J. Phys. Chem. 1981, 85,3588-3592
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indicate AGO = -6.0 kcal/mol. Unfortunately, the enthalpy change in this reaction is not known. In converting his Volta potential measurement to a single-ion free energy of hydration, Randles overlooked two items. The tabulated16gas-phase enthalpies of monovalent (16) "Selected Values of Chemical Thermodynamic Properties", Circular of the National Bureau of Standards NO.500, U.S. Government Printing Office, Washington, DC, 1952.
cations, which he made use of, include a term 6/2kTapparently representing the enthalpy of an associated electron. This should have been discounted. On the other hand, there is a change in the electronic degeneracy of the atom upon ionization. For the potassium atom (of the calomel half-cell) this introduces a term RT In 2 into the free energy. Incorporation of these (partially offsetting) corrections yields the real energies of hydration cited in the text.
Dependence of Specific Rates for Excess-Electron Scavenging in Nonpolar Liquids on Electron Energy William H. Hamlll Department of Chemistry and the Radiation Laboratory, Unlversity of Notre Dame, Notre Dame, Indiana 46556 (Received: May 4, 1981; In Final Form: June 25, 198 1)
Electron scavenging reactions in nonpolar molecular liquids exhibit anomalous solvent effects which appear to depend principally upon Vo, the quasi-free electron energy. The rate constant k, will be described in terms of the following general mechanism: (1,-1) e- + A (e-, A) and (2) (e-, A) X-, where X- is the stabilized product for either dissociative or nondissociative attachment. Important variations of the general kinetic scheme are due in part to the electron transport mechanism and the state of the reacting electron, including its energy. In neopentane, e.g., a gaslike mechanism of collisions is proposed, inefficient acceptors requiring thermal activation. It is necessary to distinguish between electron potential energy and kinetic energy. For an efficient acceptor k , a klis approximately constant in all high-mobility media. The trap model and the hopping model determine kinetic differences which are quite distinguishablein principal but not yet resolved in practice. Both k , = b p with b constant and k, a 1.1" for 0 < n < 1follow from the general scheme. Most results are expressible in terms of ( b p / k , - l)-l = k2/k-1 with k2 depending exponentially on temperature.
-
Introduction Rate constants k, for excess-electron scavenging in nonpolar liquids ranging from 1O1O to 1014M-l s-l have been reported by Allen et and Bakale et al.3 The temperature coefficient may be positive, zero, or negative. Usually k, increases with mobility, 1.1, but the reverse may occur.lA The energy of the electron in the delocalized state ed- relative to vacuum, Vo, is an important parameter, and both k,2 and p6-' usually increase as Vo decreases. Maxima in k, vs. Vo have been explained by analogy with vertical attachment processes in the gas phase where they are common.2 Both reaction and transport have been attributed to the delocalized or extended state of the e1ectron.'p2 Accordingly, the energy level of the localized electron, Val, has been assigned an approximate value below Vo shifted by the observed activation energy for mobility.E
-
-
(1)Allen, A. 0.;Holroyd, R. A. J. Phys. Chem. 1974, 78, 796. (2) Allen, A. 0.; Gangwer, T. E.; Holroyd, R. A. J.Phys. Chem. 1975, 79, 25. (3) Bakale, G.; Sowada, U.; Schmidt, W. F. J.Phys. Chem. 1975, 79, 3041; 1976,80, 2556. (4) Wada, T.; Shinsaka, K.; Narnba, H.; Hatano, Y. Can J. Chem. 1977,55, 2144. (5) Holroyd, R. A.; Allen, A. 0. J. Chem. Phys. 1971,54, 5104. (6) Kestner, N. R.; Jortner, J. J. Chern. Phys. 1973, 59, 26. (7) Holroyd, R. A.; Tauchert, N. J. Chern. Phys. 1974, 60, 3715. (8) Holroyd, R. A,; Gangwer, T. E. Allen, A. 0. Chem. Phys. Lett. 1976, 31, 520.
