J . Phys. Chem. 1989, 93, 7031-7036 laxation period. In particular, during this short period the motion may be torsional (librations) or inertial (dipolarons), motions which then might be detected in longitudinal but probably not in transverse relaxation times.
7031
Acknowledgment. We are grateful to Dr. Paul Madden for interesting discussions on these and other topics related to dielectric relaxation. We would like to thank the National Science Foundation for supporting this work.
Radical Recombination Rate Constants from Gas to Liquid Phaset Anthony G. Zawadzki* AT& T Bell Laboratories, Holmdel, New Jersey 07733
and James T. Hynes* Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-021 5 (Received: February 14, 1989; In Final Form: July 17, 1989)
A simple connection formula is successfully used to predict radical recombination rate constants, k,, from gas- to liquid-phase densities. The formula ingredients are the low-pressure termolecular rate constant, kIo,which in general has radical complex (RC) and energy-transfer (ET) contributions, the transition state theory rate constant, kfST, and the rate constant for diffusion-controlled recombination, k:. Application to iodine recombination accounts well for experimental observations of k, variation with buffer gas density (and complexity) and temperature and emphasizes the importance of the RC mechanism. Application to methyl radical recombination highlights the importance of internal degrees of freedom coupled to the reaction coordinate in accounting for broad plateau behavior in experimental rates.
1. Introduction
There has been considerable recent interest in the theoretical prediction and experimental observation of unimolecular reaction rate constants over wide ranges of solvent or buffer gas One object of such studies is to account for the “tumover” behavior in plots of the rate constant with increasing density. Since unimolecular processes are governed by essentially different dynamical phenomena in different density regimes, a general theory of unimolecular reaction rates from gas to liquid densities has been difficult to construct. Radical recombination reactions provide an important arena for such studies.2c One approach to the prediction of unimolecular rates, exploited by Hynes and c o - ~ o r k e r s , ’has ~ ~ *focused ~ on identifying simple asymptotically valid forms of the rate constant k and using the connection formula Here klo,kinl,and khi refer to asymptotic rate constant formulas at low, intermediate, and high solvent densities. (Similar formulas have been used e l s e ~ h e r e L ~to-interpret ~ * ~ ~ experimental data for unimolecular processes and atom and radical recombination reactions.) Each formula (klo,kinl,and khi)applies over a restricted density range, and proper use of the connection formula depends on defining these rate constants so that their ranges of applicability overlap significantly. We have recently used this approach to successfully predict several isomerization reaction rates over the complete density range in which turnover is ~ b s e r v e d .In ~ this paper, we apply the technique to molecular recombination reactions. Our goals are to examine the effectiveness of simple statistical theories of low-pressure rate constants and to qualitatively illustrate the “signature” of different fundamental physical processes in the shape of the rate constant versus solvent density (k-p) curve. Accordingly, we adopt simple models and do not aim for completely quantitative agreement with experiment, although reasonable success is in fact achieved. While our statistical approach makes use of frequency-dependent (non-Markovian) solvent ‘Portions of this work have been presented at the Symposium on Chemical Kinetics of Radical-Radical Reactions, 188th Annual Meeting of the American Chemical Society, August 1984.
friction,’ we apply it in the context of zero-frequency friction for comparison to other methods. Our reactive mode potential is a simple one-dimensional Morse function, thus neglecting rotational contributions to energy transfer. Our formulations of energytransfer contributions to the rate involves a generalization of an early important contribution of Zwanzig.s We consider two systems as prototypes for recombination reactions. We first treat iodine atom recombination, focusing on the relative roles of energy-transfer and radical-solvent complex formation at low densities and on the interplay between these processes and diffusion control in the intermediate- to high-density regime. We then consider methyl radical recombination, focusing on the role of nonreactive degrees of freedom in qualitatively changing the shape of the k-p curve. It should be stressed that the general topic of this paper is an old one, with many earlier (and continuing) contributions by several groups, including for example those of Troe2,6e32*45 and B u ~ ~ s . ~However, ~ , ~ some ~ ~ *more ~ ~recent , ~ efforts ~ for radical recombination have focused on statistical theories in the vein of Kramers’ theoryg and generalizations thereof to describe the overall density behavior of the rate. The particular contributions of the present paper are to account for, within such a simple statistical (1) For recent reviews, see: (a) Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; p 171. (b) Hynes, J. T. Annu. Rev. Phys. Chem. 1985, 36, 573. (c) Fleming, G. R.; Courtney, S. H.; Balk, M. W. J . Stat. Phys. 1986, 42, 83. (2) (a) Troe, J. In Physical Chemistry: An Advanced Treatise; Eyring, H., Jost, W., Henderson, D., Eds.; Academic Press: New York, 1975; Vol. 6B, p 835. (b) Troe, J. J . Phys. Chem. 1986,90, 357, and references therein. (c) Troe, J. Annu. Rev. Phys. Chem. 1985, 38, 163, and references therein. (3) (a) Borkovec, M.; Berne, B. J. J. Phys. Chem. 1985, 89, 3994. (b) Berne, B. J.; Borkovec, M.; Straub, J. B. J. Phys. Chem. 1988,92, 3711, and references therein. (c) Straub, J. B.; Borkovec, M.; Berne, B. J. J . Chem. Phys. 1988, 89, 4833. (4) (a) Northrup, S. H.; Hynes, J. T. J. Chem. Phys. 1980,73,2700. (b) Grote, R. F.; Hynes, J. T. J. Chem. Phys. 1980, 73, 2715. (5) (a) Zawadzki, A. G.; Hynes, J. T. Chem. Phys. Lett. 1985,113,476. (b) Zawadzki, A. G.; Hynes, J. T. Unpublished results. (c) Hynes, J. T. J . Stat. Phys. 1986, 1,2, 149. (6) Otto, B.; Schroeder, J.; Troe, J. J. Chem. Phys. 1984, 81, 202. (7) Grote, R. F.; van der Zwan, G.; Hynes, J. T. J . Phys. Chem. 1984,88,
4676. (8) Zwanzig, R. W. Phys. Fluids 1959, 2, 12. (9) Kramers, H. A. Physira (Amsterdam) 1940, 7, 284.
