J. Phys. Chem. 1985,89, 3970-3976
3970
excited molecule is below this threshold and the energy is coupled preferentially to the 9eactive” mode, presumably the skeletal torsion. The fact that excitation of OH, OD, or CH vibrations seems to be similarly efficient in inducing the isomerization is, however, not consistent with this model. An additional problem with “IVR threshold model” is that, while it could explain the observations in 2-FE, it could not be T process used for numerous related compounds where the G is also found to be more efficient but where the trans form is lower in energy. A convincing interpretation of this trend will undoubtedly require additional work, in particular studies using continuously tunable infrared radiation sources. Effect of the Matrix. In previous studies it was observed that the rates of fluoroethanol isomerization decreased with the decreasing mass of the rare gas host3 While very fast isomerization was observed in solid Xe, the rate decreased substantially in the lighter rare gases and was found to be slower by more than an order of magnitude in solid argon. This seemed to suggest a systematic dependence upon the mass or, perhaps, polarizability of the host matrix. By extrapolation one could therefore expect the rates to be even slower in solid neon, and some experiments
-
of Schrems22apparently led to the conclusion that no IR-induced process occurs in solid neon. Actually, we find that a remarkable reversal in the trend established in the heavy rare gases occurs, and the rates in neon are in fact faster than in xenon matrix. This observation would seem to suggest that the behavior of the guest molecules depends on the nature and detailed geometry of the trapping site, and the trends probably cannot be simply modeled by considering only the bulk properties of the host matrices. This is consistent with our recent studies of halopropanes where we observed that differences between isomerization rates in different matrix “sites” are often of comparable magnitude, or even larger, than those between different hosts.
Summary Study of the infrared spectra matrix isolated 2-FE shows clearly that only the Gg’ and T t species are trapped in the 4 K rare gas solids. Infrared excitation of the O H and CH bonds is equally effective in producing the Gg’ Tt rotamerization. The isomeriyation rate is unaffected by deuteration of the OH group.
-
(22) Schrems, 0 ,private communication
FEATURE ARTICLE State-testate Theory of Unlmolecular Reactlons‘ H. 0. Pritchard Centre f o r Research in Experimental Space Science, York Uniuersity, Downsview, Ontario, Canada M3J 1 P3 (Received: March 15, 1985)
In state-testate calculations for the thermal dissociation of N 2 0and C 0 2 ,a close approximation to the strong-collisionfalloff shape is found, even though randomization among states above threshold is not permitted. Allowing rapid randomization among all types of states above threshold causes the falloff to approach the strong-collision shape in the limit, but restriction of the randomization to only those states which are dissociative gives rise to the observed strict-Lindemann shape for the falloff.
We have seen, to date, only a handful of attempts to calculate the rate of a thermal unimolecular reaction as a sum over the rates of individual state-to-state processes. Among them are the calculation of the rates of dissociation of N 2 0 and C 0 2 by Yau2 and of N 2 0 and dioxetane by Lorquet et al.3 and the calculation of the rates of isomerization of methyl isocyanide by Clarkson4 and of H N C by Perk et aL6 Clearly, more such calculations will be forthcoming, but equally clearly, we can see that unless there is some simplifying feature (such as symmetry), the number of matrix elements required for any realistic calculation is going to ( 1 ) An abbreviated version of this paper was presented at the Symposium on The Interface between Theory and Experiment, Canberra, Australia, Feb 1985, to mark the retirement of Professor D. P.Craig, F.R.S. (2) Yau, A. W.: Pritchard, H. 0. Con. J . Chem. 1979, 57, 1731. (3) Lorquet, A, J.; Lorquet. J . C.: Forst, W. Chem. Phys. 1980, 51, 2.53, 261. (4) Clarkson, M. E., unpublished work (cf. ref 5). (5) pritchard. H. 0. ’Quantum Theory of Unimolecular Reactions*; Cambridge University Press: Cambridge, England, 1984. (6) Peric, M.: Mladenovic, M.: Peyerimhoff, S . D.; Buenker, R. J. Chem. Phys. 1984, 86, 85.
0022-3654/85/2089-3970$01.50/0
be astronomical for molecules containing more than three atoms. It is the pufpose of this article to illustrate how such a state-testate calculation proceeds once the matrix elements between the initial reactant and final product states have been determined and to draw attention to some unresolved difficulties. The two isomerization studies are very similar: the former treats the methyl radical as a rigid mas’s and the energy levels and lifetimes are determined for the (MI,)-N--C bending/restricted rotation/free rotation with an empirical potential and a rigidbender Hamiltonian,’ whereas the latter solves for the same properties for H N C with a theoretically computed potential; both, however, make the same assumption that the isomerization reaction is irreversible, whereas the physical process itself is c y c l i ~ . ~ Of the dissociation reaction calculations, the earlier one2 is the simpler. It solves for the matrix elements connecting various initial (7) Bunker, P.R.; Howe, D. J. J . Mol. Spectrosc. 1980, 83, 288. (8) In fact, this assumption would not be necessary if the problem were to be treated fully, by including the reverse reaction from the product states (cf. ref 9 and 10). (9) Widom, B. J . Chem. Phys. 1971, 55, 44. (10) Quack, M. Ber. Bunsenges. Phys. Chem. 1984, 88. 94.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 19, 1985 3971
Feature Article
TABLE I:
.. ..
