J . Phys. Chem. 1989, 93, 5189-5195 importance in the atmosphere. Nevertheless, the heterogeneous nature of reaction 4 on liquid water droplets, which may be of potential importance, needs to be investigated further. Acknowledgment. The research described in this paper was performed by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics
5789
and Space Administration. We are grateful to an anonymous reviewer for helpful criticism of the manuscript and to other members of JPL Chemical Kinetics and Photochemistry Group for useful discussions. Registry No. HzO, 7732-18-5; CIONO2, 14545-72-3; Nz05, 1010203-1; 03,10028-15-6; C0Cl2, 75-44-5.
Studies of a Model of Thermal Unimolecular Reactions with Slow Randomization Neil Snider Department of Chemistry, Queen's University, Kingston, Ontario, Canada K7L 3N6 (Received: November 28, 1988; In Final Form: March 7 , 1989)
A model of thermal unimolecular reactions was analyzed. The model allows not only for unimolecular reaction but also
for unimolecular interconversion of two groups of states of the reactive molecule. This interconversion may be viewed as a randomization process. A solution valid at long time was obtained for the kinetic equations that characterize the model. Said solution consists of a weighted sum of exponentials. One of these exponentials must decay much more rapidly than the other if the time evolution of the model is to be characterized meaningfully by a rate coefficient. A sufficient condition for relatively rapid time decay of one of the exponentials was shown to be that the threshold energy for randomization is lower than that for reaction by several kT. The rate coefficient as a function of collision frequency, w , was found, when plotted logarithmically to have an upward-concave segment under some conditions. The feature is pronounced if both the fraction of reactive states and the randomization frequency are sufficiently small. A rudimentary analysis of weak-colIision effects was carried out. It was found that the effect of weak collisions is to shift the falloff curve parallel to the log w axis in the direction of higher w and to make the upward concave feature somewhat less pronounced.
Introduction Interpretation of the kinetics of unimolecular reactions owes a great deal to RRKM A basic assumption of this theory is that the reactant molecule, after it has undergone an energizing collision, rapidly redistributes its newly acquired energy among those of its internal states that are accessible. By "rapidly" is meant at a rate that is large compared to the unimolecular reaction rate. It is certain that the process of intramolecular vibrational energy redistribution, also called IVR or randomization, is sometimes slow. The problem of how RRKM theory must be corrected in the event of slow randomization has received much attention. Approaches to this problem have been not only numerous but also variedS3Much of this work, both experimenta13s4and theor e t i ~ a l has , ~ dealt with delayed randomization brought about by deliberate selection of a particular subset of states of the reactive molecule A. Even thermal excitation of A may give rise to non-RRKM behavior, if there are groups of states of A that interconvert slowly and that are not all reactive. Such a situation gives rise to what has been called "intrinsic" non-RRKM behavior! In what follows, attention is focused on this sort of behavior. Theoretical interpretation of intrinsic non-RRKM behavior has proceeded on two levels. On one level dynamical models of reactive molecules have been largely by classical mechanics. Nonrandom lifetime distributions have sometimes been found, and these are indicative of non-RRKM behavior. Understanding
of the effects of intramolecular forces and of mass ratios of nuclei on the lifetime distribution has been gained from these computations. Progress in computing lifetime distributions quantum mechanically has been limited by technical difficulties.8-10 On another level, kinetic equations have been formulated and solved for models for which randomization may be S I O W . ' ~ ' ~ Such models have been relatively simple in order to facilitate the solving of the kinetic equations. The present article is devoted to the analysis of a model of this type. In particular, attention is paid to the time evolution of the model when reactive molecules are energized by collisions with inert molecules M at a given temperature. Also considered is the behavior of the rate coefficient K (if this coefficient is well-defined) as a function of collision frequency w . Quite early in the development of the theory of unimolecular reactions it was conjectured that slow randomization would result in a second plateau in the K vs w curve.ls It was later shown that, provided K is expressible in the generalized Lindemann form, the K vs w curve must be everywhere monotonically increasing and downward concave.I6 This result was later shown to hold whenever K is expressible as the smallest eigenvalue of the relaxation matrix for a standard set of kinetic e q ~ a t i 0 n s . I ~It was pointed out in ref 16 that inflections may occur in the graph of log K as a function of log w . A strong-collision model with slow randomization has been shown to give log K vs log w curves with upward-concave sections, Le., with pairs of inflection points. Such
( I ) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley-Interscience: New York, 1972. (2) Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973. (3) Oref, I.; Rabinovitch, B. S . Ace. Chem. Res. 1979,12, 166. (4) Meagher, J. F.; Chao, K.J.; Barker, J. R.; Rabinovitch, B. 9.J . Phys. Chem. 1974,78,2535. (5) Gelbart, W. M.; Rice, S . A,; Freed, K. F. J . Chem. Phys. 197b,52, 5718. (6) Bunker, D. L.; Hase, W. L. J . Chem. Phys. 1973,59, 4621. (7) See,for example: Bunker, D. L. J . Chem. Phys. 1964.40,1946. Hase, W. L.; Duchovic, R. J.; Swamy, K. N.; Wolf, R. J. J . Chem. Phys. 1984,80, 714.