0022-3654/89/2093-703 1$01.50/0 0 1989 American Chemical Society
7032 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
framework, important realistic aspects of radical recombination processes, e.g., radical complex formation and the involvement of internal degrees of freedom, and to show that reasonable agreement with experiment for prototype examples of radical recombination is thereby achieved. The outline of this paper is as follows. Section 2 describes the connection formula and its components. Sections 3 and 4 present our models and calculations for iodine and methyl radical recombination, while section 5 concludes.
Zawadzki and Hynes depends on formation of a vibrationally excited molecule through direct collision of two radicals. The rate-limiting step is the subsequent stabilization of this species through collisions with solvent molecules. We now outline our approaches to computing the component rate constants of eq 3. 2.1. k,": RC Mechanism. The R C mechanism can be most simply written as follows: k+
R + M Y R M
2. Theory
Radical recombination may be schematically represented by R
k, + R (+M) 7 R2 (+M)
(2)
where k , and kd are overall bimolecular recombination and unimolecular dissociation rate constants. The connection formula for recombination is
k;' = (k,Op)-'
+ (kTST)-' + (kF)-'
RM
+ R-R2kl
(Sa)
+M
Although there has been speculation as to the role of multiple solvent complexes at intermediate densities,I5 we adopt the simple RC model above for the asymptotic low-pressure regime. We use the simple and successful theory of Bunker and Davidson (BD)I2 to compute k,O
(3)
where kr0 is the low-pressure termolecular rate constant, p the solvent density, kfs' the classical transition state theory (TST) rate constant,I0 and k: the rate constant for diffusion-controlled recombination." Equation 3 illustrates the familiar "rate limiting step" concept-that is, k, is essential1 determined by the smallest of the components. While k,O and k3;"' are independent of p , k: declines with increasing p. Each of these rate constants determines the shape of a particular portion of the k,-p curve, and each depends differently on the nature of reactant and solvent. We now briefly describe the effects of each term of eq 3 on the overall shape of the k,-p curve. At gas-phase densities, the first term of eq 3 is smallest and determines the overall rate. This leads to a linear rise in the observed k , with p , with slope k,O. With increasing density, k begins to approach either the (density-independent) value k$ or the (declining) value k:. In this intermediate-density regime, typical of highly compressed gases or of liquids, the k,-p curve flattens. Finally, at high densities, the declining term k: dominates, producing a decline in k, and yielding the "turnover" in the k,-p curve. Depending on the relative sizes of k,", k f S Tand , k;, two general types of k,-p curve can be observed. (See, e.g., ref 2.) If k,' is small and k F T relatively large, k, may not reach k Y T before it begins to track kp. The k,-p curve then shows a gradual rise, a relatively sharp turnover at high solvent densities, and then a decline. In this case, eq 3 can be approximated as (4)
A second type of curve is typical when k,O is large. Here the initial rise in k, with p is rapid. Well before k: can become dominant, k , reaches k y T and ceases to rise with increasing p. At high densities, of course, k , ultimately tracks k: and declines. The overall k,-p curve in this case shows a steep initial increase, a broad plateau which may start in the gas phase and extend into the liquid phase, and a final decline. The primary determinant of the shape of the k,-p curve is thus k , O , the low-pressure termolecular rate constant. Two major mechanisms are believed to contribute to k,". The radical complex (RC) mechanism, applicable primarily to atomic recombination, is based on initial formation of a weakly bound radical-solvent c o m p l e ~ . ~This ~ - ~complex ~ is then decomposed in a rate-limiting step by a second radical. The second major contributor to k , O is the energy-transfer (ET) mechanism.I4 The ET mechanism (10) For a review, see: Truhlar, D. G.;Hase, W. L.; Hynes, J . T. J . Phys. Chem. 1983,87, 2664. (1 1) (a) Smoluchowski, M. Z.Phys. Chem. 1918,92, 129. (b) Calef, D. F.; Deutch, J. M. Annu. Rev. Phys. Chem. 1983, 34, 493. (c) Northrup, S. H.; Hynes, J. T. J . Chem. Phys. 1979, 71, 871. (12) Bunker, D. L.; Davidson, N . J . A m . Chem. SOC.1958, 80, 5090. (13) (a) Ip, J . K. K.; Burns, G. J. Chem. Phys. 1969, 51, 3414. (b) Blake, J. A,; Burns, G . J . Chem. Phys. 1971, 54, 1480.
