I. OREFAND B. S. RABINOV~TCH
4488
The Experimental Evaluation of k m in Unirnolecular Reaction Systems1 by I. Oref and B. S. Rabinovitch Department of Chemistry, University of Washington, Seattle, Washinoton 98106
(Received June 17,1968)
Empirical methods described in the literature for the extrapolation of lower pressure data to IC, in thermal unimolecular reaction systems are examined. A systematic procedure is proposed wherein the plot of IC-’ us. p - “ L is extrapolated; LYL is a parameter to be found from the data. Illustration of the method with experimental data is provided.
Introduction There are several procedures by which the high-pressure limiting unimolecular rate constant k , may be evaluated from rate data which do not extend to sufficiently high pressure. The most common of these, the extrapolation of k-‘ us. p - l , is rooted in historical precedent2 and is not based on any currently accepted theoretical concepts. We propose here a systematic method by which k , may be estimated from lower pressure data; the characteristics of this method are discussed and are related to the (empirical) methods which are conventionally employed for this purpose. Two general approaches may be used for the evaluation of k,, one “theoretical” and the other practical and empirical. The extrapolation to k, by the “theoretical” method may be made by fitting some rate function, such as that of Slater (S),2* Rice, Ramsperger, and KassellZbMarcus and Ricelsor the quasi-theoretical form of Powell4 to the curvature of the observed rate constant-pressure plot in the region in question. Obviously, the difficulty here is the accuracy and reliability of the fit obtained, else the problem posed were solved; in fact, even when experimental data do range all the way to IC,, the experimental curvature, e.g., as expressed by Kassel’s s, can be determined only roughly from typical data, e.g., As = 1 or 2. When the data do not extend to the high-pressure limit, or only extend over a limited range, it can be very difficult to establish the appropriate magnitude of Kassel’s s or Slater’s n (and hence of the range of lc/k, involved) because the same range of slopes of the plot of log le us. log p , from unity to zero, is traversed by all systems, whatever s or n, and because of the insensitivity of the functions. Hence, for incomplete data that may extend only over a limited range, the error in evaluation of the curvature increases and the value so obtained for k , may be very inaccurate. I n practice, this procedure has been scarcely used; its characteristics follow from standard statistical analysis and this note will not deal further with these other than t,o give some indirect insight below into the errors that may be generated. The empirical procedures are very simple and easy to perform; they involve plotting the data in some form,
*
The Journal of Physical Chemistry
usually k-’ us. p-’ (Hinshelwood, H) or le-’ us. p-OJ (Schlag and Rabinovitch, SR).5 The H plot is especially inaccurate because for most systems the plot approaches the ordinate in a near-asymptotic manner and the intercept is extremely difficult to evaluate; this extrapolation tends always to give an underestimate of k,. The SR plot usually has less curvature than the H plot; it may give rise to an overestimate (usually) or underestimate of k,, depending on the curvature of the fall-off data, i.e., on the true value of n or s which characterizes the data. We wish now to examine the problems of extrapolation and the nature and relation between the various procedures. We suggest an extension of these methods by which k, can be determined in a more general way if the empirical method is chosen.
Rationale of Procedure Consider the plot, IC-’ us. p-“, where 0 < LY < 1. This graphical form when applied to a given set of experimental data will be found to be concave upward or downward depending on the value of CY used. However, for every set of data there is an aL which will We have roughly linearize the plot, lew1 vs. p - L . systematically evaluated LYLwith the use of the Slater quantity I&) to generate k/k,. The function I,(@ was chosen because of its simple form: for this function the fall-off “curvature” of the dependence of k on p depends only on a single parameter n, by contrast with RRK2 or RRKMs treatments. The Slater function does not necessarily give precisely the same shape as experimental data, but then neither does any theory, e.g., RRKM, even though more soundly based. (1) This work was supported by the National Science Foundation. (2). (a) N. B. Slater, “Theory of Unimolecular Reactions,” Cornel1 University Press, Ithaca, N. Y.,1959; (b) L. S. Kassel, “Kinetics of Homogeneous Gas Reactions,” Chemical Catalog Co., New York, N. Y.,1932. (3) R. A. Marcus and 0. K. Rice, J . Phys. Colloid Chem., 55, 894 (1951);R. A. Marcus, J . Chem. Phys., 20, 359 (1952). (4) R. E. Powell, ibid., 30, 724 (1959). (5) E. W.Schlag and B. 8. Rabinovitch, J . Amer. Chem. SOC.,82, 5996 (1960); B. S. Rabinovitch and K. W. Michel, ibid., 81, 5065
(1959).
EVALUATION OF k, 60
IN
UNIMOLECULAR REACTION SYSTEMS
4489
1
I
50-
-
40
-
09
-
0.8 0
2
4
6
8
IO
12
14
16
18
20
5
IO
15
n
22
e-axIO‘
Figure 3. Plot of the value of the extrapolated intercept of In(e)-l as a function of n, for various values of a. The range of k / k co used was 0.05-0.99.
Figure 1. Plot of the inverse function os. e-“, for the case n = 11 and for various values of a ; a~ = 0.65.
LO4 1.03
t
1.02 1.01
1.00 .99
.98 5
10
15
n Figure 4. Siiiiilar to Figure 3 but for a range of k / k o3 from 0.5 to 0.99.
n Figure 2. Variation of OIL with n for various ranges of k / k , 0.05-0.99; 0.05-0.5; * * * , 0.1-0.99.
-
.