(8) Nordholm, S.; Rice, S . A. J . Chem. Phys. 1975,62, 157. (9) Pritchard, H. 0. J . Phys. Chem. 1985,89, 3970. (10) Pritchard, H. 0. The Quantum Theory of Unimolecular Reactions; Cambridge University Press: Cambridge, 1984. (11) Wilson, D. J. J . Phys. Chem. 1960,64, 323. Brauner, J. W.; Wilson, D. J. J. Phys. Chem. 1963,67, 1134. (12) Pritchard, H. 0. Can. J . Chem. 1980,58, 2236. (13) Pritchard, H. 0.;Vatsya, S . R. Can. J . Chem. 1981,59, 2575. (14) Marcus, R. A,; Hase, W. L.; Swamy, K. N . J . Phys. Chem. 1984.88, 6717. (15) Rice, 0. K. 2. Phys. Chem. 1930,8 7 , 226. (16) Johnston, H. S.; White, J . R. J . Chem. Phys. 1954,22, 1969 (17) Vatsya, S. R.; Pritchard, H . 0. Mol. Phys. 1985,54, 203.
0022-3654/89/2093-5789$01.50/0
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
behavior has been found experimentally for methyl isocyanide isomerization under both strong-collisionl* and weak-colli~ion~~ conditions and for 1,l-difluorocyclobutane decompositionZounder strong-collision conditions. It has been suggested that in the latter case the double inflection may be due to experimental complications.21 The double inflection in the log K vs log w curves for methyl isocyanide isomerization is not so pronounced, but it has been taken as indicative of slow r a n d o m i z a t i ~ n . ~ ~ ~ ~ ~ ~ ’ It has been shown that strong-collision models with allowance made for slow randomization do give rise to log K vs log w curves with double inflection^.^^*^^ As yet there has been only one theoretical study of a model for which randomization may be slow and, in addition, collisions may be weak.22 In this work the interplay between slow randomization and weak collisions was not investigated in detail. The present article is devoted in part to an investigation of another such model, one based on physical assumptions quite different from those set forth in ref 22. Another question which has not been addressed theoretically is that of whether or not a rate coefficient is meaningful in cases for which randomization is slow. Given that the rate coefficient is the smallest eigenvalue of a relaxation matrix, that eigenvalue must be well separated in magnitude from the other eigenvalues of the matrix in order for the time evolution of the model to be describable in terms of a rate coefficient.’s2 Presumably, there are at least two small eigenvalues, one characteristic of reaction and the other characteristic of randomization, in cases for which randomization is slow. The present article is also concerned with elucidating the conditions under which these eigenvalues are well separated in magnitude. Description of the Model As in the Lindemann model for unimolecular reactions, A is assumed to become reactive when it has acquired a certain energy, here denoted eo. Energy is transferred to and from A through collisions with a nonreactive molecule M which is present in large excess. The states of A are either of type c or of type d. States of type c are unreactive at all energies. States of type d are reactive at energies in excess of eo. Interconversion between states of type c and states of type d is assumed to be unimolecular. States are further partitioned according to energy. Groups of states denoted nc and nd have energy in some small neighborhood of an energy