.' ...
i 1
L
?',
'
a
2 3 4 5 6 7 8 9 10 11 12 13 14
.
. '
. .
..
.
t
m
.
The 14 States of
N20Having the Highest Decay Rate
Constants'
'
'
..
UI
u2
Ut
26 27 28 29 30 34 35 37 38 42 43 44 34 37
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1
1 1
Ei 68.337 69.926 71.439 72.874 74.232 78.897 79.870 81.587 82.329 84.530 84.888 85.169 85.338 88.028
lo-lld, 7.4 9.2 9.0 8.2 6.6 8.7 9.9 11.0 6.3 7.5 7.0 6.8 7.0 5.8
'Energy Ei in kcalmol-I; rate constant di in s-l.
B .
' .
Energy
'*
( k c a l /mol
)
91 +)
Figure 1. Plot of the energies and rates of decay for the 527 reactive states used in this calculation. Each symbol denotes a different final vibrational state of the N2 molecule as follows: dots, uf = 0; circles, uf = 1; squares, uf = 2; plus signs, uf = 3; crosses, uf = 4; asterisks, ut = 5; triangles, uf = 6; inverted triangles, uf = 7. Sets of similar points may be joined into sequences according to the initial value of u2, but the sequences are highly oscillatory and the diagram becomes illegible; some of these sequences can be seen in Figure 2 of ref 2 or Figure 6.1 of ref 5.
vibrational states of N-N-0 (ul, u2) with various final states of N2 (uf) recoiling from the 0 atom, on the assumption that the coupled oscillator system can be regarded as being linear and that the matrix elements (and therefore the total rate) are independent of how strongly the molecule is bending or how fast it is rotating; the required potential surface was constructed from spectroscopic data. This model is, in fact, somewhat of an oversimplification, for it is known3*"that the bending motion makes a definite contribution to reaction and, of course, centrifugal distortion must also play a part in bringing about di~sociation;~ for present purposes, however, these are matters of detail which must be resolved before a true a b initio calculation of the rate can be achieved.l2
From State-to-State Rates to the Macroscopic Rate It is considered that the linear vibrational motion of the N 2 0 molecule can be characterized by two quantum numbers: u2 representing the simple harmonic oscillator motion of the two N atoms with respect to each other and uI representing the Morse oscillator motion of the 0 atom with respect to the Nz pair; products are formed with N2in state uf, and the energy remainder [E(u,) E(u2) - E(uf)]appears as translational energy. In the original calculation; 527 such transitions were identified as having rate constants greater than lo5 s-l; these are shown in Figure 1,
+
(11) Miret-Artes, S.;Delgado-Barrio, G.; Atabek, 0.; Beswick, J. A. Chem. Phys. Lett. 1983, 98, 554. (12) Is the model too simple for a proper discussion of unimolecular phenomena? Jumping ahead somewhat, if the rate were to vary with the bending quantum number, than the heads of the vertical lines in Figure 2 would simply not form linear progressions; the number of matrix elements required would only be about an order of magnitude greater, but of course, their evaluation would also be more complex. Similar considerationswould apply if the rates were to depend upon the initial value of J-the heads of the lines in Figure 3 would not form such a regular pattern; also, if rotation-vibration interaction were included, then (to afirsr approximation) the vertical lines in Figure 3 would be spaced differently, and the diagram would terminate at some energy beyond the presmt range of intemt. Each of these embellishments might alter the final numerical result but would not change the qualitative picture appreciably; basically, the zeroth-order model assumes that all rotation-vibration states of the N20molecule would be distinct in the absence of the predissociation process that lead to fragmentation.
D l
&
x 2 0 1
0 60
90
70
Energy
I 00
( k c o l /mol
Figure 2. Contributions from the 14 states listed in Table I to the rate of reaction at various energies, due to convolution with the ({uknd + 11 degenerate) bending energy levels; in the range from threshold to 95 kcal-mol-', these 14 states can be found at 136 separate values of the energy.
where each symbol denotes a different vibrational state of the product N2 molecule. The simplest thing we can do is to calculate the rate constant at infinite pressure because here all the states can be considered to be populated a t their equilibrium values for the temperature in question: the rate is usually written as p ( E ) k(E)e-E/RT dE
where E is the energy, Eo is some threshold below which reaction does not occur, Q( T ) is the total partition function at temperature T . Le.