where K& = k + / k - is the equilibrium constant for complex is an formation, Z2is a solvent-complex collision rate, and arbitrary steric factor for reaction 5b. Following BD, we assume E: = ' 1 2 . was determined by BD using classical statistical mechanics. For a Lennard-Jones (LJ) interaction between radical and solvent, they give the approximate result
e:
where uRM and tRMare the relevant LJ parameters and appropriate normalization by NA, Avogadro's constant, is understood. L J parameters for R M complexes are determined by using the combining rules URM = (uR 4- aM)/2 and ~ R M= ( c R c M ) ' / ~ . We calculate Z2 using the Lennard-Jones collision rate:
Our treatment differs from that of BD in that we use temperature-dependent effective collision cross sections,I6 SRM-M2 = UR"n(2'2''(kT/~RM-M), where Q(2,2)' is a reduced gas kinetic collision integral," while BD use a fixed cross section. The net effect is that our collision rate is only weakly temperature dependent (0:F','),while the original BD collision rate is 0:70,5. All reduced collision integrals required for gas kinetic calculations in this study are computed by using the empirical formulas of Neufeld et a1.I8 Note that, for a given reactant, increases in solvent complexity generally lead to higher values for LJ parameters (and therefore for k,") and, as we shall see, to somewhat lower turnover densities. 2.2. k,': ET Mechanism. We focus on the key step of stabilization of a nascent "hot" diradical, which is often treated as occurring in a single (or few) c ~ l l i s i o n ( s ) . ~This ~ is an oversimplification, however, and leads to an incorrect temperature dependence for kro.20 A more general assumption is that energy (14) (a) Rabinowitch, E. Trans. Faraday SOC.1937, 33, 283. (b) Rice, 0. K. J. Chem. Phys. 1941,9,258. (c) Bunker, D. L. J . Chem. Phys. 1960, 32, 1001. (d) Benson, S. W.; Fueno, T. J . Chem. Phys. 1962, 36, 1597. (15) (a) Stace, A. J.; Murrell, J. N. Mol. Phys. 1977,33, 1 . (b) Burns, G.; Wong, W. H. J . A m . Chem. SOC.1975,97, 710. (16) Chan, S.C.; Rabinovitch, B. S.;Bryant, J. T.; Spicer, L. D.; Fujimoto, T.; Lin, Y. N.; Pavion, S.P. J . Phys. Chem. 1970, 74, 3160. Curtiss, C. F.; Bird, R. B. Molecular Theory of (17) Hirschfelder, J. 0.; Gases and Liquids; Wiley: New York, 1954. (18) Neufeld, P. D.; Janzen, A. R.; Aziz, R. A . J . Chem. Phys. 1972, 57, 1100. (19) For discussions of this 'strong collision" hypothesis, see: Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley-Interscience: London, 1972. Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973. Troe, J . J . Chem. Phys. 1977, 66, 4745, 4758.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 7033
Radical Recombination Rate Constants transfer between the diradical and solvent molecules occurs in a stepwise fashion. When individual energy transfers are thermally small, vibrational energy diffusion (VED) theory can be applied.8*21-24To treat radical recombination with VED theory, we first compute the bimolecular VED rate constant for molecular dissociation, kd". This is converted to k," by k," = P e l e c e k d o , where Pclw is an electronic degeneracy factor and is the equilibrium constant for eq 2. The low-pressure VED dissociation rate constant is given by22-24
kd"p =
[ x E d d E[D(E) exp(-PE)I-'
x E d E ' p(E? exp(-PE?]-l
(9)
Here Ed is the dissociation energy of the reactive mode, P = (kT)-l, p ( E ) is the molecular density of states at energy E , and D ( E ) is the VED coefficient at energy E . D(E) contains the explicit reactant-solvent dynamics and is defined by22923
sY:(t) = (2/*)1/2(L,e/7c01J exp(-t2/27,1A
(12)
D(E) for a single Morse oscillator (s = 1) linearly coupled to a solvent with the above Gaussian friction has been derived by Grote and Hynes as23 Dl(E) = (4fcollEd/~wO)W(e)Sl(e,A) Sl(e,A) =
[(l N>- 1
(13a)
+ W ( E ) ) - ~ Cexp(-pA2W(t)Z) ]~ (13b)
where wo is the fundamental Morse oscillator frequency, t = E / & is the ratio of energy above the Morse well minimum to Ed, W(e) = (1 is the ratio w ( e ) / ~ and ~ , A ~ ,~the adiabaticity parameter for solvent friction, is defined as A = 2-'/2W07,11. For the simple model of (s - 1) identical harmonic modes of frequency wo coupled to a reactive mode of the above form, D ( E ) may be derived as follows. For moderate to large values of s, p(E) and N ( E ) may be approximated a ~ ~ ~ 9 ~ ~
