J. Phys. Chem. 1986, 90, 6063
6063
COMMENTS We start with
Supercrlticai Carbon Dioxide. 2.' R* and the Dielectric Function for Supercritical CO, Media at Various Densities Sir: Equation 1, in which the terms alAl and a2A2represent dipolar and polarizability effects, was shown by Abboud et al. to correlate with the Kamlet-Taft A* parameter for a set (SSSG) consisting of the gas phase plus a select set (SSS) of nonprotic, nonaromatic, and nonhalogenated solvents.2 A* = A * ~ aIA1 ~ ~ a2A2 (1) Al = ( t - 1)/(t 2) - (n2 - l)/(n2 2) (la)
+
+
+
+
k(E) = u*(E - Eo)/hp(E)
(1)
the familiar2v3transition-state expression for a unimolecular rate constant, k(E). Here p ( E ) is the density of states of the unstable molecule at energy E, w*(E - E,) the sum over states of the transition state at energy E - E,, and h is Planck's constant. w*(E - E,) is obtained by summing over a density of states, including those of a velocity-weighted one-dimensional reaction coordinate, and hence is itself equal to a density of states of a system comprising the transition state plus a two-dimensional degree of freedom whose partition function is q = kT. We also make use of the steepest-descent r e l a t i ~ n : ~ , ~
o(E)= p(E)kT (2) (1b) where Tis the temperature at which a system of interest happens We have included supercritical C 0 2 media in a similar treatto have an average energy E(kT) equal to the particular E of ment, Since C 0 2 has no permanent dipole, t and n2 are e q ~ a l , ~ . ~ interest. Except for terms which are of the order N1,compared the alAl term drops out, and eq 1 reduces to with unity, where N is the number of degrees of freedom of the molecule, simple differentiation yields: A* = ?f*gas az(t - 1)/(2€ 1) (2) d In k/dE = (kP)-I - (kT)-' (3) If we use a set of media consisting of the experimental gas-phase5 A* of -1.05 and four values for different densities of supercritical Here P is the "temperature" of the transition state at energy E C02,' the parameters for the best fit to eq 2 are A * ~ , , = -1.03 - Eo,and T i s that of the molecule at energy E . f 0.04 and a2 = 6.86 f 0.3. The correlation coefficient is 0.996, Similarly, the change in rate constant due to a small change and the standard error of the estimate is 0.03. in the activation energy is given by We'have also compared the observed A* C02values with those predicted by eq 1 using the SSSG parameters. The parameters d In k/dE, N -(kP)-l (4) for the SSSG set are A * ~ , , = -1.04, al = 1.20 f 0.04, and a2 = From eq 3-4 one then obtains the felicitous result 5.46 f 0.27, with a standard error of the estimate for r* equal to 0.052. The observed ?r* values for the C 0 2 media are higher (5) (dE/dEo)k = T / ( T - P ) (less negative) than those predicted by from 2.8 to 3.8 standard error. The value of a2in eq 1, at 5.46 f 0.27, is also significantly Thus the incremental heat which must be supplied to overcome different from that in eq 2. The A* values for the fluorocarbons2 an increment in the potential barrier-the time scale remaining also deviate from the SSSG correlation line, but in the opposite constant-is given by a Carnot-Kelvin factor. direction from C 0 2 . It is parhaps not surprising to find a property of reversible steam engines emerging in the context of a quasi-equilibrium theory of Acknowledgment. We thank the National Science Foundation rate processes. We also observe that, when E >> Eoand hence for its support under Grant CHE-8217287. fl CY T, eq 3 becomes d In k/d(l/T) = -Eo/R (6) (1) Sigman, M. E.; Lindley, S. M.; Leffler, J. E. J. Am. Chem. SOC.1985,
A2 = (n2 - 1)/(2n2
+ 1)
+
+
.
107, 1471. ( 2 ) Abboud, J.-L. M.; Guiheneuf, G.; Essfar, M.; Taft, R. W. J . Phys. Chem. 1984,88, 4414. (3) Michels, A.; Kleerekoper, L. Physica (Amsterdam) 1939, 6, 586. (4) Keyes, F. G.; Kirkwood, J. G.Phys. Reu. 1930, 36, 754. (5) Essfar, M.; Guiheneuf, G.; Abboud, J.-L. M. J. Am. Chem. SOC.1982, 104. 6786.
Department of Chemistry Florida State University Tallahassee, Florida 32306-3006
Michael E. Sigman John E. Leffler*
In the large molecule limit we thus recover another familiar result. Equations 3 and 5 are apt to be especially useful in the context of "metastable" decompositions, where the parameter d In k/dE appears directly,6 and have already proven so in a discussion of evaporation from molecular clusters.' Acknowledgment. This research was sponsored by the Office of Health and Environmental Research, U.S. Department of Energy under Contract DE-AC05-840R-21400 with the Martin Marietta Energy Systems, Inc.
Received: June 6, 1986
On the Energy Dependence of Unimoiecuiar Rate Constants Sir: Explicit expressions for the rate constants of unimolecular reactions date from the pioneering work of Rice and Ramsberger.' They may be evaluated as a function of energy, and their implicit energy dependence exposed. Nevertheless there are occasions when it is useful to have the functional relationship rendered explicit. We present here a formulation of this dependence which should prove both useful and instructive. (1) Rice, 0. K.; Ramsberger, H. C. J . Am. Chem. SOC.1927, 49, 1617.
(1) Rice, 0. K.;Ramsberger, H. C. J . Am. Chem. SOC.1927, 49, 1617. (2) Rosenstock, H. M.;Wallenstein, M. B.; Wahrhaftig, A. L.; Eyring, H. Proc. Narl. Acad. Sei. U S A . 1952, 38, 667. (3) Marcus, R. A. J . Chem. Phys. 1952, 20, 359. (4) Hoare, M. R.; Ruijgrok, Th. W. J. Chem. Phys. 1970, 52, 113. (5) Hoare, M. R. J . Chem. Phys. 1970, 52, 5695. 1971, 55, 3058. (6) Klots, C. E. J. Chem. Phys. 1973,58, 5364. (7) Klots, C. E. Z.Phys. D , submitted for publication.
Chemical Physics Section Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 Received: July 30, 1986
Cornelius E. Klots