Q( T ) = l m0 p ( E ) e - E / RdTE p ( E ) is the density of (internal) states at energy E, and k ( E ) is
the average rate constant to be applied to any molecule having energy E, regardless of the manner in which that energy is distributed among the available degrees of freedom. We can equally well write5
k, = D i d i i
(3)
where iii is the equilibrium population of state i (normalized so that Cifii= 1) and di is the rate constant for decay of that state to products; there is no averaging involved in eq 3, unlike that used in constructing the specific rate constant k ( E ) in eq 1. We must first, therefore, establish the connection between k ( E ) and the individual state-to-state rates di. To understand how this is done, 527 states are far too many to consider, and for the purpose of illustration, we will use only the 14 fastest processes, whose ~haracteristics'~ are listed in Table
3912
The Journal of Physical Chemisfry, Vol. 89, No. 19. 1985
Pritchard
-
TABLE II: Syntbesk of k (E) at Five Values of E from the Indindual State-to-State Rate Constants for the 14-State Model no. of occurrences of state i within the range E (E + A€) = p ( , E.. P, = kcalmolV Up(€) 1 2 3 4 5 6 7 8 9 IO I1 12 70.00-70.05 75.00-75.05 80.00-80.05 85.00-85.05 90.00-90.05
E, kcabmol-' 70.00-70.05 75.00-75.05 80.00-80.05 85.00-85.05 90.00-90.05
9.9 1.3 1.7 2.1 2.6
IO' X IO6
75
X
X
IO6
258 I118 2555 8092
IO6 IO6 P, = UPW) 9.9 x 105 X X
1.3 X IO6 1.7 X IO6 2.1 x 106 2.6 X IO6
60
80
Energy
36 225 873 889 3556
227 1037 1930 3811
6 = XPg, 111 924 4386 I1197 24456
90
163 425 2836 2861
51 828 I333 2140
+, = X P A 8.8 x 7.7 x 3.6 X 9.1 X 2.0 x
IO" 10'6
215 785
209 265
41 389
W E ) = 4 = Alp, 9.0 x 6.0 X 2.1 x 4.3 x 7.5 x
44 634
207
8, at 2000 K 3.9 x 10.' 1.4 x 10-7 5.3 x 104
107
IO' 109 109 109
1.9 X IO4 6.7 X IO*
320
62
0, = $.IG
8.0 x 8.3 x 8.1 x 8.2 x 8.1 x
IO"
IO" IO" IO" IO"
im
I. Consider the first entry in this table, the threshold level for our 14-state calculation: the decay rate listed is for a molecule of N20 in states uI = 26, u2 = 0 without any bending or rotation. But we are assuming that if the reactant molecule possesses this particular combination of quantum numhers, its rate of decay is independent of how vigorously it is bending. If it happens to have one quantum of bending (Le. AEhd = 600 cm-' = 1.714 kcal. mol-') in it, its energy will be 70.051 instead of 68.337 kcal.mol-l, but since the bending is doubly degenerate,14 there are two states a t this energy; likewise, there are three states possessing this particular combination of uI and u2 at 71.567 kcal.moltl, and uM + 1 states a t 68.337 ubcndAEbcndkcal-mol-I. Thus, Figure 2 shows all the energies at which, and the relative chance that, the combination uI = 26, u2 = 0 can make a contribution to the overall rate of reaction: it may do so only at certain quantized energies, and these contributions are joined together a t their heads to form a progression. Analogous contributions for the other 13 processes listed in Table I are also shown in this diagram, which is reaching the limit of legibility due to congestion at the highest energies considered here, i.e. 95 kcal.mol-'. Our molecule in uI = 26, u2 = 0 may also be rotating, and so a molecule possessing this pair of quantum numbers (and with
+
(13) The last two proasscs leave the product N, in the state u, = I, and so we have thc capacity to discuss the falloff propertics of competing sets of unimolecular reactions and their mutual interference ( i s . we can treat N20 NAuyO) + 0 and N,O N 2 ( u iI ) + 0 as different chemical reactions) and also to examine translational energy distributions under different conditions of pressure; neither of t h s e aspects is addrcssd in the prcSent review. This model, howcver. does not contain sufficient information to permit the examination of product rotational-state distributions. Notice also that the effect of isotopic substitution fallows in a natural manner: all that one has to do is to reevaluate the matrix elements for the same electronic potential surface, but with the new atomic masses; small diffmenccswill m r bsdusc the initial energy levels, the bending frequency. and the moment of inmis will all be slightly different in the new calculation. (14) Herzberg, G. 'Infrared and Raman Spectra"; Van Nostrand New York, 1945. (IS) Yau. A. W.; Pritchard, H.0. Can. J. Chcm. 1979,57,2458. (16) Olchmki, H. A,; T m , J.: Wagner. H.Gg. Be,. Bunsenges. Phys. Chem. 1966, 70,450.
-
1013 10''
44 489 963
14
(kcol/mol>
Fig" 3. Contributions from state i = 1 of Table I to the rate of reaction at various energies. due to convolution with the ((U + I] degenerate) rotational energy levels; notice the much larger vertical scale on this diagram, compared with that in Figure 2.