1 aAEyt) D ( E ) = lim -p(E)r-or 2 at
(e)[ = (14b)
N ( E ) = s E0 p ( E ' )dE' = s - 1.5 which involves the rate of mean-square energy transfer between reactant and solvent on time scales long compared to the solvent force correlation time. Here the summations are over the participating reactant degrees of freedom (dofs), including the reactive mode itself, vi is the ith molecular mode generalized velocity, and F,is the force or torque on that mode. For a reactant with s weakly coupled but rapidly equilibrated intramolecular dofs, in line with the RRKM assumption,'*2p22 we have shown that5
(!)p(E)
s!w0s
where &,a&?) is the density of states for s coupled harmonic oscillators of frequency a,,.Since n(t)for harmonic modes is cos the friction-velocity T C F integrals (eq 1 l a ) for these exp(-AZ) to D ( E ) . modes contribute s - 1 terms of the form The friction-velocity TCF integral for the reactive mode may be obtained from the Morse n(t)given in ref 23 or by combining eq 13 and l l a and noting that p(e)-l = w ( e ) = wow(€). The resulting formula is
call
r
(ljb) where N ( E ) is the number of states up to energy E , mi is the ith molecular mode generalized mass, and we assume cross force. terms (FiF'(t))vanish. n,(t),the normalized oscillator velocity time correlation function (TCF), involves reactant dynamics at energy E in the absence of solvent. Similar formulas have been derived for the Markovian limit by Borkovec and Berne.24aSolvent effects enter through the non-Markovian friction coefficients {,(t), which we shall assume for simplicity are the same for all modes: Ci(t) = {(t). The related (time-independent) friction constant is { = dt. To determine D ( E ) , the functional form and numerical values of the solvent friction must be defined. Molecular dynamics simulations of LJ fluids indicate that solvent friction {(to typically contains a short-lived, collisional component Cau(t) and a long-time As we have pointed out,5 "hydrodynamic" component {hhyd(t)." it is the short-lived (high-frequency) portion of T(t)which typically effects the energy transfer required for reaction. Particularly at low densities, the "effective" solvent friction for reaction is collisional and can be approximately modeled with a Gaussian expres~ion:l*~,~
s;w
(20) Tardy, D. C.; Rabinovitch, B. S. Chem. Rev. 1977, 77, 369. (21) (a) Nielson, S. E.; Bak, T. A. J . Chem. Phys. 1964, 41, 665. (b) Keck, J. C.; Carrier, G. J . Chem. Phys. 1964, 43, 2284. (22) Nikitin, E. E. Theory of Elementary Aromic and Molecular Processes in Gases; Clarendon: Oxford, 1974. (23) Grote, R. F.; Hynes, J. T. J. Chem. Phys. 1982, 77, 3736. Note that there is a misprint in eq 3.11 of this paper, which is corrected in our eq 13a. (24) (a) Borkovec, M.; Berne, B. J. J . Chem. Phys. 1985, 82, 794. (b) Straub, J. E.; Berne, B. J. J . Chem. Phys. 1986, 85, 2999. (c) Nitzan, A. J . Chem. Phys. 1987, 86, 2734. (25) Levesque, D.; Verlet, L. Phys. Reo. A 1970, 2, 2514.
where F(s) = (s - l ) / ( s - 1.5). The VED rate constants obtained from eq 13-15 are
kd.1 =
kdJ = {mIIZ-l exp(-Z) X I sexp[Z(e- l ) ] Ss(e,A)
[x
1dx I
xsl exp(-Zex)
I
(16b)
where 2 = @Ed,and we have substituted x = e'/€ in writing eq 16b. To evaluate RC and ET processes on an equal footing, we shall make the simplification of replacing the full time-dependent collisional friction by the time-independent collisional friction constant, as in standard Langevin theories of reaction rates1 (This is a reasonable approximation for energies near d i s s o c i a t i ~ n . ~ ~ ) { ( t ) therefore has a delta function form, obtained as the limit of eq 12 as A approaches 0. Resulting formulas suitable for numerical evaluation of kdZ8in the l -dimensional and s-dimensional cases are ~~~~~~~
~~
(26) DeMarcus, W. C. A m . J . Phys. 1978, 46, 733. (27) Thiele, E. J . Chem. Phys. 1963, 38, 1959. (28) A common approximation to the exact one-dimensional Langevin rate in eq 17a was first obtained by Kramers (ref 9) as kd(Kramers) = 2rZ exp(-2). The error in this approximation varies from about 15% at Z = 50 to 40% at Z = 15. For I2 dissociation, these correspond to temperatures of 360 and 1200 K, respectively. In this paper, we evaluate kd numerically from eq 17a and 17b, rather than making use of the Kramers rate constant.
