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S. H. Bauer Cornell University Ithaca. New York 14853
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Molecular versus System Relaxation Times A careful definition leads to a clear distinction
All symbols which are introduced in deriving relationships between experimental and/or theoretical quantities should he defined precisely, hut one must he particularly careful in formulating definitions of probability functions. For emphasis, I shall first call attention to an apparent paradox which arises when one is not sufficiently precise in defining a rate constant in unimolecular reaction rate theory, and then show how the difficulty is resolved by the addition of a few well chosen words. For simplicity I have used in the following example the RRK formulation for s independent, hut weakly coupled, oscillators a t the classical limit. Recall the well known expressions ( 1 ) . It is understood that in
","
E+k) ,... will convert from M* to h l . , which then rnpidlg transform;i t o the product speri(,s. , the fractwn of mulecules u,hich react ner u n i t Thus f . ~ gives time. ~ h e r i c i ~ r o cofa lqintis then the mean lifetime'(r,J of molecules M* which incorporate energy E. This is a mean value since the accumulation of the threshold amount of energy in the reaction coordinate is a random process for which, [M*]t>@= [M*],=,,
. exp (-
2)
Equation (4) allows us to calculate the mean-life for any specified E and s. One could select the most probable value, given by that magnitude of E for which the f qintproduct has a maximum (since it is a hell-shaped curve), or the mean or the median. These differ only slightly from each other. For Emax = Eth + (S - 1)RT (3),
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The accompanying table lists values for the azirdine example introduced above. Mean Life-Times (at Em,,) for Aziridine, a t 350°K Assumed s ~ i ~ t ( E ~ d rml(Ernax) 3 4 5 10
4.69 X lo9 s-1 1.19 X lo9 3.86 X los 9.52 X lo6
2.13 X 10-'0s 8.39 X 10-lo 2.59 X 10W 1.05 X
While it is not surprising that q,,, and hence 7,) are both functions of s while is not, it is at first sight distuhing that r,! is many orders of magnitude smaller than rll2.Note that the disparity between these two lifetimes increases with increasing Eth/RT. Of course, these represent different types of lifetimes, hut where in the derivation did this exceptionally large factor creep in? The answer is that the definition given ahove (and in most textbooks) for qint(E;Eth;s) is incomplete. I t was correctly given by N. B. Slater (41, who stated "qint(E)At is the prohability that M* (with internal energy E ) convert to Mf within a short time interual At." This definition compells us to inquire: How short is "short"? The answer is that it be short compared to the mean time between collisions, given by
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{vd.[X]1-', since it is meaningless to discuss the probability for the transition M* M* after M* had been deexcited by collision with X. We now see why the high pressure limit r 1 / 2 is so much longer than r,~. When Ivintl-' >> {rld,[X])-' many M*'s are generated, but their sojourn in that state is cut short hv -" collisions and these do not react. Onlv a tinv fraction which transforms prior to colliding with X i s counted toward r l , ~ , which clearlv measures the svstem relaxation: that orobabilitv is weightedUbythe inverseuf the f(E > ~ i h ; ~ ;function. sj [Note that the f function is normalized over the interval 0 5 E < -,not over Eth 5 E < m.] In contrast, at the low pressure limit. the 'mean lifetimes aiven bv T,I do measure the conversibn probability, but thin the net obserued rate is limited by v..[X], unless excitation is generated by absorption of a short pulse of radiation (rather than molecular collisions). I t is now clear why the system relaxation should he measured ~~~~~~
~
~
~
by r l n , hut the "fall-off" condition by T,]. The "fall off" regime in unimolecular kinetics provides an observable transition from a system relaxation time to a molecular relaxation time. Professor Hammes pointed out that the above example is a member of a large class of cases wherein the system relaxation time is long because it is determined by a low population of the reagents (or precursors), even though the specific reaction rate constants are very large. Literature Cited (11 Hammea. G. G.."Prineiples of Chemical Kinetiu: Academic Press. New York, 1978: Chapter 4. Wwton, R. E., and Schwsrtz, H. A.,"Chemieal Kinetics," Prentice-Hall, Englewmd C1iffs.N. J.. 1972. (21 Caner, R. E., Drakenberg. T., and Bergman, N-A,. J. Am. Chom. Sor., 98. 69W ,,-c> ,,=,*,. I31 Benson, S. W., "Foundations of Chemical Kinetics," McGraw-Hill, 1960, p. 230-232. 141 Slater, N. B.,"Thaoryof Unimoleeular Reaetions,"CornellPresb, 1959.
Volume 56. Number 6, June 1979 1 383