-
61 656 371
13
4. Plot of the function $,for the 14-state (lower set) and 527-state (upper set) calculations; notice that, for the sake of clarity, the 14-state results have been shifted downward by 0.25 lag unit and the 527-state results upward by the same amount. The solid lines show the continuous representation of the function 4, deduced as the inverse Laplace transfarm" of the rate law given by these calculations.' Qure
w = 0) may occur a t any of the specific energies 68.337 + J(J
+ l)AE, where AEJ = 0.419 cn-l
= 1.198 X kcal.mol-'; since the degeneracy of any rotational state is 2J + 1, the chance that = 0 can make a contribution the combination uI = 26, u2 = 0, to the overall rate is shown in Figure 3. Notice that the vertical scale in Figure 3 is 10 times larger than in Figure 2 what is true of the combination uI = 26, u2 = 0, uhd = 0 alone in Figure 3 is equally true for each of the 136 combinations depicted in Figure 2, and so when we consider all bending and rotational possibilities for the 14 states in Table I, we will have a fan-shaped diagram like Figure 3 with an origin at each of the energies shown in Figure 2 and with wntributions of (U+ I)(+ l)djfalling a t energies E, + uMAEknd + J ( J + l)AEJ. The degree of congestion of the resulting diagram already defies any attempt to represent it in a meaningful way. For any real molecule, the functions k(E) and p ( E ) cannot be expressed in any analytic form, and so the integrations (1) and (2) must be performed numerically: we therefore divide up the energy spearum Eo 5 E < into small grains, labeled r, of width AE and count the total contribution from each basic state in Table I to the rate in each energy range E, 5 E < E, + a selection of results for a grain width of 0.05 kcal.mol-l is shown in Table 11. The entries in the early part of the table are straightforward in the second column is the total number of rotation-vibration states in the grain; this is the only quantity that was not derived by direct counting hut was estimated by the usual methods for calculating sums and densities of states? Then follow 14 columns showing the number of times a contribution from state i (in Table I) falls inside the grain boundary as a result of superposing all possible combinations of Figures 2 and 3. The next two give the quantities o, the total number of reactive states lying within each grain r, and @,, the total decay rate the grain would have if the population of each state within were unity. Since we have assigned to all molecular energy levels their propcr state degeneracies, all states have equal a priori probability, and if the grain width is
-
The Journal of Physical Chemistry, Vol. 89, No. 19, 1985 3913
Feature Article
Energy
(kcol /mol)
Figure 5. Plots of the quantity a, the mean decay rate constant of the reactive states, for the 14- and 527-state calculations. Also marked on this diagram are the values of the Arrhenius parameters A , and E , calculated for these reactions and the average energy E of the reacting molecules at 2000 K. 'I
60
I
I
I
I
I
70
80
90
IM
Energy (kcol / m o 1 )
Figure 6. Plots of the fraction of reactive states within each grain as a function of energy for the 14- and 527-state calculations. Notice that, in the 14-state case, because the d, are almost constant, there are some strong resemblances with Figure 4 at low energies. sufficiently small, then all states within the grain will have equal populations at equilibrium. Given this simplification, we can identify 4, with the quantity A E [ p ( E ) k ( E ) ] , in which case the value of k ( E ) to be used in eq 1 becomes +,/P,, where P, = hE[p(E)],is the total number of states within grain r; these values of k(E) are shown in the next column of Table 11. The penultimate column gives values of p,, the equilibrium population of grain r for the temperature of 2000 K, the infinite-pressure rate constant for this temperature is then simply k , = C,.@,d,The final column of Table I1 gives the values of a, the mean rates of decay of the reactive states within the grain. Having established these connections, we can now inspect the forms of the functions generated. First, we look at the function 4, in Figure 4: the lower set of points extend the results presented in Table 11, giving the value of 9,for the 14-state calculation for every grain between threshold and 95 kcal-mol-'; the upper set of points is the analogous plot for the case when all 527 reactive states are counted-there is a lower threshold and the magnitude of 4, is larger at any given energy, both for obvious reasons. The important point to notice is that although the patterns of reaction rate in Figure 1 were highly oscillatory and the energies at which they occurred were discrete and quite sparse, by the process of convoluting these rates with all possible combinations of bending and rotation, we have arrived at a set of decay rates for each grain, &, which form a relatively smooth sequence, and the more reactive states we consider to begin with, the narrower is the spread of the computed points; in a sense, the behavior has become statistical. This is illustrated further in Figures 5 and 6 which show respectively the mean rate of decay, D,, for all the reactive states within the grain and the fraction of reactive states within each grain. Figure 5 shows that the mean rate of decay of the reactive states is virtually constant by 10 kcabmol-I above threshold-this is not surprising in the 14-state calculation where (Table I) all reaction rates are within a span of a factor of 2, but in the 527-state case, the original di(Figure 1) can lie anywhere between los and lo'* s-l; also, Figure 6 shows that the fraction of reactive states
2
3
4
Log p
5
6
7
(Torr)
Figure 7. Comparison of the results of the 14- and 527-state calculations with experiment. The experimental points are from ref 16, and the dotted curve which passes through them is a strict-Lindemann function: klindomann = k,(l + k,/k@) with k , = 1.0 X lo5s-I and kO= 4.0 torr-I s-'. The heavy broken line for each case is the strong-collisioncalculation, i.e. eq 4a, whereas the dashed lines were calculated by using eq 5; the pair of solid lines are the strict-Lindemann curve shifted down so as to make its high- and low-pressure limits coincide with those of the computed curves. The values for the internal relaxation rate used in these calculations were the following: eq 4a, 14 states, p = 2.0 X lo3 torr-] s-I; eq 4a, 527 states, p = 1.1 X lo4 torr-' 8;eq 5, any number of states, p = 2.5 X lo6 torr-' SKI.
