The Journal of
Physical Chemistry
0 Copyright, 1986, by the American Chemical Society
VOLUME 90, NUMBER 19 SEPTEMBER 11, 1986
LETTERS Decoupllng Vibrational Relaxation from Collisional Deactivation in the Theory of Unimolecular Reactions H. 0. Pritcbard Centre for Research in Experimental Space Science, York University, Downsview, Ontario, Canada M3J 1 P3 (Received: April 18, 1986; In Final Form: June 24, 1986)
A modified rate expression, which allows for the fact that the vibrational relaxation rate may be smaller than the collisional deactivation rate, is derived for a strong-collisionunimolecular reaction; for this model at the high-pressure limit, the incubation time for the reaction is the inverse of the vibrational relaxation rate rather than the inverse of the collisional deactivation rate, as given by standard strong-collision theory.
Introduction In the standard treatment of a strong-collision thermal unimolecular reaction, the location of the falloff of the unimolecular rate on the pressure axis indicates the pressure range for which the collisional deactivation rate is commensurate with the mean reactive decay rate. For strong-collision reactions, this deactivation rate is usually within about an order of magnitude of the (hard-sphere) collision rate' and, following the notation of ref 1, can be written as p = pXZ, (units SI), where p is the pressure (Torr), 2, is the collision rate constant (Torr-' s-I), and X is the collision efficiency.* To a first approximation, we can regard the position of the falloff as being defined3 by the pressure at which pko becomes equal to k , a n d for the special case where the shape of the falloff is strict-Lindemann, this occurs when fi = d , i.e., at p l l Z ,the half-pressure; here, k, and k, are the low- and (1) Pritchard, H. 0. Quantum Theory of Unimolecular Reactions; Cam-
bridge University Press: Cambridge, England, 1984. (2) More elaborate treatments use collision rates calculated from the Lennard-Jones potential for the colliding systems,) but we can defer consideration of this refinement at the present stage; if, instead of Z,we use ZLJ, then X becomes 6,. (3) Troe, J. J. Chem. Phys. 1977,66, 4745, 4158.
high-pressure limiting rate constants, respectively, and d is the effective decay rate constant for the reactive molecular states. For molecules that undergo a unimolecular reaction that may be classed as a strong-collision one, vibrational relaxation data at reaction temperature are scarce. Most convincing, however, is the case of cyclopropane, for which the collisional deactivation rate p is some 80-100 times faster than the vibrational relaxation rate (PZ,,) at reaction temperature.IA Thus, we have the paradox that the rate a t which the molecule accumulates energy (PZ,,,) is much slower than the rate p at which the reactive states appear to be replenished by collisional activation: this paradox has never been resolved until now. Moreover, when the strong-collision formalism is used to calculate incubation times, one obtains the results that6,' T~~~ l/p as fi m and rinc constant as p 0; neither is physically
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(4) Also (although at a temperature 500 'C higher than that at which the falloff was measured), the incubation time for isomerization of cyclopropane appears to be about the same as the vibrational relaxation time measured in the same experiment.'.' (5) Dorko, E. A.; Crossley, R. W.; Grimm, U. W.; Mueller, G. W.; Scheller, K.J. Phys. Chem. 1973,77, 143. (6) Vatsya, S. R.; Pritchard, H. 0. Chem. Phys. 1982, 72, 447. (7) Pritchard, H. 0. J. Phys. Chem. 1983,87, 3179.
