J . Phys. Chem. 1986, 90, 3501-3502 distribution at u = 7, and (b) 300 K Boltzmann distribution. The results are shown in Figures 1-4. Equation 20, which is the solution of eq 15, yields the moments as a function of time (Figure 1). Note that, as expected, case a shows the system relaxing by losing energy to the heat bath, while in case b it gains energy from the heat bath. Knowledge of the moments as function of time then yields the population distribution n,(t) from eq 10 (Figure 2). It can be seen that both initial distributions are completely relaxed near wt = 100. The bulk average energy transfer ( ( A E ) ) ,calculated from eq 25 for k = 1, is shown in Figure 3. Observe that ( ( A E ) ) is negative for the u = 7 initial &function distribution since the overall direction of energy transfer in the bulk system is from the substrate to the heat bath, whereas ( ( A E ) ) is positive for the initial room temperature Boltzmann distribution, since in this case the overall direction of energy transfer in the bulk system is from the heat bath to the substrate. Finally, Figure 4 demonstrates the exponential relaxation of three linear combinations of the moments. The coefficients of the k-th linear combination ( k = 1, 2, 3) are given by the k-th row of matrix V-' (eq 32). Results in the anharmonic case were not sufficiently different to make them distinguishable on the scale of the graphs. Conclusions We have shown in a previous p~blication'~ how to transform the master equation in terms of bulk observables. This concept is further developed in the present paper, the principal theme of which is the analysis of the relaxation of moments in a gaseous system in contact with a heat bath. These moments represent
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actual or potential experimental (bulk) observables and are considered to be the primary information from which the time evolution of the population distribution can be obtained in analytical form if the nature of the collisional transition probability is known. It is demonstrated in this paper that the nature of the relaxation of moments depends on the form of a matrix A given by eq 17 and 18 in the continuous and discrete cases, respectively. The relaxation in the general case is not exponential; it is determined by the coupling of the moments due to the elements of the matrix A which are not in general zero. For any probability function q(x,y) or pij,the condition which is sufficient, but not necessary, for the relaxation to be exponential, is given by relation 30. If this conditions is not satisfied, then, since the condition is not necessary, exponential relaxation may nevertheless occur in cases where the transition probabilities q(x,y) or p i j have some special form, as in the SWA case. We also show that for any q ( x y ) or pijthere are, quite generally, always N linear combinations of the moments (eq 32) which will all relax exponentially (eq 35). One particular case examined are the discrete pij that satisfy the SWA conditions (eq 43). It is shown that the SWA system of equations is a special case of the more general system discussed here (eq 15). Acknowledgment. The authors are grateful for financial assistance received from the National Sciences and Engineering Council of Canada and the Province of Quebec Ministry of Education.
Inflections in Unlmolecular Falloff Curves H.0. Pritchard Centre for Research in Experimental Space Science, York University, Downsview, Ontario, Canada, M3J 1 P3 (Received: December 5, 1985; In Final Form: February 7, 1986)
Improvements to the consistency tests for thermal unimolecular reaction falloff curves, originally proved by Johnston and White, are noted. It is also shown that the Occurrence of an inflection in the plot of log kunivs. log p (where kuniis the rate constant at pressure p ) does not violate these consistency tests.
Introduction The idea of a false high-pressure limit, caused by a failure of communication between different domains in a reacting molecule, was first suggested by Rice in 1930: "... at high pressures, one would thus find a reaction of first order, then, a t somewhat lower pressures a transition region in which the rate constant decreases, thereupon a second reaction of first order, and, finally a further decrease. Were the molecule to be divided into more than two parts, this sequence could occur several times in succession."',* However, in 1954, Johnston and White derived a number of consistency tests for unimolecular falloff curves, among which they stated "... a plot of k vs. [MI has a positive slope and a negative curvature everywhere; there can be no inflection points... Similarly, (1) Rice, 0.K. 2.Phys. Chem. B 1930, 7, 226. (2) Rice actually cites and gives credit to K a s ~ e l but , ~ Kassel appears to have foresecn the (almost imperceptible) steplike behavior that occurs in diatomic dissociation;' Kassel's idea carried through into the 1950's in a unimolecular reaction context, as "induced decomposition", but this concept has now fallen from the vocabulary of thermal unimolecular reaction theory.s (3) Kassel, L. S. J. Am. Chem. Soc. 1928, 50, 1344. (4) Ashton, T.;McElwain, D. L. S.; Pritchard, H. 0. Can. J . Chem. 1973, 51, 237. (5) Shaw, D.H.; Pritchard, H. 0. Can. J . Chem. 1968, 46, 2721.