Snider which has been used in the interpretation of lifetime distributions obtained from dynamical calculation^.^,^^^^ If C-type states and d-type states are grouped only according to whether their energy is greater than or less than eo, it reduces to the model that was used to interpret deviations from RRK theory in large molecule decomposition reactions23 and, more recently, to interpret the observed pressure dependence of the rate of inversion of a ~ i r i d i n e . ~ ~ As was pointed out in the Introduction, the important new feature of this model is its allowance both for weak collisions and for slow randomization. The Rate Coefficient General Results. Solutions of eq 1 are weighted sums of exponentials of the form exp(XicMt). The X i s are eigenvalues of the matrix B - KCM-I where the matrix B has elements
for m = n
= - C k,,,, mfn
and similarly for Bnd,md. Matrix elements Bnc,mdare zero. The matrix K is also block diagonal. It consists of 2 X 2 blocks, one for each n such that tn is greater than e l . The nth block has the form
where
f i n , pn, and qn are bln
given by =
Kndc
+ Kncd + K n
P n = Kndc/(Kndc+ qn =
Kncd)
(2)
Kn/Pn
It should be noted that pn is the fraction of states in the nth energy interval which are of type d. The eigenvalues Xi are all negative. The unimolecular rate coefficient K is given by
.6,
The kinetic equations for this model have the following form: dcnc -= dt
E (knc,mccmc - kmc,ncCnc)CM+ KncdCnd - KndcCnc m#n
where c, and cA are concentration of molecules in groups of states nc and nd, cMis the concentration of M, the k’s are rate constants for collisionally induced transitions between designated groups of states, K~ and K , , ~ are ~ unimolecular rate constants for interconversion of C-type states and d-type states in the interval n, and K, is the unimolecular rate constant for reaction of molecules in interval n . It is to be understood that K, is zero for en less than to. The rate constants Kd, and Kndc are zero for t, below some threshold energy t l which is less than to. This model bears some similarity to models that have been treated previously. In the strong-collision limit with K, independent of n it reduces to the model for which it was demonstrated that log K vs log w curves have upward-concave portions.1°J3 In the strong-collision limit at a given n it becomes the two-state model (18) Schneider, F. W.; Rabinovitch, B. S. J . A m . Chem. SOC.1962, 84, 4215. (19) Wang, F. M.; Rabinovitch, B. S. J. Phys. Chem. 1974, 78, 863. (20) Conlin, R. T.; Frey, H . M. J. Chem. SOC.,Faraday Trans. I 1979, 75, 2556. (21) Pritchard, H. 0. J. Phys. Chem. 1986, 90, 3501. (22) Pritchard, H . 0. J . Phys. Chem. 1988, 92, 4333.
K
=
-XOCM
where & is the Xi that is smallest in magnitude. If randomization is slow, which in this model means slow interconversion between states of type c and states of type d, there is another Xi, call it A,, which is also relatively small in magnitude. The eigenvectors corresponding to Xo and XI, call them I u d ) and Iul’), differ only slightly from eigenvectors Iwo) and I w l ) of B, these being the eigenvectors that have zero eigenvalues Blwo) = B ~ w ] = ) 0 The components of these vectors are given by
where x , is the fractional population of the nth interval when the distribution of molecules among the groups of states is a Boltzmann distribution. The constant K is given by K = ( CPnxne) / ( I - EPnXne) n
n
Under conditions that are characteristic of most unimolecular reactions, only quite small components of I u d ) and I u l ’ ) differ significantly from those of Iwo) and l w l ) , respectively. This feature allows one to make important simplifications of the equations for ho and A , . (23) Gill, E. K.; Laidler, K . J . Proc. R. SOC.London 1959, A250, 121. (24) Borchart, D. B.; Bauer, S. H . J . Chem. Phys. 1986, 85, 4980.