Zawadzki and Hynes
7034 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
TABLE I: solvent H2 He Ne Ar
kdp = 1
lco~~Z-l exp(-Z)[
& dx 1
dt exp[Z(t - l ) ]
Xs-l exp(-Zex)]-'
(17b) For consistency with RC rates, we adopt the gas kinetic form for lmll
P,,,,measures the "effectiveness" of solvent collisional coupling to the reactant mode and should be less than unity for strictly vibrational energy transfer. For our model, however, P,,,,also incorporates the effects of rotational energy transfer from the solvent to the molecule and subsequent intramolecular rotational-vibrational coupling. This process should increase the overall reactant-solvent energy transfer, particularly in small diatomics. We thus take P,,,,= 1 for iodine recombination and P,,,,= '14 for methyl recombination. 2.3. Transition State Theory Rate Constant kf". We will derive TST recombination rates from TST dissociation rate constants and equilibrium data, in a fashion analogous to our treatment of the ET case above. TST is a rich field, which has attained a degree of precision and sophistication not yet present in stochastic theories such as VED. In line with our goals we therefore do not attempt to provide improvements on previous TST calculations of diatomic or molecular dissociation rate constants. Instead, we will simply adopt reasonable values of k F T calculated from these and from thermodynamic data for use in eq 3. These will be described in sections 3 and 4. 2.4. Diffusion-Controlled Rate Constant kp. At high solvent densities, k: is essentially the inverse of the average time needed for two radicals to diffuse to within a critical contact distance RRM of each other and is given by"
where DRM is the radical diffusion constant and the contact distance is6 RRM = 2-'/'(aR + uM). Lack of experimental diffusion data at high densities has hindered the development of good formulas for DRMover wide density ranges.29 However, Hippler et have proposed a useful interpolation formula for the functionflp) = kT/(&M), which we adopt. f ( p ) can be calculated from gas kinetic theory in the low-density limit asfo(p) and is assumed to approach the density-independent Stokes value, f o = 3mR, at high densities. At intermediate densities, the interpolation formula is30 f(P)/f"
= 1 - expE-f0b)/f"1
(20)
This approach essentially replaces the calculation of DRM as a function of p with the calculation of 7 as a function of p , a more thoroughly characterized problem.2g To compute 7,we use the method of Jossi et which fits residual viscosity, 11 - v0, versus density data as a truncated power series 4
[t(7-70)+ 111/4= CA,~: n=O
(21)
where the A's are tabulated constants. Here, pr is the density p divided by its critical value p,, ( is defined as Tc1/6P;2/3M-1/2 (M (29) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (30) Hippler, H.; Schubert, V.;Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 760. (31) Jossi, J . A.; Stiel, L. 1.; Thodos, G. AZChE J. 1961, 7, 625.
k,' for Iodine Recombination at 314 Kasb uT t, kRC keT k.' expt' 59.7 2.67 3.60 (4.0-4.93) 1.84 0.85 2.83 0.78 1.27 (1.16-1.47) 2.83 10.2 0.35 0.44 1.30 1.18 (1.6) 32.8 1.05 0.25 2.82 3.76 0.26 4.02 2.18 (2.4-3.2) 3.54 93.3 3.36 (3.9) 3.66 179 6.88 0.23 7.11 Kr 0.23 10.4 5.26 (4.48-5.18) 10.2 Xe 4.05 231 4.37 (3.8-4.37) N2 3.80 71.4 3.41 0.31 3.72 4.42 4.40 (5.9-6.5) 4.13 0.29 O2 3.47 107 Cod 3.69 91.7 4.34 0.32 4.66 5.4 C02 3.94 195 8.95 0.33 9.28 12.3 (12.5-13) 12.3 12.7 0.44 13.1 C2H6 4.44 216 18 (29) 18.3 0.44 18.7 C3Hs 5.12 237
"Units for rate constants: io9 L~ mol+ s-'. b~~ parameters, in A and K, taken from ref 29. eGiven values from ref 32, unless otherwise noted. Values in parentheses are the range of other literature values quoted in that reference. d A t 298 K. Experimental data from ref 34. = solvent molecular weight), and v0 is the (density-independent) gas kinetic viscosity: 7 ' = ( 5 / 16 ) ( ~ ~ k T ) 1 / 2 [ * ~ ~ 2k ~ T (/ e2~. )2] )- '' ( (22)
Solvent critical constants are as follows:29 T,, P,,and pc for argon are 150.8 K, 48.7 bar, and 13.4 mol L-I; for propane, they are 369.8 K, 42.5 bar, and 4.93 mol L-I. 3. Iodine Recombination Iodine atom recombination rates have been measured over pressure ranges large enough to observe turnover, for a variety of solvents, near 300 K.6932 Low-pressure rate constants have also been determined up to about 1200 K for several, chiefly inert, buffer gases.13b,33,34In comparing our model predictions with these results, we seek to answer two questions. First, can the simple statistical theories given in the last section predict RC and ET contributions to k,O at and above room temperature? To answer, we will compare predicted values of kro to the data of T r ~ ate ~ ~ ~ ~298-1173 ~ , ~ ~K. Second, can our 314 K and of B ~ r n s for connection formula, using the proper low-pressure rate constant, accurately predict the turnover in overall rate? Here we will compare to the high-pressure data of Troe.6 3.1. Low-Pressure Termolecular Rate Constant k,". kRCand kETare calculated by using the techniques of section 2, with the following parameters. LJ data for solvents (Table I), as well as solvent critical constants, were taken from ref 29. For I, we use UR = 4.32 A and €R/k = 231 Ki6 for 12, UR = 5.16 A and €R2/k and equilibrium = 474 K.29 Ed for 1, is 35.87 kcal constants p (T ) are calculated from thermodynamic data in the latest JANAF tables.36 As noted above, we choose PSt,,= 1 