a plot of log k vs. log [MI has a positive slope, varying from one A Plot of 1% (k/[MI) vs. 1% [MI has everywhere a to zero negative varying from zero to One."' Recent Developments The derivation of Johnston and White is incomplete, in the sense that they assumed the unimolecular rate to be expressible as a sum of Lindemann shapes: this is only true for the case of pure exponential relaxatjon of the internal state distribution because then, there is no interference between the competing reaction channel^.'.^ However, the results of Johnston and White do, in fact, hold for a much more general form of the relaxation model. Let the master equation for a unimolecular reaction be cast in the form of an eigenvalue equation* (PA + B + D - yo)+o = 0 (1) where A is a matrix representing collision processes with rates proportional to the pressure p , B is a matrix representing first-order (6) Johnston, H.S.; White, J. R. J . Chem. Phys. 1954, 22, 1969. (7) Vatsya, S.R.; Pritchard, H. 0.Proc. R. Soc. London, Ser. A 1981,375, 409. (8) Pritchard, H.0. Quantum Theory of Unimolecular Reaction& Cambridge University Press: Cambridge, England, 1984.
0022-3654/86/2090-3501$01.50/00 1986 American Chemical Society
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Pritchard butane;I6 all of the isocyanide data have been replotted in order to reveal these apparent inflections in ref 8 and 17, and the 1,l -difluorocyclobutane data are replotted here, in Figure 1. Likewise, many theoretical attempts to treat the problem originally posed by Rice, that of restricted intramolecular vibrational relaxation in the decomposing molecule, lead to (logarithmic) falloff curves exhibiting very marked inflection^,^^'^.^^ but never a false high-pressure limit; inflections also appear when tunneling is assumed to be important.*O
I
I
I
I
-2
-I
0
J
Log p
(Torr)
Figure 1. Plot of the unimolecular falloff in the thermal decomposition of 1,l-difluorocyclobutane:crosses, data of Conlin and Frey;I6 solid line, expected strong collision falloff shape; lower dashed line, numerical fit to the experimental points; upper dashed line, d log k,,,/d log p for strong-collision curve; points, d log k,,,/d log p for fit to experimental data. Also shown in the lower panel are two hypothetical falloff curves for k , and k,, see text for explanation.