For intermediate and high-mobility alkanes in which1 k , 1.1' and 0 < n < 1,there has been no generally accepted mechanism. According to the model of Yakovlev et al.,9 the localized electron reacts, while Allen et a1.2 attribute reaction to the electron in the quasi-free state. Excess electrons in molecular liquids are not understood theoretically, but they can be treated phenomenologically. There is no conduction band, and transport has been described as electronic Brownian motion.1° The upper and lower energy states of electrons will be designated simply as delocalized and localized, ed- and ei. It is the purpose of this work to develop a phenomenological kinetic model for electron scavenging in nonpolar liquids. It should be sufficiently general to comprehend simple, clearly recognizable kinetic types. Electron transport must be an integral component. The states of the reacting electron, including their energy, are to be distinguished. Kinetic Model In order to examine the mechanism of electron scavenging phenomenologically, it is necessary to adopt first a transport model. In low-mobility media, e.g., n-hexane, the Debye-Einstein relation has been demonstrated to apply to SF6,e.g., k , / p = b, a c o n ~ t a n t .If~ the electron (9) Yakovlev, B. S.; Boriev, I. A.; Balakin, A. A. Int. J.Radiat. Phys. Chem. 1974,6, 23. (IO) Cohen, M. H. Can. J. Chem. 1977,55,1906.
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The Journal of Physical Chemistry, Vol. 85,
Excess-Electron Scavenging in Nonpolar Liquids
spends less than one vibrational period at an acceptor site, k&p) is no longer applicable.’l The dependence p( V,) has been examined for the hopping model12which will be considered in this work. Vo is the minimum energy of the electron in the extended state. It varies among alkanes as the associated entropy changes, and dVo/dT = -ASo. Also, Vo T AS, = 0.74 eV at 298 K. Electron localization is not determined by V, itself but by the entropy of incoherent scattering in the extended state.12 It will be shown that k,(V,) is not due only to p( Val, in general. The electron affinity, x, is involved, and the energy of the electron must also be considered in the chemistry. Reaction may involve not only ed- or el- but also eTOB- at the top of the barrier for hopping. The substantial independence of k , / p with regard to temperature is common and argues for reaction in the conducting state (ed- or eTOB-) in those instances when it occurs. For alkanes at 296 K, p = 0.36 exp[-O).35Vo/(kBT)]~ The activation energy for the same alkanes is E, = 0.38V0 + 0.16 in eV.4 This is of the form E, = a(V, - Val) with 0 < cy < 1, which corresponds to hopping12 with a barrier intermediate between Voand Vo, if Vo,is constant. However, these regularities can also be described by a linear dependence of Val on V,. In the latter case it would be impossible to distinguish between a = 1 and a < 1, i.e., the trap model and the hopping model. The following general reaction scheme
i1
RX
No. 24, 1981 3589
t
e-
m
+
e-
+A
(e-,A)
2
X-
(1)
will be used. Step 1 describes transport in all media and requires elaboration. The entity (e-,A) may involve ed-, e l , or eTOB-. In the last instance, as well as the first, = 0. In any event, (e-,A) is the potentially reactive complex with a transmission coefficient often less than unity. Step 2 covers both dissociative and nondissociative processes. For exoergic attachment A- is initially vibrationally excited, A-*. When the acceptor is polyatomic and A- is stable in the medium, quenching is expected to be efficient, at least in molecular solvents. As the number of internal degrees of freedom of A decreases, the lifetime for autoionization of A-* decreases. SF6 and O2 may be examples of these two types. The general scheme allows important variations in microscopic detail. As an example, in n-hexane the entity (e-,A) may involve e7 with a relatively long lifetime. In neopentane if refers to a collision involving ed-. The conventional rate equation k , = ki[kz/(k-i +
(2)
involves the stationary-state approximation. When eq 2 applies, k,(T) is not always a useful function. Whenever k,(p) is important, eq 3 may be useful.