The Journal of Physical Chemistry, Vol. 93, No, 15, 1989 5791
Model of Thermal Unimolecular Reactions The working equations for Xo and XI are here derived by a method closely akin to one that was used to determine Xo under conditions of rapid r a n d o m i z a t i ~ n . ~The ~ eigenvalue equations to be solved are
Xol~o')
(B - KCM-I)IUO)) = (B
- KcM-I)JuI') = X l ( ~ l ' )
(3)
Equation 11 is of the generalized Lindemann form. Hence, all of the conclusions of ref 16 regarding derivatives of K ( W ) apply. In particular, the first derivative is always positive, and the second derivative is always negative. For log K as a function of log w , the first derivative is always positive, but the second derivative may be of either sign. Further exploration of this point is in order. Logarithmic differentiation of eq 11 gives
It is convenient to normalize Iud) and Iul') such that they satisfy = 1
(WilUj')
(4)
where i is either 0 or 1. One then has from eq 3 and 4 the equation Xi = -(
~il~1~j')~M-l
(5)
Typically, the elements of K are nonzero only for n such that x, is extremely small. It follows that only the very small components of Iu;) and Iul') differ significantly from those of Iwo) and Iwl). Thus, one has, to a good approximation (WilUj')
=0
(6)
where ij is either 01 or IO. Let the orthogonal projector Q be defined by
Q = I - Iwo)(wol
and ( ), denotes an average over the distribution function p t ( p ) given by
- Iwi)(wil
where I is the unit matrix. Note that B and Q commute and that B(I - Q) is the null matrix. It follows that eq 3 can be written
(BQ - KcM-')Iu,')
= Xil~j')
(7)
Next let a be some number that is greater in magnitude than any of the eigenvalues of B. Adding a(I - Q)luj') to both sides of eq 7 , rearranging, and making use of eq 4-6, one obtains [a- X i
+ ( B - a)Q - KCi,$-I]IUj')
= alWi)
(8)
Equation 8 pertains just to i = 0 and 1. Equations 5 and 8 are equivalent to eq 5 and 7 of ref 25. Thus one may proceed exactly as in ref 25 to obtain for the eigenvalues Xo and A, the formula
+
-XicM = ( w ~ ~ K ~ / C~ C( MI - I ) - ~ K ~ / ~ I W ~ )
(9)
where the matrix C is given by
c = -,l/2(BQ)-IKl/2
(10)
For i = 0, eq 9 gives the rate coefficient K for the reaction. For i = 1, eq 9 gives the phenomenological randomization frequency. If X,and XI are not well separated in magnitude, two exponentials are required to describe the long-time evolution of the model. If these exponentials make approximately equal contributions, then the longtime evolution of the system cannot be characterized by a rate coefficient. The matrix BQ is negative definite. It follows from eq 10 that C is positive definite. Let the eigenvalues and the normalized eigenvectors of C be denoted p l / k * and It-!), respectively, where k* is a positive constant with the units of a second-order rate constant. If k* is properly chosen, then k*cM is u, the frequency of collisions between A and M. In terms of p / , Ir!), and w , eq 9 has the form I(wiI~~/~Ir/)1~
-XFM
where u is given by
+
p/
In the continuum limit I( wolK1/zlr/)lzis expressible in terms of pr(p), a distribution function in frequency space
I ( @lK1/zlr/)I2 -,K,P,(w) dp where K , 1s the high-pressure limiting rate coefficient. Thus, in the continuum limit the rate coefficient as given by eq 9 can be expressed in terms of pr
( 2 5 ) Singh, S.R.;Pritchard, H . 0.Chem. Phys. Lett. 1980, 73, 191.