in each grain varies very slowly (and, again, remarkably smoothly) with energy. Clearly, then, it seems permissible to say that we can assign the same rate constant, on average, to any reactive state in the grain; at high energies, this average value will be a constant, but it is a function of the energy for (crudely) molecules reacting with less than the average energy above threshold. Equally clearly, since the fraction of reactive states within each grain can be well represented as a smooth function of the energy, there can be nothing seriously wrong with assigning an average rate of decay [ k ( E ) ] ,= &/P, which can be applied to all states in the grain (as we have done in one of the columns of Table 11). From a computational point of view, the latter statement is true, but there is, nevertheless, a logical problem: if a molecule is produced in an unreactive state, then it makes no contribution to the rate in eq 3; however, in eq 1, it does make a contribution because all states have been assigned an equal average rate k(E). How then does a molecule created in an unreactive state in grain r get into a reactive state so that it can make its proper contribution to the rate in eq l ? Since it has been eq 1, rather than (3), that has been the basis of the successful description of unimolecular reaction and falloff over the past 60 years, then we must postulate some process, or processes, taking place much more quickly than the rates at which the states decay, which replenish the decaying states from the much larger pool of unreactive ones; such processes are variously called intramolecular vibrational relaxation (IVR), intramolecular energy transfer (IET), or just simply randomization.
Calculation of the Falloff with Pressure The 14 states listed in Table I make the dominant contribution to the infinite-pressure rate at 2000 K, and in fact, since the average energy of all molecules reacting (see Figure 5) is between 70 and 71 kcal-mol-', only the first five of them are really important. With 14 states k , is 3.8 X lo4 s-I and with 527 states k , is 9.1 X lo4 s-l, whereas the highest experimental value for the ratel6 at 2000 K is 9.1 X lo4 s-I, with k , probablySclose to 1.0 X lo5 s-I. Figure 7 shows the experimental rates at 2000 K as a function of the pressure p , and the dotted line which passes through them is a strict-Lindemann curve which fits them reasonably well. If we have some knowledge of the nature of the collisional relaxation process, we can solve the master equation numerically
3974
The Journal of Physical Chemistry, Vol. 89, No. 19, 1985
for the coupled reaction-relaxation system; modern laser studies of vibrational relaxation have the potential to generate the required information, but so far, we have had to rely on the construction of plausible models"J8 which has not been very productive. For the time being, let us make the crude assumption that the relaxation can be considered to be a pure exponential decay, for no other reason than that it simplifies the computations drastically and, at the same time, leads to some better understanding of the prob1e1n.I~ In the exponential relaxation case, the expressions for the unimolecular rate as a function of pressure are very simple, viz.
in the usual continuous form, or (5) in discrete form. Here, p is the rate of relaxation, i.e. P = XZ[M] where 2 is the collision number, [MI is the concentration of particles, and X is an efficiency factor; usually, p = w = Z[M] is written in eq 4, Le. X = 1. With the information available in Table 11, we may now proceed in one of two ways, the standard way (via eq 4) or the state-to-state way (via eq 5); however, for computational purposes, eq 4 will be written as a sum
where d, = [ k ( E ) ] ,is the grained approximation to k ( E ) and (3, is the equilibrium population of the grain, Le.
Figure 7 shows a set of three falloff curves labeled "14 states": the bottom one (broken line) is obtained by using eq 4-actually (4a)-whereas the next one (dashed line) is obtained by using eq 5; the upper one of the three is the strict-Lindemann curve (Le. the experimental shape) shifted down so that it has the same limits ko, k , as do the other two curves. Equation 4a, of course, is just the ordinary strong-collision expression except that, usually, for d, is substituted a k(E) function calculated by R R K M or some other similar prescription; here, we have constructed the d,, i.e. [ k ( E ) ] , ,by averaging over the appropriate state-t+state processes. Equation 5, on the other hand, gives almost strict-Lindemann behavior: this is because one of the two conditions for the occurrence of strict-Lindemann behavior22 is almost met in this calculation, Le. that with a pure exponential relaxation, the decay rates of all reacting states are the same; in this case (Table I), all decay rates lie between 6 X 10" and 11 X 10" s-I. In contrast, the averaged d, calculated in Table I1 from the same data vary strongly with E , leading to the greater spread in the falloff range when eq 4 is used;this point has been explained in much more detail elsewhere.ss22 If we now go to the full 527-state calculation, we lose the simplification that all the di are (almost) the same (cf. Figure l ) , (17) Vatsya, S.R.; Pritchard, H. 0. Chem. Phys. 1984,87, 233. (18) Pritchard, H. 0. Can. J. Chem. 1984,62, 157. (19) It would appear that, except for very high temperatures, we can always make the assumption of exponential relaxation. The difference between the steady-state population distribution during the reaction and the Boltmann distribution can be expressed (cf. footnote 1 of ref 20) as a sum of terms in (A, - yo)", where A, are the eigenvalues of the unperturbed matrix and yois the rate constant; hence, the largest term is (A, - yo)-' where X I = 2.5 X lo3 and yo = 4.0 torr-' s-I. Also, it has been shown recently that the complementary assumption, that the decay processes themselves are exponential when, in fact, they may be more complex, has no discernible effect on the bulk reaction rate.*' (20) Pritchard, H. 0.;Vatsya, S . R. Can. J . Chem. 1981, 59, 2575. (21) Hase, W. L. Chem. Phys. Leu. 1985, 116, 312. (22) Vatsya, S. R.; Pritchard, H. 0. Proc. R. SOC.London,A 1981, 375, 409.