(b/.t/k, - 1)-l = kz/k-l
(3)
The probability for (e-,A) to form X-, encased in square brackets of eq 2, will be represented by P and can also be expressed by eq 4,where T is the lifetime of (e-,A) with P = 1 - exp(-~/8) (4)
-
respect to separation and 8 is the mean time for (e-,A) A-. In order to examine ks(/.t)above the diffusion-controlled limit, k , has been resolved into its components, /.t and P. To consider ed- and eTOB-within a common scheme, one (11)Funabashi, K.; Magee, J. L. J. Chem. Phys. 1975, 62, 4428. (12) Hamill, W. H. J. Phys. Chem., in press.
Separation R - X
Figure 1. Schematic potential-energy profiles for the scavenger RX as functions of the separation R of R-X. Curve I is for the reactants (e-, RX) in a medium in which reaction is approximately diffusion controlled. Curve I1 applies to the same reaction in a medium of lower V, or V, shown by broken lines, which requires thermal activation. Curve 111 describes RX-.
invokes the continuity provided by the random walk of Brownian motion for ed- in molecular liquids and hopping for el-. Both will be represented by k l = bp. This is not to be confused with k , 0~ p in general. In high-mobility liquids one may expect a resemblance to gaslike behavior and kl = uuth in terms of collision cross section, u,and thermal electron velocity, uth. It follows that
-
k , = auth[l - eXp(-T/8)]
(5)
Since ks/(uuth) = P, P T / O for large p and T may be as small as s. Collisions in, e.g., Me,&, are very inefficient in terms of k/p1v3,but k, as large as loi4 M-ls-l can be observed because collision frequency is high. In the limit whenever 8 >> T , by letting T = L/Uth, eq 6 follows, where L is the distance along the electron trajectory within which reaction can occur. Equation 6 can also be expressed by k , = k1k2/k-,. The approximation of using mean uth in eq 5 does not affect eq 6. When u is appreciably greater than Uth, the parameters are expected to change, e.g., by applying an electric field.3 For an efficient acceptor and the conditions chosen, k, is not expected to depend strongly on V,, p, and T. Reactions which follow the behavior described by eq 6 will be called type i. The other extreme, type ii, applies to diffusion-controlled reactions involving el- and k2 >> k-1, or k, = k l .= b p , As mobility decreases, type i gives rise to type 11. Alternatively, for type iii, if reaction occurs only while electrons are moving through the liquid outside traps and if, in low-mobility media, the trap model applies,2then only the fraction pL/& of electrons are delocalized. For electrons at the Vo level, the specific rate would be kiii = k,p,/p.2 Electron hopping, type iv, is expected to occur by analogy with transport in disordered solids,13but ed- always contributes to the extent that the upper state is populated thermally. Since the barrier for hopping lies between Val and V,, the regularity p a exp(-0.37Vo) for 1 2 alkanes4 requires k l ( p ) and consequently k,( V,). The coefficient of V , is important, and the distinction between types iii and iv is primarily a problem in electron transport,12not kinetics, and requires more data for Val to resolve the mechanism of transport. (13)Mott, N.F.;Davis, E. A. “Electron Processes in Non-Crystalline Materials”; Claredon Press: Oxford, England, 1971;p 32. (14)Christophorou, L. G.;McCorkle, D. L.; Carter, J. G. J. Chem. Phys. 1971,54, 253.