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acceptable, but the formulation described below does not suffer from either of these defects. A Two-p Reaction Rate Formula A useful clue to this resolution comes from the results of a recent state-to-state study of the thermal decomposition of nitrous oxide,8 where the same situation appears to e x k g When a count is made of the states above threshold, most of them are unreactive; Le., they possess too little energy in directions required to cause reaction and correspondingly too much in passive form-in this case, rotation and bending. At the average reaction energy for this process at 2000 K, only about 1% of the states are reactive (cf. Figure 6 of ref 8). Clearly, it is the communication between these unreactive states and the reactive ones that takes place on a time scale approaching every collision, since the energy deficit in each such interchange collision is minute. This leads to the following modification of the strong-collision model. Let all the states of the molecule be coupled by a collisional relaxation process that is exponential in character and that has a rate po = pZrlxrwhere Zrlxis the rate constant for vibrational relaxation. Above threshold, let the states be subdivided into those that are reactive, as usual, with equilibrium populations iii and decay rate constants, d j , i = 2, 3, ..., m and into those that are unreactive, population at equilibrium ii,; further, let these states aboue threshold be coupled by an additional collisional relaxation process of rate p l = pXZ,, where X and Z , are as defined before. Finally, let the equilibrium population of states below threshold be iio with normalization such that xlEoiii= 1 . With these definitions, the total equilibrium population of reactive states is x’iii = cr-2iii, and the total equilibrium population of all states if, for example were about above threshold is x”iii= Cltliii; then X’ii! could be as high as in a small molecule but in a large one it would be considerably less. For this model, it is possible to derive a pair of bounds1° on the eigenvalue yo, Le., the rate constant, in the form
c”iii
;t(O) = Po[l - d(0)l 5 Yo 5 P o [ l - +(O)l/d(O) = x ( 0 ) where
(1)
and
In the earlier paper, only” the properties of the upper bound x ( 0 ) were examined, although it cannot be reduced to a compact formula; on the other hand, substitution of (2a) and (2b) into (1) and some rearrangement leads to the expression
Letters
Position of the Falloff In the present discrete notation, the usual strong-collision rate constant is given by k, = PC-
P
iiidi
+ di
(4)
where p = pXZ, is the relaxation rate required to position the falloff in the correct place on the pressure axis;l at the low-pressure limit ko,s =
PC’&
(5)
On the other hand, the low-pressure limit of eq 3 is (for 11, >> go)
ko = po(iil
+ C’iii) = ygC”iii
(6)
in other words, the low-pressure rate constant is the product of the oibrational relaxation rule with the sum of all states above threshold; that the low-pressure rate should be governed by the rate at which the molecule acquires excitation is a desirable result. Whether one uses eq 5 or eq 6 to interpret the meaning of the limiting low-pressure rate makes a great deal of difference to one’s judgement as to what determines the position of the falloff of the pressure axis: for a fixed value of k,, p k , will intersect with k , at k , = pXZ,c’ii, = p Z r l x ~ f ’ i i l whence the collision efficiency
z
C”Ei z c C’A,
likewise, for Lennard-Jones collision
p
Zrix C”Ei ZLJ C’iii
=--
Equations 7a and 7b add a little to our understanding of collision efficiencies in unimolecular reactions, since these are now related directly to the vibrational relaxation rate of the molecule. However, these equations still contain only one observable and three theoretically constructed parameters and, in effect, only move our ignorance from one aspect to a different one: the crucial test of eq 3 is whether the model leads to an incubation time that is the inverse of the vibrational relaxation rate rather than the inverse of the collisional deactivation rate.
Incubation Times In the context of unimolecular reaction theory, the incubation time is usually taken to be that for a hypothetical shock-wave heating of the reactive gas from a very low temperature (so that the initial population vector may be considered as [ 1, 0, 0, ..., 019 to the temperature in question; the concept of an incubation time is, however, much more general than this, and negative incubation times are possible.6 The incubation time can be evaluated once the eigenvector fi0corresponding to the rate constant yo is known, and it may be expressed as1g7 = YO-’In [(s0,10)(E-1’2n(o),lI.,)1 (8) if the normalization of fi0is such that (Go,G0) = 1, ( , ) denoting the scalar product of two vectors; here So is the eigenvector of the symmetrized relaxation matrix, with elements ii2liZ, ...,E is a matrix required in the symmetrization of the relaxation matrix, and n(0) is the starting population vector. Before proceeding, however, it is appropriate to check the validity of eq 8, particularly in view of the fact that it is derived from the properties of a nonconservative reaction matrix, and some suspicion has been raised recently that the dissipative result may not always lead to Tinc
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In fact, $(O) is an excellent approximation to yo: from ( 1 ) 1 as po m, and (1 - c”ii,)as 0; since is the sum of all states above threshold (perhaps 10” in any practical application), eq 3 gives an exceedingly good approximation to the rate constant at all reasonable temperatures.l 2
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; t ( O ) / x ( O ) = q5(0), which
(8) Pritchard, H. 0. J . Phys. Chem. 1985, 89, 3970. (9) Here, the measured relaxation rate is about loo0 times slower than the value of p needed to position the falloff correctly on the pressure axis;* this reaction, however, is not a strong-collision one by the usual definition. (10) Vatsya, S. R.; Pritchard, H. 0 Chem. Phys. 1981, 63, 383. (1 1) With the exception that &O) was shown to tend to Cii,d,as p o m, i.e., to the correct high-pressure limit for yo.
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(12) This quality of the two approximations is confirmed by model calculations, which show that 3(0) and x ( 0 ) converge gradually as po increases; also, it was found that the true value of y o was closer to 6(0) near the low-pressure limit but closer to x(0)near the high-pressure limit. The extreme closeness of these two bounds is of considerable help in computing yo numerically by iteration.