Since u is restricted to values between 0 and 1, the right side of eq 12a is restricted to values between -1/4 and +1/4. Positive values imply that a portion of the log K vs log w curve is concave upward. Positive values come about if the mean square deviation of u from its mean is sufficiently large, a condition that may be met if pt is bimodal. Bimodality of pt implies bimodality of pr, but the area under the high-frequency peak in pt is progressively reduced as w decreases. A bimodal pr is plausible for the model under consideration. One peak might be expected to occur in the range of frequencies that characterize reaction at different energies. The other peak might be expected to occur in the range of frequencies that characterize randomization. The bimodality of pr for the model here considered is confirmed in subsequent sections. In the strong-collision limit the distribution function p r ( p ) is related to the distribution over lifetimes at a given energy. Call this distribution P ( e ; s ) . Standard formulas6,' give for K in the strong-collision limit K
= w l o m d cp e ( e ) l m d re-wrP(e;s)
(13a)
where pe is the Boltzmann distribution function. Substituting the identity
1 =
lme-(w+p)r
ds
w + p
into eq 11 and switching the order of integrations, one obtains K
= K=w&"i5r(s)e-wT d s
(13b)
where pr(s)is the Laplace transform of pr(p). Comparison of eq 13a and 13b shows that pr is, to within a factor K,, the Boltzmann average of the lifetime distribution function = ~ , - ~ l ~ ~ P ( c ; ~ de )pde)
(14)
It should be kept in mind that eq 14 pertains to the strong-collision limit. The weak-collision p , ( p ) has not such a straightforward interpretation. Strictly speaking, p is w times the ratio of a reaction frequency to an average frequency of the modes of vibrational relaxation. In the strong-collision limit all of these modes can be taken to have the same frequency,I0 and one can set that frequency equal to w . One may then identify p with K, provided that randomization is sufficiently rapid. In the weak-collision case p pertains to a group of intervals and is w times the ratio of an
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The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
average K, for this group to an average frequency of vibrational relaxation for the group. Here w is to be taken as an upper bound to the vibrational relaxation frequencies. Thus for weak collisions the y's are larger. Hence the effect of weak collisions is to shift pr to larger y and thereby to broaden this distribution. The distribution pr characterizes relative reactivities of intervals, or groups of intervals, in the limit of large w . For w small compared to y, relative reactivities are decreased by a factor y-' since it is the population of intervals by collisions, not the reaction itself, that is rate limiting. Intermediate cases are accounted for by the factor
Snider
If randomization is rapid, then q(t) is close to zero, both q ( e ) y(e) and &)(e) are very small compared to y(+)(e), and y,,-)(t) is very nearly equal to ii(c). Equation 19 then reduces to
w
w+cc
Hence one sees from eq 12c that p , ( y ) is the reactivity distribution which characterizes w's in the range of collision frequencies wherein K is falling off. The Strong-Collision Limit. It is knownI3 that a closed expression for K is obtainable for this model in the strong-collision limit. Nevertheless, a review of the strong-collision results is appropriate since the approach taken here differs considerably from that taken in ref 13. There are many ways to define the strong-collision limit, but all of them lead to the result that, in the subspace orthogonal to the null space of K, BQ is diagonal or very nearly so. In what follows, BQ in this subspace is assumed to be equal to -k*I. The just mentioned assumption reduces eq 9 to - h j C ~ = (Wjl(1 + KU-')-IKIWj)
(15)
Since K is block diagonal with none of its blocks larger than 2 X 2, one can readily diagonalize it. In terms of the spectral representation, eq 15 has the form
where y,(*) and (vi*)) are the eigenvalues and the normalized eigenvectors of the nth block of K . The eigenvalues are given by
