J. Phys. Chem. 1992, 96, 172-179
172
Shapes of Unimolecular Fall-Off Curves H. 0. Pritchard* and S. R. Vatsya’ Centre for Research in Earth and Space Science, York University, Downsview, Ontario. Canada M3J 1 P3 (Received: March 29, 1991; In Final Form: August 22, 1991) By using a smoothly varying transition probability model, for which the master equation can be solved analytically for the eigenvalue ^lo (the rate constant) and the corresponding eigenvector +, some rigorous results concerning the shapes of fall-off curves for thermal unimolecular reactions can be proved. It is shown that, for a fixed set of populations (a,) and decay rates ( d J ,the shape of the fall-off curve is determined solely by the collisional relaxation pattern among the reactive states, and the conditions for a multiexponential unimolecular reaction system to yield the usual steady-state “sum of Lindemann forms” expression are demonstrated. Also, the eigenvector corresponding to the low-pressure limiting rate is derived in analytic form for this transition probability model.
Some time ago, we proposed a transition probability model for use in weak-collision calculations, which took the form of a sum of strong-collision transition probability mat rice^.^.^^ Originally, we were only able to provide an analytic solution for the rate constant for a two relaxation rate ~ y s t e m ,but ~ , ~more recently, we have derived analytic expressions for the rate constant for systems possessing any number of distinct collisional relaxation rates.5 The power of this approach is that it enables us to vary the relaxation patterns for the unreactive and the reactive states (or, for that matter, any subset of states) independently; this is not ordinarily the case, since if some transition rates are arbitrarily altered, all of the eigenvalues of the matrix will be changed. In what follows, we describe a simplified version of this model and its application to the thermal decomposition of an artificial molecule which mimics the behavior of nitrous oxide at 2000 K, in addition, the results found for pseudo-N20 necessitated a few calculations on other molecules (methyl isocyanide ana cyclopropane) in order to clarify the consequences of certain results. Multiexponential Relaxation Model The energy level spectrum of the molecule is divided into bands of increasing energy, with populations h, El, ..., fiN, Cfij = 1. States within each b a n d j are coupled with each other, and with all states above, at a rate A,, j = 0,1, ..., N, with the restriction that A . > A . I; this is probably not a debilitating restriction, since it is t Aoigk that for most molecules the rate of collisional relaxation increases with increasing energy. The formal properties of this model have been described extensively e l s e ~ h e r e > and -~~?~ only the results will be quoted here. In the way in which the model was originally conceived, bands above reaction threshold could contain molecules that were reactive and unreactive, could decay to different products, were of different vibrational symmetry or of different J, etc.; here, we drop this degree of sophistication and assume, as in most other unimolecular calculations, that molecules at a certain energy are either reactive (with decay rate constant dj) or unreactive. The analytic expression for the lower bound to the rate constant5 then simplifies considerably to
fi,dj Aj
N
b(0) =
bZ j=O
+ dj
of the range over which each Aj operates, to be used in calculating the contribution of each elementary Lindemann form to the total rate. As an aid to understanding the conclusions of this paper without necessarily following all of the mathematical details, we note the meanings of some important terms: A is any relaxation matrix which can be used in unimolecular reaction rate calculations, and M is the smoothly varying relaxation matrix which is similar to it (there is always a similarity transformation5 that will take A into M); Sois the eigenvector of M or A corresponding to the zero eigenvalue, elements fij1I2;x(0) is an upper bound to yo (=kUni) of (M + D), and b(0) IS a lower bound, both of which are usually very tight bounds indeed; D is a diagonal matrix of decay rate constants dj; llI/so112 = fij is the population in band or grain j ; IIJPol12is the population in b a n d j and all bands or grains above j ; llI$0112 is the total population in grains having nonzero dj and, in the present approximation, is equal to the total population above threshold; X,= 1.l0 is the vibrational relaxation rate of the molecule a t the temperature and pressure of the experiment, and Aj is the corresponding relaxation rate within band j ; pj is the increment in relaxation rate for band j , Le., pj = Aj The corresponding expression for the limiting low-pressure rate constant also simplifies to j*-1
bo(())= b ~ ~ ~ k-I n%-k+1(0) ~ O ~ ~ 2 / (2) where j * is the index of the first reactive band. As before, iji-k+l(0) is defined in the original paper,5 with the considerable simplification now that [IkD- Zk]is either zero (for reactive grains) or one (for unreactive grains)-thus, all jji-k+l(0) = 1 for k 1 j * . Consequently, (2) reduces to the usual formula ko =
bllbsol12
(3)
when j * = 1. It should be noted, however, that although X,and l Z$0112 appear explicitly in eq 2, the limiting rate constant is quite insensitive to either quantity, as is shown below for X,and was shown previously5 for III$ollz. These properties can readily be examined by expanding (2), for despite its apparent recursive complexity, it expands directly into a sum of prcducts of u,bjwhere
(1)
f r ( l - PN-k+lqN-k+l)
k=l
The reader is referred to the original paper5 for the definitions of the j i and q terms (where an algorithm for their calculation is also given), but they contain the weighting factors, in terms (1) Present address: Whiteshell Laboratories, Pinawa, Manitoba, Canada ROE ILO. (2) Vatsya, S.R.; Pritchard, H. 0. Chem. Phys. 1981, 63, 383-390. (3) Pritchard, H. 0. Quantum Theory of Unimolecular Reactions; Cambridge University Press: Cambridge, England, 1984. Chapters 1-9 and the Appendix in this book are denoted by suffixes a-i and A , respectively. (4) Pritchard, H. 0. J. Phys. Chem. 1986, 90,4471-4473. ( 5 ) Vatsya, S. R.; Pritchard, H. 0. Theor. Chim. Acra 1990, 77, 63-84. (6) Pritchard, H. 0. J. Phys. Chem. 1988, 92, 4333-4339.