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There is finally type v, for which k , = ( k l / k - , ) k Zin all media. Reaction is slow, insensitive to transport, and involves the lowest state of the electron, whether e< or ed-. Quenching A-* may be rate controlling. The effect of medium on the rate of reaction is shown schematically in the configuration diagram of Figure 1, which corresponds to that of Funabashi and Magee." Curves I and I1 represent (e-,A) for one acceptor in two media. The displacement along the energy axis is considered to arise entirely from isothermal differences in V,. The crossing of curves I and I11 represents an activationless attachment process in a reference system. Curve I11 applies to both dissociative and nondissociative processes, but crossing occurs at small R for positive electron affinity x in the medium. It includes the polarization energy of the medium by the ion, both nuclear and electronic: and is taken to be constant in a series of similar 1iquids.l' As electron energy decreases from curve I to curve 11, it becomes necessary to thermally activate the electron from u * 0 vibrational level to the crossover. The energy required is y AV with AV the energy separation at the minimum and 0 < y < 1. Over a limited energy range y should be approximately constant, corresponding to the familiar kinetic effect of linear free-energy change. For reactions involving ed-, AV = VOI- VOII. For eTOB-9 AV = a(V01- Van) and 0 < a < 1,corresponding to type iv. (It will be convenient to characterize type iii by a = 1.) The net energy requirement is ay AV. Consequently,a cannot be determined from k,. The ambiguity concerning electron transport lends interest to the inverse kinetic process AA + e-. By microscopic reversibility, the same path or paths are fol10wed.l~ Type-iii scavenging requires ionization to the edstate in all media, not to the lowest state, It can only be stated with assurance that the inverse of the diffusioncontrolled reaction gives e 0 in the medium, and an adequate density of vibrational acceptor states. Since there is no experimental (15)Kauzmann, W. "Quantum Chemistry"; Academic Press: New York, 1957; p 649.
Ham111 I
,
I
,
,
"0
Figure 2. The dependence of ( b p l k , - 1)-' = k2/k-' on V,, for CCI, (0),C2HC13( O ) , and SF, (V)at 296 K, from ref 1 and 2, except CCI, from ref 4 (0)and from ref 19 (8).
evidence for contributions from higher electronic states of A- in liquids, they need not be considered." Diatomic and triatomic acceptors should tend to be inefficient because of the low density of acceptor states,16although vibrational broadening by the molecular medium contributes to crossing. The work of Allen et aL2 and Bakale et al.3 on SF6 provides the best example. In ethane, propane, butane, and n-hexane over ranges of 10-3-1 cm2 V-l s-l in mobility, 0.2 to -0.05 eV in Vo, and 110-300 K, k , was diffusion controlled, or k , = kl. The requirement of a long lifetime for (e-,A) is satisfied by e21 are described quite well (except for Me4Si) by k, 0: p0*59,which is expected to be intermediate between types i and ii, or transport dependent. The deviation of the datum for Me4Si in Figure 3 is consistent with type-i behavior. In neopentane and Me4Si,the nominal energy of activation is 0.1 eV for CC4. Unlike SF6,the molecular anion has not been observed, even in organic glasses, and probably does not exist. If the dissociative curve I11 in Figure 1 is displaced to larger R, relative to SF6, thermal activation of CC14 could be accounted for. The dependence of k, on electric field, E , is most useful for kinetic diagnostic^.^ In Ar and Xe, k, for SF6 and O2 decreases monotonically as the kinetic energy of the electron is raised by the field. This suggests that energy dissipation from A-* is inefficient in monoatomic liquids. The opposite effect was observed for N20, indicating endoergicity in these mediae3 This is supported by k,(SF,)/k,(N,O), which is -1 in n-hexane and lo4 in Xe at low field. At low Vo,NzO probably requires thermal activation. Other acceptors should be examined at high field in neopentane and Me4Si at -300 K to to improve the basis for kinetic analysis. Except for the trivial case of the diffusion-controlled reaction, it may not be possible to account adequately for k,(T). In the context of this work, it will be assumed that the primary purpose of such a study is to examine the thermal activation of the acceptor. The kinetic energy of the electron can be controlled better by the electric field E in high-mobility media. Lacking information on k,(E) for other acceptors, the