where y, q,, and pn have been defined by eq 2. One sees from is much larger than c(,(-) if qn is close to zero or eq 17 that if q, is close to one. In the former case randomization is much more rapid than reaction; in the latter case reaction is much more rapid than randomization. One can derive algebraic equations for the squares of the matrix elements ( wilv,(*)) from the fact that the Iv,(*))'s span the 2 X 2 subspace for a given n. For i = 0 one obtains
+ I(wolv,(-))12
l(wolv,,(+))l2
=
(18a)
x,e
As was to be expected, eq 21 is the standard strong-collision formula's* for the rate coefficient. One can now determine pr(y) from the distribution functions pr(*)( e) given by pr(f)(t)
=
-&CM
=
W
q(e) ~ w
( €-1f i ( - ) ( e )
+ /A(+)(€)
1
%(e) p e ( t )
de
y(+)(t) - y q 6 )
-~
( ~ ' ( 6~ )(
- y(-)(t)
e )pe(e)
(22a)
Km
Explicitly, one has
where
e(*)&)
are the e's that solve the equations y(f)(e) = y
Equation 22b strongly suggests a bimodal distribution. As will be seen, calculations confirm this suggestion. The peaks of p,(k) may or may not be widely separated, and the areas under the peaks may or may not differ greatly. Even if pt is bimodal, the right side of eq 12a may not be positive if pt weights the frequencies in the regions of the two peaks unequally. At sufficiently low w most of the weighting is invariably in the region of the low-frequency peak. This will continue to be the case at large w, where pt approaches pr, if pr puts greater weight on frequencies in the region of the low-frequency peak. The equations for ~ W ~ I ~ , which , ( * ) are ) ~ analogous ~ to those for I( wolv,,(*))12 are the following: I(Wllvn(+))l2 j&,(+'l(
+ l(Wllvn(-))12 = [K(1 - p n ) + K-'pnlxne
Wl1Vi+))[2
+ &(-)I(
(23a)
w1lv,c-))12=
K-'gn[(K + 1I2(1 - Pn)(l - q n ) + q n l p n ~ n e (23b)
Solving eq 23 for I( w,lvi*))12, combining the results with eq 16 and 17, and passing to the continuum limit, one obtains = wK-'
[ ( K + 112(1 - P N l - 9 ) + q l w +
(18'~)
In eq 18 use has been made of eq 2 and of the fact that pn is the fraction of states of type d in the nth interval. Solving eq 18 for I( ~ ~ l v , , ( * ) ) 1substituting ~. in eq 16, making use of eq 17, and passing to the continuum limit, one obtains K
* q ( t ) MU(€)
y(+)(t)
-X,C,
PnPnqAne
=
(19)
where %(e) is y(e) q(e) p(t) and y(*)(e), q ( t ) , etc., are fin(*). qn, etc., for t = e,. Equation 19 gives the following asymptotic results:
wJ,-pe
dt
small
w
where the definition of K has been used to obtain eq 24c. One sees from eq 20b and 24c that Xo and XI become asymptotically equal as w goes to zero, if the limits of integration are the same in both equations. As mentioned above, Xo and XI must be well separated in magnitude if the time evolution of the system is to be describable by a rate coefficient. As just shown, this requirement is hardest to fulfill at low values of w . A sufficient condition for its fulfillment is the condition that eo exceed t l by
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5793
Model of Thermal Unimolecular Reactions
O,Ot
-Lo+
i
- 3.0
- 5.0
-3.0
0.0
l o g 10 w
- 5.0
0.0
loglo (0
Figure 1. Logarithmic K vs w curves for a Kassel model with slow randomization for different values of the fraction p of reactive states. For each curve the other model parameters are as follows: xo = 20.0, XI =
10.0, s = 6 , and A , = 0.001.
several kT. This condition implies that the threshold energy for @(e) is considerably less than the threshold energy for q(t). Kassel Model with Slow Randomization. Some numerical results will serve to make the foregoing ideas concrete. In order to obtain numerical results, explicit formulas for z(E),p(t), p,(t), and p(t) must be assumed. Here these formulas are assumed in the spirit of the Kassel model1!*
Figure 2. Logarithmic K vs w curves for a Kassel model with slow randomization for different values of the randomization frequency A , . For both curves the other model parameters are as follows: xo = 20.0, x1 = 10.0, s = 6 , and p = 0.1.
2000
I
IO00
Pr
Figure 3. Reaction frequency distribution function p , ( p ) for a Kassel model with slow randomization. Model parameters are xo = 20.0, x , = 10.0, s = 6 , A I = 0.001, and p = 0.5.
