
-
w
Acknowledgment. This work was supported by the National Sciences and Engineering Research Council of Canada. (51) Bernstein, R. B. Adu. Chem. Phys. 1966, 10, 7 5 . (52) A direct plot of k,,, versus must be monotonic for any choice of collisional relaxation probabilities.’k) (53) Schlag, E. W. Ph.D. Thesis, University of Washington, 1958. (54) Pritchard, H. 0.;Sowden, R. G.; Trotman-Dickenson, A. F. Pror. R. SOC.London,A 1953, 217, 563.