Pritchard and the three curves labeled "527 states" in Figure 7 show the result: the bottom line is again the strong-collision calculation (4a) and the dashed line is eq 5; the solid line is the experimental curve shifted down for the shape comparison. Although the remaining 513 states have lower rates, together they make up rather more than half of the total rate, and because of this, they affect the shape of the falloff each term in (5) is a Lindemann curve, but each is centered about the pressure for which 1 = di; it is the spread of these p1 pressures which causes the falloff to broaden in this manner.2 1
Do We Really Need the Concept of Randomization? Strong-Collision Reactions. Examination of Figure 7 shows that the full 527-state calculation yields almost the strong-collision shape without the need to invoke the concept of r a n d o m i ~ a t i o n , ~ ~ and it would seem reasonable to conjecture that in more complex molecules, the state-to-state calculation, eq 5, and the strongcollision calculation, eq 4 or 4a, will converge to the same falloff shape; randomization would be unnecessary because the molecules which react would have a random distribution of reaction rates for any grain, except for those grains just above threshold. The simplest theory of unimolecular reactions is that derived from the original paper of Polanyi and Wigner, and here, it is considered that all systems which decay to products do so at the same rate;5 thus, eq 5 (no randomization) gives strict-Lindemann behavior and eq 4a (infinite randomization) gives strong-collision behavior, and almost any shape intermediate between the two can be generated on the assumption of a specific form for the supposed rate of swapping between the reactive and the unreactive This analysis is beginning to encounter difficulties now because it is necessary to postulate exceedingly high randomization rates (p, > 10ls s-l) in order to explain the strongcollision shapes observed in (say) the cyclopropane isomerization, but these high randomization rates carry with them the requirement of a step in the chemical activation D/S ratios at pressures which are a factor of lo6-lo8 too low.2s If it were not true that all reactant states decayed to products at the same rate (or with a rather narrow band of rates as in RRKM theory26)but with a highly random set of rates such as is shown in Figure 1, then the need to invoke the hypothetical process of randomization disappears and, along with that, the difficulties just mentioned in the extension of these state-to-state ideas to the chemical activation problem.2s Weak-Collision Reactions. It appears that several weak-collision thermal unimolecular reactions may exhibit a falloff behavior N 2 0seems which is close to being strict L i n d e m a ~ ~certainly, n;~~ (23) Precisely similar results were obtained for the thermal dissociation of C 0 2 ,again by using the data from Yau's calculation (ref 2), but they are not presented here for the sake of brevity. (24) Pritchard, H. 0. Can. J . Chem. 1981, 58, 2236. (25) Pritchard, H. 0.; Vatsya, S. R. J. Phys. Chem. 1984, 88, 5816. (26) RRKM theory assumes a slightly more complicated form, in which the rate of passage from reactant configurations to product configurations is proportional to the square root of the nonfixed energy (ref 27); thus, a discretestate version of RRKM theory (if it would be meaningful to contemplate one) could not give rise to strict-Lindemann behavior as the randomization rate was assumed to approach zero. (27) Robinson, P. J.; Holbrok, K. A. "Unimolecular Reactions"; WileyInterscience: New York, 1972. (28) In fact, since strong-collision unimolecular reactions often appear to be strict Arrhenius,' a very good approximation to the k ( E ) function can be obtained via the inverse Laplace transform p r o c e d ~ r e ; ~ it* 'should ~ < ~ ~ not be difficult to deduce the appropriate distribution functions for discrete d, which would yield the same strong collision falloff shape. We have to distinguish, however, between two distinct types of unimolecular reaction: (i) the nonadiabatic case, such as this, where the decay rates must scatter over a very wide range of values, and (ii) an adiabatic reaction, such as the isomerization of methyl isocyanide, or perhaps the dissociation of ethane, where the state-tostate rates may remain approximately the same once the threshold energy for the required motion is exceeded; these remarks would not apply to the latter type of process. Moreover, it has becn argued'O that even in the nonadiabatic case the decay rates may change over from being widely scattered to being approximately constant with increasing energy. (29) Yau, A. W.; Pritchard, H. 0. Can. J. Chem. 1978, 56, 1389. (30) Sundberg, R. H.; Heller, E. J. J . Chem. Phys. 1984,80, 3680. (31) Yau, A . W.; Pritchard, H. 0. Chem. Phys. Lett 1978, 60. 140.
The Journal of P h y s i c a l Chemistry, Vol. 89, No. 19, I985 3915
Feature Article
I
II
I
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I
2
3
4
5
Log p ( T o r r ) Figure 8. Comparison of the shapes of falloff curves for the 527-state calculations, including full randomization among states above threshold. Dotted lines represent second-order randomization, and dashed lines denote first-order randomization. The randomization rates a t p = 1 torr are listed to the left of the various curves; p = 2.5 X lo6 torr-l s-' in all cases.