where Ao, A I , and s are parameters and r(s) is the gamma function. The parameter A. can be regarded as a scale factor. In what follows, p , w , and A I are taken to be expressed in units of Ao. It is further assumed that t l and to - t l are large compared to kT, in order to justify assumptions made in previous sections. The simplest assumption to make for p(t) is that it is independent oft p = (b
+ 1)-l
aooot
(28)
where b is some constant that is greater than I . Substitution of eq 25-28 into eq 19 and 20a, change of the variable of integration to (e - t o ) / k T and use of eq 17 give
50 2000 io00
I 0
I
o
5 x 10-6
P
where xo and y are given by XO
bw
- 1- - I + Y(X)
= eo/kT
0)
+ A , [ ( x+ x z ) / ( x + X0)IS1
~2 = ( t o - t l ) / k T (30) If w is much smaller than A I throughout the falloff region, eq 30
gives y = 1 and eq 29 gives a standard Kassel curve. If w is much larger than A I in some portion of the falloff region, eq 30 gives y equal to (1 + b)-I, and eq 29 gives a standard Kassel curve shifted to higher w by a factor (1 b). Curves of log K as a function of log w were generated from eq 29 and 30 for selected sets of parameters. The results are displayed in Figures 1 and 2. It can be seen that, for A I equal to a value
+
5x
I 62
P
Figure 4. Reaction frequency distribution functions for a Kassel model with slow randomization. Shown are curves for a strong-collision ({ = 0) case and weak-collision ({ = 0.2) case. All other parameters for the two cases are the same and are given by xo = 20.0, x1 = 10.0, s = 6 , A I = 0.001, andp = 0.1.
of w which is well into the falloff region and for p on the order of 0.1 or less, the curve has a distinct upward-concave section. Substitution of eq 25-28 into eq 17 and 22 gives an explicit expression for p I ( p ) . Since this expression is cumbersome and not very instructive, it is not displayed here. Graphs of p I ( p ) are shown in Figures 3-5. One sees that pI is bimodal with a high, narrow peak at low and a lower, broader peak at high p. These peaks become more widely separated with decreasing p , as is illustrated by Figures 3 and 4. As A , increases, the peaks also separate, as is illustrated by Figures 3 and 5 . An increase in A , also diminishes the area under the high-frequency peak. Wide
5794
The Journal of Physical Chemistry, Vol. 93, No. 15. 1989 15.01
Snider
n
0.01
,Ooot
-1.o+
J.”
0
5
z P Figure 5. Reaction frequency distribution function for a Kassel model with slow randomization. Model parameters are given by xo = 20.0, xI = 10.0, s = 6 , A , = 0.1, andp = 0.1. separation of the peaks and near equal areas under them are both required in order to give pr’s such that the right side of eq 12a is positive. Approximate Treatment of Weak-Collision Effects. Weakcollision effects on K(W)are often characterized by a collision efficiency factor Pc.26*27 This factor is defined such that the weak-collision rate coefficient is related to the strong-collision rate coefficient K, by K(W)
= Ks(PcW)
It is well-known26that Pc is not independent of w . The evidence, both experimental19and theoretical,26s28indicates that Pc decreases with increasing w . A simple way of accounting for weak-collision effects in eq 19 is to replace w by w times an attenuation factor a ( € ) . If a is a decreasing function of e, then 0, decreases with increasing w since p ( * ) increase with increasing e . That a should decrease with increasing e is physically plausible. It reflects the fact that weak collisions become progressively less effective in deactivating molecules as the energy of these molecules increases. The object of the present treatment is to get a rough idea of how weak collisions affect the upward-concave feature which the model’s log K vs log w curves exhibit for certain values of the parameters. Hence, all that is required is to have some a ( € )which decreases with e and which gives a rate of decrease of Pc with w on the order of what is observed. The a assumed here is of the form a ( € )= exp[-{(e - d / k T l
(31)
where {is a constant. It was found that a value of {equal to 0.2 gives Pc(w+m) equal to 0.3&(w-O). Such a decrease is in accord with results of model calculations for methyl isocyanide isomeriza t ion. 26 Replacement of w in eq 19 by aw with a given by eq 31 resulted in the lower curve in Figure 6. The strong-collision curve for the same values of the other parameters is the upper curve in Figure 6. The upward-concave portion, while still present in the lower curve, is somewhat less pronounced. A consistent treatment of weak-collision effects on p , ( p ) is carried out by substituting a[e(*)(p)]11for p in this function. This change has the effect of broadening the peaks, the broadening being greater on the high-frequency side of each peak. This nonuniform broadening is illustrated in Figure 4. It results in greater weighting of the frequencies between the peaks, which in turn decreases the mean-square deviation of u from its mean. Apparently, the somewhat less positive values of d2 log K/(d log w)’ (26) Tardy, D. C.; Rabinovitch, B. S. Chem. Rev. 1977, 77, 369. (27) Troe, J. J. Chem. Phys. 19717, 66, 4745. (28) Snider, N . J. Chem. Phys. 1982, 7 7 , 789.