for VED rates. The final ingredient needed for VED calculations is Pel,. Due to the 4-fold degeneracy of the iodine atom ground state, there are 16 possible potential surfaces for recombination of two iodine atoms, five of them attractive.14a At low pressures, a purely adiabatic treatment would indicate that Pel, = 1/16, while relaxation of the adiabatic condition with increasing solvent density might permit Pel, = 5/16.'5*37 We therefore adopt the adiabatic value Pel, = 1/16 for our calculations in the low-density limit, in line with molecular dynamics (32) Hippler, H.; Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 1104. ( 3 3 ) For a review of halogen recombination rates, see: Boyd, R. K.; Burns, G. J. Phys. Chem. 1979, 83, 88. (34) Ip, J. K. K.; Burns, G. J. Chem. Phys. 1972, 56, 3155. (35) Huber, K. P.; Herzberg, G. Constants ofDiatomic Molecules; Van Nostrand Reinhold: New York, 1979. (36) JANAF Thermochemical Tables, 3rd ed. Chase, M. W., Jr.; Davies, C. A,; Downey, J. R., Jr.; Frurip, D. J.; McDonald, R. A,; Syverud, A. N. J . Phys. Chem. ReJ Data 1985, 14 (Suppl. 1). (37) Dawes, J. M.; Sceats, M. G. Chem. Phys. 1985, 96, 315. (38) (a) Wong, W. H.; Burns, G. J . Chem. Phys. 1973, 58, 4459. (b) Chang, D T ; Burns, G. Can. J . Chem. 1976, 54, 1535.
The Journal of Physical Chemistry, Vol. 93, No. 19, I989 7035
Radical Recombination Rate Constants
I -- He BUFFER GAS
I+I-I2
,
'*
9.0
E
L
.........A.........~........ T
E
9
-. -. -- - ---
-
kED
N
A
0.0
Y
J
L
--- ---
u x 9,
0
kRC
9,
-
7.0 250
500
750 T (K)
1000
71 -3
1250
--I
9.0
-
I
E
-
N
J
OL
f 8.0 9,
-
7.0 2
/'
I
-2
I
I
I
-1 0 1 log [pM(mol c')]
2
Figure 2. Variation of I recombination rate constants with solvent density near room temperature. Solid and dashed lines calculated for 314 K by using the two-term connection formula (eq 4). Experimental data of Troe and co-workers: ref 32, 314 K ( 0 and 0);ref 6, 298 K (0).
10.0
*,
r
VI
................................
0
-
2
c
N I
? & L
111
1
1
I
I
500
750
1000
1
3
T (K)
Figure 1. Temperature variation of low-pressure termolecular rate constants for I recombination in (a) He and (b) N2. Labeled curves indi-
cated calculated rate constants; solid circles are experimental data of Burns and co-workers (ref 34). Table I compares our calculated values for kr0 = kRC+ kET and its components to experimental data for iodine atom recombination in a variety of solvents at 314 and 298 K. Calculated rate constants are in reasonable overall numerical agreement with experiment and show the proper relative trend with buffer gas complexity. It is important to note that the RC component, kRC, dominates kETfor all solvents except He. To illustrate the effects of temperature on the components of k,", we examine two solvents: Nz, typical of solvents with significant R C rates, and He, where E T effects may be especially significant. Predicted values of k," are shown for these systems at a function of T from 300 to 1200 K in Figure la,b, along with relevant experimental data. Agreement between predicted and experimental rate constants is surprisingly good considering the approximate character of the theory. These calculations of low-pressure iodine recombination rates predict the following. First, both R C and ET processes play a role in low-pressure recombination of I atoms. At or near room temperature, the R C process typically strongly dominates over ET for all common solvents except He. As temperature increases, both components of kr0 decrease, but kRCdeclines significantly more rapidly than km. By about 800-1200 K, the two components become substantially equal in their contributions to k,". For I recombination in He, however, ET processes are dominant at 300 K and above. All of these predictions are borne out by the trajectory simulations of I recombination by Burns and c o - w o r k e r ~ ~ ~ and are consistent with molecular dynamics calculations'5a and related suggestions.2*12 The temperature variation of the calculated kETand kRCmay for kRCand {for kEThave be understood as follows. Both Zcoll only weak variations with temperature, on the order of Z"',', since the decrease in the collision integral with increasing temperature offsets the factor of For kRC,the remaining component,
a,
has a temperature variation which shifts from T2.5 to T1.5 with increasing temperature. For kET,there are two terms of interest besides 5: The first, K44 exp(-Z), varies as in the temperature range we treat, while the remaining integral varies as Thus, even at high temperature, kRCdeclines Overall, kET0: more rapidly than kET. 3.2. k,-p for Iodine Recombination. We now examine the complete room-temperature kr-p curves for iodine recombination in a simple solvent, argon, and a "complex" solvent, propane. Recent theoretical estimates of kPT for iodine r e c o m b i n a t i ~ n ~ ~ ~ ~ ~ are large enough (on the order of 10" L mol-' s-I) to make the use of the two-room connection formula, eq 4, realistic. Values of kr0, and LJ data used to compute kp, are given in Table I. Figure 2 displays complete k,-p curves computed from eq 4 for iodine recombination in argon and propane, compared with the experimental data of Troe and c o - w o r k e r ~ .The ~ ~ ~two-term ~ connection formula is able to well describe the sharp turnover behavior in the rate constants. The effect of increased solvent complexity in shifting the turnover to lower densities is also apparent from both experiment and theory.