:
Figure 1 presents sample plots of k,k-’ vs. e-a for the case of Ill(@). It is seen that the curvature of the plot varies with the value of a ; for a = 0.65, the plot is approximately a straight line. As shown in Figure 2, aL is a strong function of the value of n; but a~ is also a weak function of the range of k k - l . Figure 2 illustrates that LYLvaries weakly with the proximity to unity, and with the range of the values of I,@) from which it is deduced, for the sample cases I,(@) = 0.050.99, I,(e) = 0.5-0.99, and I,(0) = 0.05-0.5. For actual experimental data, n varies by at least one or two units over a wide range of falloff and consequently the best fit to the data should also vary on that account. This variability in a~ forebodes the error engendered (whether by the present method or by the “theoretical” fitting discussed first) by long extrapolation or by inaccurate or narrow range of data. It transpires that an experimental system of conventional accuracy must be close to the first-order region before any accurate conclusion regarding k, can be reached. With this limitation, a procedure may be proposed for the evaluation of k,: the quantity is found by extrapolation of the plot of k-1 va. P - Q L ; CYL may be evaluated by the
weighted linear regression of k-’ on p-”; a EaRter, and in many cases satisfactory, alternative would be evaluation of aL by graphical trial and error. Before illustrating the application of this procedure to experimental systems, it is desirable, first, to examine more generally the extrapolation of the Slater expression in order t o illustrate the behavior to be expected for “ideal” data. The inverse Slater function, treated as a set of experifor a range of n and a!. mental data, was plotted vs. @-“ The value of the intercept, ( k m k l ) , , was in each case found by fitting the Ic,k-’ vs. e- plot by least squares with a fourth-order and a tenth-order polynomial in 0-“. The choice of order was arbitrary, but the two series gave agreement and both give a good fit to I,(@. Figures 3 and 4 indicate the results of the extrapolation based on the fourth-order polynomial. They illustrate the kind of improvement in the evaluation of k, to be expected when the “data” emphasize a range closer to k, (Figure 4 relative to Figure 3) ; errors in the evaluation of a~ become unimportant, as is to be expected, Not shown is another set of intercept estimates evaluated as a function of n for the limited range 0.05-0.5 of k/k, : the errors are notably enhanced, especially for lower values ( aL,the plot of IC-’ us. p-a is concave downward; for Q < a ~the , plot is concave upward; these plots could, in one possible procedure, be made to yield identical interceptsa8 The possible advantage, if any, of this method over the first is simply the avoidance of the evaluation of aL. Trial application was made to the methyl isocyanide, nitrogen pentoxide, and cyclopropane experimental systems, with pw = 0. The data were fitted by a least-squares procedure to fourth-order polynomials with use of Grant’s reduction method9 to solve the normal equation and to obtain the coefficients of the polynomial. I n each case, three plausibly disparate values of Q were guessed; no effort was made to optimize the range of a. The large variation of the intercept with a found (Table 11),and which exceeds that suggested by Figures 3 and 4, reflects the influence of experimental error. Inasmuch as the choices for a are arbitrary, simple averaging of intercepts is here an unsatisfactory procedure. By way of contrast, the Slater curve for n = 3 was again represented by a fourth-order polynomial (this choice of n is close to the experimental value, n = 4-5, for CHJVC). Good agreement within a few per cent was found between the individual intercepts for various a, and also between the average intercept and the theoretical value. This emphasizes the fact that the (6) F. W. Schneider and E. S. Rabinovitch, J . Amer. Chem. SOC., 84, 4215 (1962).
(7) H. S. Johnston and J. R. White, J. Chem. Phys., 22, 1969 (1954). (8) L. F. Loucks and K. J. Laidler, Can. J . Chem., 45, 2795 (1967). (9) F. B. Holdebrand, “Introduction t o Numerical Analysis,” McGraw-Hill Book Co., Inc., New York, N. Y., 1956.
EVALUATION OF k,
IN
UNIMOLECULAR REACTION SYSTEMS
4491
Table I1 : Application of “Averaging Method” to Experimental Systems
System
CHsNC
C-CsHe
T,“C
No. of data points
of klk,
a
aL
199.4
50
0.99-0.012
0.60 0.80 0.99
0.77
230.4
99
0.99-0.0016
0.60 0.80 0.99
0.92
259.8
49
0.27-0.0018
0.60 0.80 0.99
0.89
500
43
0.96-0.08
0.40 0.50 0.60
0.58
0.43-0.008
0.60 0.80 0.99
0.71
OI29G
3.48G 7.85 12.0
0.13c
1-0.03
0.60 0.80 0.99
0.83
(1,OO)
1.10 1.04 0.947
1.03
NzOa
27
16b
Slater
...
15
Range
(n = 3)
b
Wreported), lod sec-1
7.506
92.56
7676
61.7l
Intercept, 10’ 8ec
6.2 10.6 17.1 0.92 1.31 2.56
k(av), sec-t
8.8
61°
0.116 0.162 0.513
375
1.45 1.58 1.71
63
Higher order polynomial or a value of pw > 0 improves the fit notably; the low value reflects the extreme range of k / k , being fitted. Smoothed values. 0 Units of sec-1 and sec.
fluctuations of the experimental data in real systems contribute heavily to errors and difficulty in the estimation of IC,. It has been as much our purpose in this note to clarify the problems of data extrapolation, and particularly to emphasize the doubtful conclusions that obtain from long extrapolation of even good data, as it has been
to suggest a particular procedure. We are not advocating the exclusive use of the empirical method. We conclude that evaluation of k, from lower pressure isothermal experimental data is a somewhat insecure bootstraps operation in which both “theoretical” and empirical procedures may be usefully employed to explore the vagaries of the data.
vohme 79,Number 18 December 1968