0.0
-5.0 loglo w
Figure 6. Logarithmic K vs w curves for a Kassel model with slow randomization. Shown are curves for a strong-collision (< = 0) case and a weak-collision ({ = 0.2) case. All other parameters for the two cases are the same and are given by xo = 20.0, x I = 10.0, s = 6 , A , = 0.001, and p = 0.1.
which are found in the weak-collision case are related to this small amount of merging of the peaks in p , ( p ) .
Discussion The model treated in this article gives rise to two possible unimolecular rate processes per energy interval. The lifetime distribution for molecules at energy 6 can be expressed as a weighted sum of two exponentials exp(-p(+)r) and exp(-p(%). If the ratio p ( - ) / p ( + )is small, then it is possible to identify one of these p’s with the frequency of interconversion of molecules of types c and d, the randomization process for the model. The other p is then identifiable with the frequency of reaction. For some choices of the model’s parameters, curves of log K vs log w have upward-concave segments. Such a feature of log K vs log w curves has already been noted for a model13 that is formally akin to the model here considered. The upward-concave segment for a given curve occurs in a range of w that is neither close to the linear portion of the curve at low w nor close to the level portion at large w . One can see from eq 12c that, at low w , pt(p) weights a narrow range of low frequencies p. Thus the positive term on the right side of eq 12a does not dominate at low w . Neither does this term dominate at high w , where it decays as w - ~ . Only the term - ( p ) \ decays as w-l. In a proper treatment of weak-collision effects for this model one would use an appropriate nondiagonal C in eq 9 to obtain the rate coefficient. To find pt, one would first find pr by diagonalization of C for a quasicontinuous set of intervals and then one would use eq 12c to derive pt from pp A simpler, but still plausible, method of determining pr was adopted in the work reported here. This method gives weak-collision pr’s for which the peaks are broader than they are in the strong-collision limit. Said broadening is greater on the high-frequency side of each peak. Consequently, the peaks are not as well separated for weak-collision pr’s as they are for corresponding strong-collision limiting ,or’s. This effect of weak collisions on pr is almost certainly related to the somewhat smaller degree of upward concavity in the weak-collision log K vs log w curves. It appears from the analysis here given that weak collisions give rise to a different degree of broadening of pr at different p. This effect is illustrated in Figure 4. Weak collisions give rise to some broadening of pr at all values of p since a higher frequency of collisions is necessary to deactivate molecules at any energy if collisions are weak. As mentioned above, the p in pr(p) is to be understood as w times the ratio of an average reaction frequency divided by an average vibrational relaxation frequency. If for weak collisions this average relaxation frequency were the same for all modes, one would be justified in assuming a weak collision pr of the form p r k ) = Pcprs(Pcp)
(32)
where prs is pr in the strong-collision limit. Such an equation is
Model of Thermal Unimolecular Reactions not physically plausible. The higher energy intervals, which generally have higher reaction frequencies, are expected to have lower average vibrational relaxation frequencies since more energy must be removed from molecules in these intervals in order to deactivate them. Thus one expects weakening of the collisions to cause larger shifts in p,(p) at large p, Le., a greater broadening of this distribution at large p . Broadening of pr to the same extent at all frequencies simply causes a shift of the log K vs log w curve along the w axis. One verifies this assertion by substitution of eq 32 for pr into eq 1 1 to give
For methyl isocyanide isomerization in He the upward-concave portion of the observed log K vs log w curve appears simply shifted to higher w relative to its position in the corresponding curve for pure CH,NC.’8,’9 No difference in the degree of upward concavity is evident from the data, but the scatter in the data is sufficient to mask any small difference such as is predicted by the present study. I n the extreme case of p very small and q close to one, eq 19 reduces to
Consider p, q, and p for t values that dominate in the foregoing integrals. Suppose w satisfies the following inequalities relative to these quantities:
P(1
- q ) F