4. Methyl Radical Recombination Methyl radical recombination and its reverse process, ethane dissociation, have, despite intense study, retained considerable mystery.39 In particular, exact descriptions of the temperature dependence of kr0 for a wide range of gases remain elusive. We do not attempt to solve this problem but simply use methyl radical recombination to illustrate the role of increasing reactant dofs in the k,-p profile for an ET-dominated process. We therefore select one temperature at which to compare calculated k,-p curves with experimental data. Recently, such data have become available over a fairly wide density range for Ar solvent near room temp e r a t ~ r e , ~so, ~we ' chose 298 K for our calculations. Due to its size, we cannot now ignore kyT,as in the atomic case, and must use the three-term connection formula, eq 3. kFT (=3.6 X 10'OL mol-' s-l) is taken from the recent work of Wagner and Wardlaw,,* which also provides the estimate 1.2 X 10l6 L2 mol-'s-' for kr0. LJ parameters for CH3 are those of CH,: ffR = 3.82 A, C R / k = 137 K.29 Other parameters are Ed = 87 kcal 39,43 and = 1.00 X los9L mol-', calculated from tabulated thermodynamic (39) (a) Wardlaw, D. M.; Marcus, R. A. J . Phys. Chem. 1986,90, 5383, and references therein. (b) Baulch, D. L.; Duxbury, J. Combust. FIame 1980, 37, 313. (c) Skinner, G. B.; Rogers, D.; Patel, K. B. Inr. J . Chem. Kinet. 1981, 13, 481. (40) Slagle, I. R.; Gutman, D.; Davies, J. W.; Pilling, M. J. J. Phys. Chem. 1988, 92, 2455. (41) Hippler, H.; Luther, K.; Ravishankara, A. R.; Troe, J. Z . Phys. Chem. (Munich) 1984, 142, 1. (42) Wagner, A. F.; Wardlaw, D. M. J . Phys. Chem. 1988, 92, 2462. (43) Hase, W. J . Chem. Phys. 1972, 57, 730.
Zawadzki and Hynes
7036 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 TABLE 11: k." for CHI Recombination in Ar at 298 K system kr0,L2mol-2 s-I system k,", L2 mol-2 SKI expta 1.2 x 10'6 VED, s = 6 1.7 X 10l6 VED, s = I 5.7 x 107 VED, s = 8 8.5 X lo'* 1.6 x 1013 VED, s = 4 X
"Determined using the extrapolation formula of ref 42: k," = 3.18 1035T7.03 exp(-1390/T).
Figure 3. Variation of CH3 recombination rate constants with Ar buffer gas density at room temperature. Curves calculated by using VED theory
for varying numbers of internally active modes s at 298 K. Experimental ref 41, 298 K ( 0 ) . data: ref 40, 296 K (0); As before, our main interest centers on k,". Energy transfer should now play a dominant role, due to the existence of coupled nonreactive dofs in ethane. We apply the VED theory of section 2, using P,,,, = 1/4 and Pelcc= 1 / 4 (due to the 2-fold degeneracy of the CH3 ground state). The remaining parameter is s, the total number of molecular vibrational modes (including the reactive mode) participating in energy transfer. In previous work,s we assumed that only those modes close in frequency to the reactive mode have a chance to couple efficiently in unimolecular reactions. The magnitude of the reactive mode frequency in ethane is 995 cm-i.4s Ethane also contains four other modes between 800 and 1200 cm-' and an additional four between 1200 and 1400 c~n-'.~' Accordingly, we perform VED calculations of k," for s = 4,6, 8 and include the equivalent atomic case, s = 1, for comparison. Table I1 compares k," for various values of s with the extrapolated low-pressure value of Wagner and Wardlaw. Our value of k,O for s = 6 is in fair agreement with experiment. Given the crudity of our VED model, which may underestimate ET efficiency, we do not propose that the actual number of contributing modes is precisely 6. A more accurate treatment might show better agreement for s = 4. Our point is that a significant number of total dofs are involved and that a one-dimensional theory fails dramatically in accounting for the experimental broad plateau, consistent with previous work on isomerization and other unimolecular p r o c e ~ s e s . ~ , ~ ~ * ~ These additional dofs and their influence on k,O have a dramatic effect on the k,-p curve, as compared to that for iodine recombination. Figure 3 shows predicted k,-p curves for s = 1 , 4,6, and 8, together with the experimental data. For s 1 4, the predicted broad plateau is observed, while, in contrast, the onedimensional system never reaches the TST value. As for isomerizations with fairly high barriers,ls5 the coupling of nonreactive dofs to the reaction coordinate is predicted to shift the onset of turnover of radical recombination rate constants into the normal gas phase from the very dense gas, or liquid, phases. Observation (44) Chao, J.; Wilhoit, R. C.; Zwolinski,
B. J. J . Phys. Chem. ReJ Data
1973, 2, 427. (45) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240.