The Journal of Physical Chemistry, Vol. 90, No. 19, 1986 4413
Letters the correct inter~retati0n.l~The following is a rederivation of the incubation time for a conservative reaction matrix. The conservative reaction matrix is easily constructed by the usual rules,' by treating the reaction product as one state of the system, with equilibrium population E,. Let be the eigenvector of this matrix corresponding to its zero eigenvalue: the unnormalized 4o = (So f i p l / z ) ; also, let be the eigenvector corresponding to y l , the lowest nonzero eigenvalue, and let all eigenvectors be normalized so that (&, q5J = 1. At long times, after all transients have died, we can write the evolution of the product population as (eq 2.14 of ref 1)
where the pth element of any vector is the one corresponding to product. But (40)p(40)p = ii, and (E-'/2n(0),40)= 1 (cf. ref 1); therefore
As is well-known, y l calculated from a conservative reaction matrix is not the rate constant but is the sum of the rate constants for the forward and reverse reaction^;'^.'^ likewise, for a first-order reaction approaching an equilibrium, the behavior is16
where is also the sum of the forward and reverse rate constants. With allowance for the existence of an incubation period, (10) has to be rewritten as
or
Equating (9) and (loa), we have
and identifying
with y 1
No analytical work along the previous lines6,' has yet been attempted on eq 11, but numerical experiments show that eq 8 and 11 yield the same values of the incubation time at any pressure for the standard strong-collision reaction matrix. Likewise, a selection of twc-p reaction matrices of order between 4 and 40 was constructed (cf. ref 10 or the exact form of the matrix elements), and it was found numerically that rincwas the same whether calculated by eq 8 or eq 11. Also, as expected, T~~ became decoupled from p I and exhibited the property that a s p 0 3 , T~~~ l/po, in other words to the inverse of the true vibrational
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(13) Quack, M. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 94. (14) Snider, N. S. J . Chem. Phys. 1965, 42, 548. (15) Widom, B. Science (Washington, D.C.)1965, 148, 1555. (16) Glasstone, S.Textbook ojPhysical Chemistry, 2nd ed.; Macmillan: London,England, 1947.
relaxation rate. In the falloff region, T ~ , , ~ / T , lOOC'2, (as is probably the case in most reactions), the disparity found in these model calculations was less than a factor of 3.
Conclusions The rate constant expression (3) has the advantage that it is derived for a model in which the molecular relaxation and the collisional activation-deactivation processes are regarded as being distinct; moreover, (3) is an extremely accurate solution to the given problem. Thus, for the first time, we now have a strongcollision model of a unimolecular reaction in which the incubation time is connected to the molecular relaxation rate and not to the deactivation rate for the reactive states; this eliminates the inconsistency exemplified by the cyclopropane results. Nevertheless, the model is still too simple to account properly for observed incubation time behavior. At the high-pressure limit, it yields the result that the incubation time is the same as the bulk relaxation time, which is obviously sensible. However, in the falloff region, and at the low-pressure limit, it gives an incubation time a little shorter than the relaxation time, whereas the observed behavior for N20,albeit a weak-collision reaction, is that the incubation time is slightly longer;" moreover, incubation times longer than relaxation times seem to be reproduced rather easily in purely numerical model The reason, qualitatively, is straightforward-that the present model, like the conventional strong-collision one, treats all states below threshold as a single entity, always in equilibrium with each other, whereas the true relaxation must exhibit some stepladder qualities. It appears feasible, though, that eq 3 may be extended to more than two values of p , in which case, it should be possible to include multiple relaxation processes among the unreactive states. Nevertheless, it seems that a fuller understanding of the relationship between vibrational relaxation time and incubation time will come from numerical experiment, since the theoretical analysis of the relationship between the two observables is rather difficult: Equation 3, however, also presents us with something of a dilemma, since it tells us that the position of the falloff is determined principally by the bulk relaxation rate-and not by the collisional deactivation rate as we have believed throughout the history of unimolecular reaction theory: numerical experiments show that for a given set of energy levels and specific decay rate constants, the position of the falloff is determined by pol the bulk relaxation rate and it is quite insensitive to the value of p l , the collisional deactivation rate. Thus, we are now able, through magnificent experiments:' to measure the rate at which molecules in the reactive energy range lose their energy in collisions, but it appears to be a quantity of secondary significance in positioning the falloff on the pressure axis. On the other hand, a great deal of work undertaken in the 1950s to study vibrational relaxation rates as a means of understanding unimolecular falloff and later abandoned as not being directly relevant now appears to be somewhat vindicated-although the connection still remains a rather indirect one. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada. (17) Dove, J. E.; Nip, W. S.;Teitelbaum, H. Fifteenth Symposium (Internarionaf) on Combustion;The Combustion Institute: Pittsburgh, PA, 1974; p 680. (18) Dove, J. E.; Troe, J. Chem. Phys. 1978, 35, 1. (19) Yau, A. W.; Pritchard, H. 0. Can. J . Chem. 1979, 57, 1723. (20) Forst, W.; Penner, A. W. J . Chem. Phys. 1980, 72, 1435. (21) Heymann, M.; Hippler, H.; Troe, J. J . Chem. Phys. 1984.80, 1853.