Figure 9. Comparison of the shapes of falloff curves for the 527-state calculation, including only limited first-order randomization, with experiment. From top to bottom, the values of pr are lOI3, lo1', lo", lo1", lo9, and 0 s-l; p = 2.5 X lo6 torr-' s-' in all cases.
to be a good example of such behavior (cf. Figure 7). The occurrence of strict-Lindemann behavior requires some rather stringent conditions to be met: either there must be some kind of drastic bottleneck in the activation ladder1s*32 or else all grains (eq 4a) or all s t a t e s (eq 5 ) must have the same decay rate constant.5*22,33I have abandoned the expectation of finding a suitable form for the vibration-rotation relaxation processes which will give rise to strict-Lindemann behavior,''^'^ and yet, clearly from Figure 1 (cf. also Table 4 of ref 3b), the di are not constant. Perhaps we should not be too hasty in discarding the concept of randomization-for instance, could it be that if the randomization were to be sufficiently fast, all reactive states could, in effect, appear to decay at the same rate, and thus we would achieve the I I I desired Lindemann shape? 2 3 4 5 The investigation of the effect of randomization is particularly Log p ( T o r r ) easy if the collisional relaxation is assumed to be e ~ p o n e n t i a l , ~ ~ i.e. Figure 10. Comparison of the shapes of falloff curves for the 527-state calculation, including only limited second-order randomization, with N Prjdj -7 I
experiment. From top to bottom, the values of p, are lo9, lo', and 0 torr-' s-l; p = 2.5 X lo6 torr-' s-' in all cases.
'
',
where p, d,, and /3, have their previous meanings, pr is the randomization rate for grain r, and P,. is the equilibrium population within grain r of thejth reactive state. In the present calculation, r runs over 721 grains from 59.0 to 95.0 kcal-mol-', and j runs up to N = 527 possible reactive states;37also, I will treat pr as being the same for all grains (although this is by no means necessary) and will examine two simple assumptions, that the randomization is either first order (intrinsic) or second order (collision induced). The results are shown in Figure 8, for various choices of pr with p / p kept constant. With the second-order assumption, the falloff shape changes gradually from the almost strong-collision shape of the 527-state calculation to the pure strong-collision shape as K~--*a. With the first-order assumption, the same transition (32) Vatsya, S. R.; Pritchard, H. 0. Chem. Phys. 1981, 63, 383. (33) Notice that strict-Lindemann behavior is a limiting form (ref 34 and 35): it is not permissible to have a falloff shape which has a sharper curvature than the Lindemann shape.. (34) Snider, N. J. Chem. Phys. 1982, 77, 789. (35) Vatsya, S. R.; Pritchard, H. 0. J. Chem. Phys. 1983, 78, 1624. (36) Vatsya, S. R.;Pritchard, H. 0. Can. J . Chem. 1984, 62, 2879. (37) The present calculation is not sensitive to the choice of grain width: doubling the width to 0.1 kcabmol-' only affects the data presented in Figures 8-10 in the fourth or fifth significant figures
occurs between the two limiting forms, but for intermediate values of pr, the falloff exhibits inflections when plotted in this manner. Clearly, no constant value of pr, either first or second order, could yield the observed Lindemann shape. The large increase in rate which occurs when randomization is included arises because the nonreactive states are allowed to participate in the reaction-they can be cycled into reactive states to replace those molecules that have decayed to products, without waiting for the next collision to occur.38 What if we exclude this possibility that the nonreactive states may be cycled into the reactive ones on this much faster time scale, between collisions? All that is necessary is to change the lower limit of the summations in eq 6 from j = 0 to j = 1 . Figure 9 shows the result for the assumption of first-order randomization, and Figure 10 for the (38) The magnitude of this increase in rate at the low-pressure limit is the reciprocal of what is usually called the weak-collision efficiency, p, (ref 39): if the fully randomized (strong-collision) curve is shifted so as to coincide at the low-pressure limit with the nonrandomized (weak-collision)curve, this shift defines j3,. We see that 8, is simply the fraction of the reactive states out of the total number above threshold: this is because k"",,O= Pstrong"
kun,,o= ~
fmm eq 4a
, i , r ~ ~ t Z %from
5
where E' denotes summation over only reactive grains, or states, respectively. (39) Troe, J. J. Chem. Phys. 1977, 66, 4745, 4758.