relaxation processes (e.g., radiative and/or surface activation processes), D is the diagonal matrix of state-specific decay rate constants for reaction, yo (= kuni)is the eigenvalue, and is the corresponding eigenvector. It can be shown, by an extension of the Hellmann-Feynman t h e ~ r e m , ~that , ' ~ d2yo/dp2 is strictly negative for all 0 5 p < m; ihis result was proved originally for B = 0 only,I0 but the same proof goes through for B f 0-all that we need is d2X/dp2 I0, where X = (PA + B + D). Inflections: Observed and Computed As a matter of expediency, kineticists studying thermal unimolecular reactions usually plot falloff curves in log kUnivs. log p form and, occasionally, these curves exhibit an inflection. This is not an inconsistency, since
and with 0 < dk/dp < I , d2k/dp2 < 0, everywhere, it can have either sign. The most obvious Occurrence of an inflection in a (logarithmic) falloff curve is the attainment of a low-pressure first-order region when the mean free path exceeds the vessel i.e. p A 0 in eq 1. Possible features also appear in falloff curves in systems where the activation and deactivation are thought to be purely collisional (Le,, B = 0): these include, perhaps ethyl isocyanide,'* certainly methyl isocyanide (either alone,13 pressurized by helium,I4 or d e ~ t e r a t e d ' ~ and ) , 1, l-difluorocyclo-
-
Numerical Tests All of the experimental data for homogeneous systems cited in the last paragraphl3-I6 (Le,, with the exception of ethyl isocyanide where the effect is quite marginal) were examined numerically, by fitting them in either numerical or polynomial form. In all cases, dyo/dp (and thus d log yo/d log p ) were positive and d(y,/p)/dp was negative2' throughout the experimental pressure range: thus, in no case can we rule out the existence of these features on the grounds that they are inconsistent with unimolecular reaction theory. In the case of CD3NC, the experimental data are too sparse,15and it is unclear whether or not an inflection occurs in the (logarithmic) falloff plot; for the other two methyl isocyanide experiments where there are plenty of data points with very little ~ c a t t e r , ~the ~ .numerical '~ differentiations reveal a very weak inflection, hardly any more than a stationary point, between 5 and 6 Torr for CHSNC, and between 40 and 50 Torr for (CH3NC + He). However, in the 1,l-difluorocyclobutane experiment,16 an inflection seems to Occur between about 0.04 and 0.2 Torr. It might be thought, superficially, that if there were really two different reactions taking place, such as two shown dotted in the lower panel of Figure 1, the (logarithmic) falloff curve for the total rate might exhibit an inflection. However, this is not so: if the two curves to be added are such that neither exhibits an inflection when plotted by itself in logarithmic form, then examination of eq 2 for k = kl + k , reveals that it will not show an inflection eitherSz2 Summary Four sets of experimental data, all displaying significant departures from the expected shape of the (logarithmic) falloff curve, have been examined numerically and shown not to be inconsistent with the tests originally proposed by Johnston and White, these tests having been put on a much sounder footing in this note. In the three isocyanide cases, the occurrence of these features has been tentatively associated with a failure in the randomization processes in the reacting molecules.8 However, 1,l -difluorocyclobutane is a much more complicated molecule, and one would not expect to find the reaction being impeded by intramolecular vibrational relaxation; also, attempts to formulate a reason for this behavior should be tempered by the realization that we have relatively few experimental points with considerable scatter derived from a difficult experiment in which the reaction of a fluorinated hydrocarbon is studied at high temperature in a glass vessel. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada. (16) Conlin, R. T.; Frey, H. M. J. Chem. SOC.,Faraday Trans. 1 1979, 75, 2556.
(17) Yau, A. W.; Pritchard, H. 0. Can.J . Chem. 1978, 56, 1389. (18) Thiele, E.; Goodman, M. E.; Stone, J. Chem. Phys. Leu. 1980, 69, 18.
(9) Silverman, J. N. In?. J . Quantum Chem. 1984, 25, 915. (10) Vatsya, S. R.; Pritchard, H. 0. Mol. Phys. 1985, 54, 203. (11) Kennedy, A. D.; Pritchard, H. 0. J. Phys. Chem. 1963, 67, 161. (12) Maloney, K. M.; Rabinovitch, B. S. J. Phys. Chem. 1969, 73, 1652. (13) Schneider, F. W.; Rabinovitch, B. S. J . Am. Chem. SOC.1962, 84, 4215. (14) Wang, F-M.; Rabinovitch, B. S. J. Phys. Chem. 1974, 78, 863. (15) Schneider, F. W.; Rabinovitch, B. S . J. Am. Chem. SOC.1963, 85, 2365.
(19) Pritchard, H. 0.;Vatsya, S.R. Can. J . Chem. 1982, 59, 2575. (20) Forst, W. J. Phys. Chem. 1983,87, 4489. (21) We have not yet shown that Johnston and White's conclusion about y o / p is rigorously true, but it is certainly approximately so, cf. footnote on p 207 of ref 10. (22) One might add that the plotting of experimental results in log-log form is often regarded as a subterfuge to obscure the scatter of the data, but in this case, it reveals a structure in the results which otherwise would be difficult to detect.