For the case of j * = 5 , the one most extensively explored in this paper, 24 such terms are obtained upon expanding ijd0)iiN-l(0) jjN-2(0) ijN-3(0), which occurs in the denominator of (2). Computational Method A state-density function p(E) and a specific rate function k(E) were generated for the thermal dissociation of N 2 0 into N2 0 by standard methods;’ the original functions were tabulated at
+
(7) Pritchard, H. 0. J. Phys. Chem. 1985.89, 3970-3976.
QO22-3654/92/2096-172%03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 173
Unimolecular Fall-Off Curves 35-cm-I (0.1 kcal mol-I) intervals. These grains, below threshold, were assembled into bands 5-10 kcal mol-' wide and assigned relaxation rates A,,, AI,..., A,, ..., A,*-l, increasing with increasing j , and with A,, set equal to the observed vibrational relaxation rate. These bandwidths and relaxation rates A,, 1 Ij < j * , were then adjusted so as to cause the low-pressure limiting rate constant to coincide with the experimental value; while there is considerable latitude in choosing parameters to effect this identity, there are (as shown below) some significant constraints too. The relaxation matrix M was constructed according to the rules described previously:2 First, all off-diagonal elements M,J were filled with the elements Aoii,, representihg the strong-collision probability of a transition into state i from any other state. Then, to all off-diagonal elements with i, j L 1 were added the quantities (A, - &)E,/( 1 - ii,,), then for i, j 2 2, were added (A, - Al)fil/(l - ii,, - El), and so on, up to A,.-1. Call this Mo; notice that, for clarity, p, is written out here as (A, - A,-l). The matrix elements connecting states above threshold were then constructed in one of two ways, forming equivalent matrices5 A and M, respectively. For one, the reactive energy range was divided into suitable grains and relaxation was assumed to be governed by one of the standard transition probability models, exponential (exp), Gaussian, or stepladder, with ( AE) as a parameter and with normalization of the p r o b a b i l i t i e ~ . ~The ~~ eigenvalues A e x p of this system were calculated by standard numerical methods, and they were used to continue the construction of the matrix M from Mo by using A, = A,.-1 + A;?$+', on the reasonable assumption that Ayp>> A,.-'; = 0, of course, as required for the upper subsystem. At the same time, the elements corresponding to i, j 1 j * of a second matrix A were constructed by adding the exponential (Gaussian, stepladder) probabilities t o those already assembled in M,,; note that since the off-diagonal elements of M,, satisfy the constraint of detailed balancing, so too do those of M and A. The diagonal elements of M and A were then formed in the usual manner from the sum of the elements in the column,3band it was shown by numerical calculation that M and A shared the same set of eigenvalues to within the rounding errors of the calculations.I0 In what follows, the lower bound $(O) to the eigenvalue of (M D)was calculated by using the generalized Lindemann expression (l), and the limiting low-pressure rate constants were calculated by using eq 2 for i o ( 0 ) or by numerical calculation of the smallest eigenvalue of M truncated" between j * - 1 and
ep
+
j*.
Position of the Fail-Off The two features of interest in experiments on thermal unimolecular reactions are the shape of the fall-off curve, and its position on the pressure (p) axis: Throughout this discussion, we will assume that the fall-off curve is plotted as log k,,, vs log p . We will deal with the position first, as it is the simpler of the two questions. First, we consider the strict Lindemann expression kLindemann
--
k, = w'- %dl 1+k,/kg w+dI
(4)
where w is the deactivation rate. In this log-log form, it possesses some simple geometric properties: If the low-pressure limiting line is extended upward to higher pressure and the high-pressure limiting line is extended to lower pressure, they intersect a t the pressure known aspll2, i.e., the pressure at which k,,, is exactly half of the infinite-pressure at this point, the slope. of the (8) Tardy, D. C.; Rabinovitch, B. S.J . Chem. Phys. 1966,45,3720-3730. (9) Gilbert, R. G.; King, K. D. Chem. Phys. 1980, 49, 367-375. (10) Ca. when using Fortran Real.16 variables. An alternative procedure in forming A would be to substitute the exponential (Gaussian, stepladder) transition probability elements for those already present in &: The eigenvalues of this matrix are those of & and then a series of values slightly smaller than The simple additivity is lost. Also, for low (AE), some of the A TP fall below the assumed value of A,.-l, which is inconsistent with the assumption of the model. (11) Gilbert, R. G.; Ross, I. G. Ausf. J. Chem. 1971, 24, 1541-1565. (12) Oref, I.; Tardy, D. C. Chem. Rev. 1990, 90, 1407-1445.