3976 The Journal of Physical Chemistry, Vol. 89, No. 19, 1985
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assumption of second-order randomization; in either case, as the m, the falloff shape tends to the desired strictvalue of wr Lindemann shape without changing the reaction rate of either end of the pressure range. We may summarize the results of this calculation as follows: (i) a state-to-state calculation, at an intermediate level of complexity, for the rate of the thermal dissociation of N20reproduces the infinite-pressure rate constant, k,, to within about 10%or 20%; (ii) on the assumption that the internal relaxation is exponential and has a rate equal to the hard-sphere collision rate40 of 2.9 X lo6 torr-I s-I, it reproduces the limiting low-pressure rate constant, ko, embarrassingly well; (iii) further, if it is assumed that only those states which are connected to the dissociation continuum may participate in the randomization process, the observed strict-Lindemann shape for the falloff can be achieved with an assumption of wur> l O I 3 s-' (first order) or pur> lo8 torr-' s-I (second order). We are thus left the following questions to answer: (1) Why are the unreactive states to be excluded from the randomization processes, when throughout the history of the theory of unimolecular reactions it has been tacitly assumed that all states were to be considered equally? It may still be true that, for large molecules, all states (including unreactive ones) must participate in the randomization process, and so it is important to know (either in terms of molecular complexity or of energy-level density) where the division lies between the small-molecule behavior described here and the large-molecule behavior which has hitherto been I should caution also that these conclusions universally concerning restricted randomization depend rather heavily upon the correctness of one set of shock tube dissociation measurements for N,O; other experimental results on small molecules are not nearly as definiti~e,~' and it would be reassuring to see more measurements of unimolecular reactions for small molecules in order to confirm that strict-Lindemann behavior is the norm.44 (40) For a collision diameter of about 3.85 and 3.45 A, respectively (ref 41), the collision rate at 2000 K for N 2 0with Ar is 2.9 X lo6 torr-' 8. With this value for p, the computed curves in Figures 7.9, and 10 would coincide with the experimental data in the region below lo4 torr and obscure some of the finer details; the value of p = 2.5 X lo6 torr-' s-l used in these calculations was chosen simply in order to retain the clarity of the diagrams in the lowpressure region-the choice of is arbitrary, anyway. (41) Hirschfelder, J. 0.; Curtiss, C. F.; Byrd, R. B. "Molecular Theory of Gases and Liquids"; Wiley: New York, 1954. (42) Certainly, there have been shown to exist real intramolecular relaxation processes in large molecules, taking place on a time scale between picoseconds and nanosecond^;^^ the rates that appear to be required in unimolecular reactions, on the other hand, would seem to be in the range of picoseconds to femtoseconds, or faster. (43) Gobeli, D. A,; Morgan, J. R.; St Pierre, R. J.; El-Sayed, M. A. J . Phys. Chem. 1984,88, 178. (44) Added June 1985: a study of the reaction SO2 M SO + 0 M has recently been published.45 For the case where M = N1.the results are well represented by a strict-Lindemann curve, although this was not the form chosen by the authors themselves; it would require a somewhat longer pressure range to distinguish convincingly between the two shapes.
+
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+
Pritchard (2) Is this randomization a first-order process, i.e. intrinsic or assisted by the black body radiation, or is it collision i n d ~ c e d ? ~ Recall that Mies and K r a ~ s s ~have ~ , ~ already ' established the formalism necessary for the treatment of this coupling, if it is intrinsic, and have provided some model calculations; in this system, we are concerned with the more complicated of their two cases, the one where the decaying states are sufficiently wide that they overlap e x t e n s i ~ e l y . ~ ~ (3) How do we account for the fact that the collisional relaxation rate p is approximately equal to the collision rate, which is roughly 3 orders of magnitude greater than the observed vibrational relaxation rate:9 2.5 X lo3torr-' s-', at this temperature? First, the depopulation of states above threshold is miniscule, even in the second-order region; there still remains a large preponderance of unreactive states (see Figure 6) at all energies. Thus, one does not have to contemplate the repopulation of depleted states directly from below threshold by large energy jumps, but only the collisional redistribution among states of about the same energy, which one would expect to occur at a high rate, commensurate with the collision rate. The answer to the second part of the question is also relatively straightforward. The rate of vibrational relaxation is closely related to the incubation time for the r e a c t i ~ n and , ~ ~falloff ~~~ curves like those shown in Figure 7 are for conditions where the observation time is significantly longer than the incubation time. For us to be able to make the assumption of pure exponential relaxation, all that we need is for the following condition to hold: if the steady distribution at times t >> qncis disturbed by the disappearance of some reactive molecules, the replenishment of that steady state can be considered as being exponential-the total relaxation does not need to be a pure exponential one (nor is it ever so in practice). No inconsistency arises because at 2000 K k,,,, = 4.0 torr-' s-l, much lower than the internal relaxation rate of 2.5 X lo3 torr-l s-l, so that the system can always replenish the perturbed steady state with considerable time to spare.
Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada; it also owes much to Andrew Yau, upon whose 1979 study2 this development is based. Registry No. N20, 10024-97-2; C 0 2 , 124-38-9. (45) Cobos, C. J.; Hippler, H.; Troe, J. J . Phys. Chem. 1985, 89, 1778. (46) Mies, F. H.; Krauss, M. J . Chem. Phys. 1966, 45, 4455. (47) Mies, F. H. J . Chem. Phys. 1969, 51, 787, 798. (48) The most reactive states have decay rate constants (cf. Table I) in the range of (5-10) X 10" s-I; in other words, their widths are about 3-5 cm-'. Since the grain width of 0.05 kcabmol-I is equivalent to about 17.5 cm-', Table I1 shows that even in the grains quite close to threshold there are likely to be frequent overlappings between reactive states. (49) Dove, J. E.; Nip, W. S.; Teitelbaum, H. Symp. ( I n t . ) Combust., [Proc.],15rh, 1974 1975, 903. (50) Pritchard, H. 0.;Vatsya, S. R. Chem. Phys. 1982, 72, 447.