qualitatively similar behavior of N20, CO,, CzH5Br,and C2HC13suggests that all require thermal activation for k2 at low electron energy. The Vo threshold for activation may be the primary cause for, and coincide with, the maxima of k,( Vo) in liquids. The customary purpose for measuring k,(T) is invalidated by the greater effect of Vo(T),even in neopentane. For neopentane dVo/dT = -9 X 10" eV/K and at 300 K, T ASo = 0.27 eV while kBT = 0.026 eV and the kinetic energy of the electron is equivalent to 10 classical oscillators. The Arrhenius dependence is inappropriate. The wide separation of vibrational levels in O2 and 02restricts gas-phase attachment to one level, u' = 4. The effect of high pressure observed by Goans and Christophorou22has been attributed to van der Waals dimers by Shimamori and Fe~senden,~ and by Huo.16 Electron attachment by 0, in liquids, as well as by other acceptors considered here, has been discussed by Christ o p h ~ r o u . ~More ~ recent data show that electron attachment by oxygen is characterized by relatively small values of k, in all solvents. Reaction is always thermally activated although xvac= 0.43 eV,21,25 Increasing the kinetic energy of the electron in an electric field to ca. 1 eV3or increasing T ASo by increasing the potential energy Vo isothermally from -0.55 to 0.1 eV causes k, to decrease tenfold. Such changes in the medium leave u = 0 unaffected but v' must change, imposing discontinuous effects on k,. The total energy for attachment in alkanes (x,Vo,and polarization) is Et L 2 eV20 and u' = 15 for Oz-is quite (21)Baxendale, J. H.; Geelen, B. P. M.; Sharpe, P. H. J.Radiat. Phys. Chern. 1976,8, 371. (22)Goans, R.E.;Christophorou,L. G. J. Chern. Phys. 1974,60,1036. (23)Shimamori, H.; Fessenden, R. W. J . Chern. Phys. 1981,74,453. (24)Christophorou, L.G.Chern. Rev. 1976, 76, 409. (25)Holroyd, R.A.;Gangwer, T. E. Radiat. Phys. Chern. 1980,15,283.
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J. Phys. Chem. 1981, 85,3592-3599
possible with turning points at -1 and -2 A,26 Such large changes in the size and shape of 02-* must considerably change the polarization energy for large v’which is contrary to formation of a quantized state. Strong coupling of 02-* with phonons provides the only mechanism for stabilizing the anion. The comparative efficiency of attachment in xenon cannot be accounted for if an acceptor state with large v’in strongly polarized media is forbidden. In gross terms the kinetics fit type v, but the details are not understood.
polar media: electron transport to and from an encounter or collision pair (e-, A) and ion formation. One must consider also the parameter Vo, distinguish between localized and delocalized electrons, and also allow for different transport mechanisms. The kinetics of detachment may provide a more sensitive probe for identifying the state of the reactive electron. There may be about five distinguishable types of overall kinetics. Within this general model, k,(Vo), k,(E), and k,(p) are important isothermally, but k,(T) is not useful.
Conclusions A minimum of three mechanistic steps are required to describe the specific rate of electron attachment in non-
Acknowledgment. I am indebted to K. Funabashi for extensive discussions and to A. Mozumder and D. M. Chipman for comments. The research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-2151 from the Notre Dame Radiation Laboratory.
(26) Das, G.; Wahl, A. C.; Stwalley, W. C. J. Chem. Phys. 1978, 68, 4262.
Intramolecular Vibrational Energy Redistribution Paul R. Stannard+ and William M. Gelbart*t Department of Chemistry, University of California,Los Angeles, California 90024 (Received: M y 12, 198 I; In Final Form: July 15, 1981)
A simple quantum-mechanicalscheme is presented which describes the flow of vibrational energy from an initially prepared distribution into the rest of the molecule. Strong and weak couplings lead to at least two time scales for decay of the initial state. Computational studies for highly vibrationally excited water (H,O) and benzene (C6H6)are used to illustrate the differences between large- and small-molecule behaviors within this theoretical scheme. In both cases we discuss also the nature of the breakdown, with increasing energy, of “local-mode’’ separability. This context allows us to examine the dynamical consequences of “overtone” vs. “combination” distributions of vibrational energy. Some recent experiments-in particular, near-infrared overtone studies of benzene, and transient hot-band electronic spectra of glyoxal-are treated in light of these results.