of complete turnover behavior, however, may require measurements over extremely large density ranges. 5. Concluding Remarks
We have successfully applied the connection formula eq 3 to predict k,-p curves for both atomic and more complex radical recombinations. The shapes of these curves, in particular the location and sharpness of the turnover, are strongly influenced by k,", the limiting low-pressure termolecular rate constant. For iodine recombination, we have demonstrated the utility of two simple statistical theories, Bunker-Davidson (BD) and vibrational energy diffusion (VED), in predicting the two major components of k,", kRC,and kETas functions of temperature. For recombination of methyl, a prototypical "complex" radical, we have illustrated the strong effect on k," of s, the total number of molecular modes involved in reactant-solvent energy transfer. Recently, a number of theoretical studies using stochastic or ~~?~' Langevin models for I recombination have a p ~ e a r e d . ~ , Such models, which treat only energy-transfer (ET) processes, typically yield values for k," which are significantly larger than we have calculated by inclusion of the dominant radical complex mechanism and functional forms for k," with an approximate T1/' dependence near room temperature. This 7'dependence disagrees with both experimental data and trajectory calculations, in contrast to our predictions. Accurate models of I recombination must treat ET and RC processes, most especially the latter. For methyl radical recombination, the agreement of our predictions with experimental results emphasizes the import a n ~ e ~ ~ ~ of ~ ,the ~ ~coupling * " @ of a finite number of intramolecular modes to the reaction coordinate, in the vein of RRKM theory, for turnover behavior. For this system, there are no indications of the solvent-induced breakdown of RRKM theory suggested to O C C U ~ in ~ ~some . ~ ~unimolecular reactions in the presence of buffer gas collisions. We expect that further experimental and theoretical studies of polyatomic radical recombination reactions will be revealing on the validity of RRKM theory at intermediate and high densities. We do not wish to imply that either the BD or VED formulations we have presented are quantitative theories. Both, for example, contain an uncertain steric factor. Although they are well able to predict experimental data, their main utility lies in the easy and consistent comparison they provide of the relative contributions of various dynamical components to unimolecular rate constants. The VED model we have employed, for example, might be improved in several ways. First, a more realistic model of the solvent friction might be incorporated. Since our theory already explicitly treats time-dependent friction, this requires only the appropriate ((t)-a not insignificant problem. Quantitative calculations in this regard have recently attracted some interest.'*% Another obvious step would be the incorporation of rotational effects on our reactant potential. Although such effects are likely not significant for I recombination kinetics in most solvents, due to the large moment of inertia of Iz,sithey will certainly play a role in recombination of lighter species such as H atoms.''
Acknowledgment. This work was supported in part by N S F Grants C H E 84-19830and C H E 88-07852. (46) Borkovec, M.; Berne, B. J. J . Chem. Phys. 1986, 84, 4327. (47) Sceats, M. G.;Dawes, J. M.; Millar, D. P. Chem. Phys. Lett. 1985, 114, 63. (48) Nitzan, A. J . Chem. Phys. 1984, 82, 1614. (49) (a) Rice, 0. K. Z . Phys. Chem. 1930, 7,226. (b) Spring, C. A.; True, N.S . J . Am. Chem. SOC.1983, 105, 7231. (c) Lazaar, K. I.; Bauer, S . H. J . Phys. Chem. 1984,88, 3052. (d) Kuharski, R. A,; Chandler, D.; Montgomery, J. A,, Jr.; Rabii, F.; Singer, S . J. J . Phys. Chem. 1988, 92, 3261. (50) (a) Rodger, P. M.; Scats, M. G.;Gilbert, R. G. J . Chem. Phys. 1988, 83, 6448. (b) Straub, J. B.; Borkovec, M.; Berne, B. J. J . Phys. Chem. 1987, 91, 4995. (c) Gertner, B. G.;Wilson, K. R.; Hynes, J. T. J . Chem. Phys. 1989, 90, 3537, and references therein. (51) Nesbitt, D. J.; Hynes, J. T. J . Chem. Phys. 1982, 77, 2130; 1982, 76, 6002. (52) See, for example: Blais, N. C.; Truhlar, D. G . J . Chem. Phys. 1979, 70, 2962.