ATP.
-18
1,
3
c1
:
'10
"
' 20
1
" 30 '
1
1 40
50
~
80
L i
f
8
1
r-h/P (a)
5 -
4
--
\
-
3 '
A,/p
AQ/P
Al/p
,
"
-
,
"
"
"
2000 1000 600 BOO 400
threshold
AS/P '
-
'
1, *
log-log plot is and its curvature is a maximum. Since fall-off curves, if they are not strict Lindemann, must lie below a strict Lindemann curve having the same pair of asymptotic rates,I3J4 the geometry is modified somewhat: p112 occurs at a higher pressure than the intersection of these two lines, but the maximum curvature and a slope of for the log-log plot both occur near this intersection point.'$ Therefore, we propose to use this point as a useful working definition of the position of the fall-off (assuming, of course, that both ko and k, are known for the reaction in question).l6 Thus,the position of the fall-off is determined by the condition that k,g = k , (assuming that ko has dimensions of pressure-' time-!). Since ko always has a smaller Arrhenius temperature coefficient than does k,? the higher the temperature, the higher the pressure at which this intersection occurs;i.e., the fall-off moves to higher pressure with increasing temperature (with the shift having an Arrhenius temperature coefficient of E , - Eo). The meaning of k , has been well understood since T ~ l m a n , ~ ' ~viz. '* k , = c,E,dj, and in the next section, we examine some of the properties of ko for the thermal reaction of pseudo-N20 a t 2000
K. Examination of the Low-Pressure Limiting Rate Constant ko The notion that in a ladder-climbing process the second-order rate constant for dissociation is determined cooperatively by all of the collisional relaxation steps is well established, not only for diatomic d i s s o c i a t i o r ~but ~ ~in~ unimolecular ~~ reactions too. For the case of the stepladder probability model, one can find explicit formulae not only for the low-pressure rate3h,2'-23but for the rate ~~
~~~~
(13) Snider, N. S.J . Chem. Phys. 1982, 77, 789-797. (14) Vatsya, S.R.; Pritchard, H.0. J. Chem. Phys. 1983,78, 1624-1625. (15) An alternative description of this intersection is that, for the strict Lindemann (one reactive state) case, it occurs when the deactivation rate w (dimensions s-l) equals the mean decay rate, for the usual sum of Lindemann forms expression (eq 20), when w equals the weighted average (in the Tolman sense) of the d,. (16) In the case of N20, these coincidences are virtually exact since the fall-off shape is approximately strict Lindemann. In the case of C3H6, which conforms to the usual extended Lindemann shape, maximum curvature occurs near 0.1 Torr, a slope of ' / 2 near 0.2 Torr, and k,,i = '/2k., near 5 Torr, whereas the intersection occurs at 0.62 Torr. Thus, characterization of the position by using this coincidence is only a qualitative one. (17) Tolman, R. C. J . Am. Chem. SOC.1920, 42, 2506-2528. (18) Kassel, L. S.Kinetics of Homogeneous Gas Reactions; The Chemical Catalog Co.: New York, 1932. (19) Pritchard, H.0.Spec. Period. Rep.: React. Kiner. 1975,1,243-290. (20) Yau, A. W.;Ritchard, H.0. J . Phys. Chem. 1979,83,134-149,896.
174 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
at any pressure.24 An important result is that the low-pressure rate is independent of the decay rates of the reactive At 2000 K, the limiting low-pressure rate constant for the thermal dissociation of N20 is about7 4.0 Torr-' s-I, and we now examine what kinds of relaxation patterns can sustain that rate, given the fact that the observed26vibrational relaxation rate constant A,,/p = 2.5 X lo3 Torr-' s-I. Two typical behaviors are shown in Figure 1, for a model with five relaxation zones below threshold, threshold being taken to be at 59 kcal mol-'. The relaxation patterns were as follows:
Pritchard and Vatsya is resolved by a set of ortho projections (Po, p,), p j = [(Jj- p j ) (Jj+i -Pj+l)],j = 0, 1, ..., N - 1, p N = J N - p p Eigenvalues Aj > 0 of M are given by
and Mpo = 0. Since dj = 0 for 0 5 j < j * and dj > 0 for j* 5 j I N, we have ID = Jim. The rate constant yo is the fixed point of x ( x ) given byZ5v2*
relaxation rate A,/p
i
energy band, (kcal mol-')
0
OIE