I. Introduction Much attention has recently been focused on the problem of vibrational energy redistribution in isolated polyatomic molecules. Experimental studies have included time-resolved fluorescence spectra via “cold” molecular beam1 and “chemical timing”2 techniques, multiphoton infrared3 and single-photon near-infrared4excitation, and chemical a~tivation.~ Most theoretical work has involved stability and ergodicity analysis of the classical and semiclassical dynamics associated with model HamiltoniansPg Numerical simulations (classical trajectory analyses) of “real” molecules have been carried out on only a few threeand four-atom species.1° Quantum-mechanical studies of model systems have also been reported.l1-l2 In the present work we present results of our numerical solutions to the quantum-mechanical equations of motion for some realistic vibrational Hamiltonians. In section I1 we outline briefly the basic phenomenology relating vibrational eigenstates and energies to the time evolution of prepared nonstationary states. We find that the time evolution of the initial state is characterized by several time scales and modes of behavior. These general features are illustrated in sections I11 and IV by two numerical examples, H20 and C6H6, which exhibit the ‘Systems, Science a n d Software, P.O. Box 1620, L a Jolla, CA 92038.
* Camille and Henry Dreyfus Foundation Teacher-Scholar. 0022-3654/81/2085-3592$01.25/0
“small”- and “large”-molecule limits of vibrational energy redistribution dynamics. In the closing section (V) we discuss qualitatively the following: multiple-time-scale relaxation, the role of energy delocalization (i.e., combi(1) J. B. Hopkins, D. E. Powers, and R. E. Smalley, J. Chem. Phys., 72, 2905, 5039, 5049 (1980); 73, 683 (1980). (2) R. A. Coveleskie, D.A. Dolson, and C. S. Parmenter, J. Chem. Phys.,72,5774 (1980); D.A. Dolson, C. S. Parmenter, and B. M. Stone
in “Proceedines of the NATO Advanced Studv Institute on Fast Reactions in Energetic Systems”, Ioannina, Greece, July 1980. (3) (a) P. A. Schulz, Aq. S. Sudbs, D. J. Krajnovich, H. S. Kwok, Y. R. Shen, and Y. T. Lee, Annu. Reu. Phys.Chem., 30,379 (1979); (b) N. Bloembergen and E. Yablonovitch, Phys.Today,31,23 (1978); (c) J. C. Stephenson and D. S. King, J. Chem. Phys.,69,1485 (1978). (4) (a) R. G. Bray and M. J. Berry, J. Chem. Phys.,71,4909 (1979); (b) R. L. Swofford, M. E. Long, and A. C. Albrecht, ibid.,65, 179 (1976); (c) J. S. Wong and C. B. Moore in “Proceedings of the Sergio Porto Memorial Conference”, Rio de Janeiro, Brazil, July 1980. (5) See the review discussion by I. Oref and B. S. Rabinovitch, Acc. Chem. Res., 12, 166 (1979). (6) E. J. Heller, E. B. Stechel, and M. J. Davis, J. Chem. Phys.,73, 4720 (1980). (7) Y. Weissman and J. Jortner, submitted for publication in J. Chem. Phys.and Phys.Lett. A. (8) D.W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys., 71, 2864 (1979). (9) C. Jaffe and P. Brumer, J. Chem. Phys.,73, 5646 (1980); R. T. Lawton and M. S. Child, Mol. Phys.,37, 1799 (1979). (10) R. J. Wolf and W. L. Hase, J. Chem. Phys., 72, 316 (1980); 73, 3779 (1980), and references contained therein. (11) (a) S. Nordholm and S. A. Rice, J. Chem. Phys.,61, 203, 768 (1974); (b) K. G. Kay, ibid.,72,5955 (1980), and references cited therein. (12) E. Thiele, M. F. Goodman, and J. Stone, Chem. Phys.Lett., 69, 18 (1980).
0 1981 American Chemical Society
