8310
J . Phys. Chem. 1989, 93, 8310-8316
Application of the Kramers Equation to Stilbene Photoisomerization in n -Alkanes Using Translational Diffusion Coefficients To Define Microviscosity Ya-Ping Sun and Jack Sakiel* Department of Chemistry, The Florida State University, Tallahassee, Florida 32306-3006 (Received: March 31, 1989; In Final Form: June 19, 1989) Translational diffusion coefficients for toluene in n-alkanes, calculated with the Spernol and Wirtz empirical relationship (Spernol, A,; Wirtz, K. Z . Naturforsch., A 1953, 8, 522), are used as a probe for the microfriction experienced by the SI state of trans-stilbene in the It* Ip* twisting process. A solute/solvent-specific property, the microviscosity, is defined whose substitution for shear viscosity in the hydrodynamic form of the Kramers equation gives an excellent fit of It* Ip* rate constants in n-alkanes. The use of rotational reorientation times as a measure of microfriction coefficients in the Kramers equation is shown to be consistent with this approach.
-
-
Introduction The use of stilbene as the prototypic system for cis-trans photoisomerization continues to focus much attention on the photochemistry and spectroscopy of the two isomers.' The reaction has been investigated more thoroughly in the trans cis direction because the longer lifetime of It*, the lowest excited singlet state of trans-stilbene, makes its study easier. It is generally agreed that twisting about the central bond to a perpendicular intermediate, Ip*, follows electronic excitation as the first step leading to photoisomerization. The It* Ip* process occurs through crossing of a small internal potential energy barrier,2 Eo or AH,, whose height in the vapor phase under isolated molecule conditions has been estimated at 3.4 f 0.3 kcal/mol based on the experimental threshold energy for radiationless d e ~ a y . ~Al.~ ternative values of 2.65 and 3.7 f 0.2 kcal/mo16 for Eo have been based on the optimal fitting of RRK and RRKM theories to the excess energy dependence of fluorescence quantum yields and decay rate constants, respectively. Rate constants for It* 'p* twisting, kobsd,in solution have been inferred from fluorescence quantum yields, &, and fluorescence lifetimes, 7f
-
-
-
= kf(+? - +f)/+?$f (1) k o w = 7;' - kf - ki, (2) where &' is the limiting fluorescence quantum yield at low T and kf and ki, are rate constants for radiative decay and intersystem crossing from It*, re~pectively.~Activation parameters for the twisting process are solvent dependent. Medium viscosity affects the rate at which solvent cages can rearrange to accommodate the motion of the solute and thus effectively imposes an additional barrier to twisting, AH", which is solvent The polarity and the polarizability of the medium affect the relative energies of relevant excited states responsible for the barrier and may affect the intrinsic barrier height @,.Ic In terms of the model suggested by Saltiel and D'Agostino,* twisting rate constants are given by kobsd
kokd = K ( kT/h)a,eaS'i/Re-.W'a~lRT
where a, is a medium-related entropy term and AH*oM = AH*, + AHv. Rate constants from several laboratories have been shown to adhere to eq 3 with K exp(AS*,/R) = 0.47, a, = 0.284B,,,and AH,, = (2.85 0.39EJ kcal/mol, where E,,, the activation energy for viscous flow, is defined for each alkane by the Andrade equation.'
+
In
qs =
In A ,
+ E,/RT
t 4)
The analysis in ref 7 established that, conclusions to the contrary notwithstanding?J0 the inherent potential energy barrier to twisting AH*,= 2.85 f 0.04 kcal/mol, remains constant throughout the n-alkane solvent series. Since Hochstrasser's initial application," attempts to account theoretically for the effect of the medium on kOMhave focused on Kramers' proposal12J3 for diffusive barrier crossing. The Kramers model is based on the concept of a Brownian particle escaping over a one-dimensionalpotential energy barrier that exists in a potential energy curve that is piecewise parabolic, Figure 1. The Kramers equation derived as an approximate solution to Langevin's equation, applied to the It* Ip* rate constant gives
-
koM = (w/27r)(@/2w')([l
+ ( 2 ~ ' / 8 ) * ] -~ /I)e+o/RT ~
(5)
where Eo is the barrier height, w and w' are the frequency of the initial well and the curvature at the top of the barrier, respectively, and 8, the reduced friction coefficient, is equal to the ratio of lT, the friction coefficient for the twisting portion of the stilbene molecule, to I , its moment of inertia. It is important to realize at the outset that in applications of the Kramers equation the twisting motion of the stilbene molecule is assumed to be equivalent to translational diffusion of a molecular group of radius r along a curved pathway defined by the radius of gyration rg.
(3)
(1) For reviews see: (a) Saltiel, J.; DAgostino, J.; Megarity, E. D.; Metts, L.; Neuberger, K. R.; Wrighton, M.; Zafiriou, 0. C. Org. Photochem. 1973, 3, 1. (b) Saltiel, J.; Charlton, J. L. In Rearrangements in Ground and Excited States; de Mayo, P., Ed.; Academic: New York, 1980; Vol. 3, p 25. (c) Saltiel, J.; Sun, Y.-P. In Photochromic Materials and Systems; Durr, H., Bouas-Laurent, H., Eds.; Elsevier: Amsterdam, in press. (2) Greene, B. I.; Hochstrasser, R. M.; Weisman, R. B. J . Chem. Phys. 1979, 71, 544; Chem. Phys. 1980, 48, 289. (3) Syage, J . A.; Lambert, W. R.; Felker, P. M.; Zewail, A. H.; Hochstrasser, R. M. Chem. Phys. Lett. 1982, 88, 266. (4) (a) Syage, J. A.; Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1984, 81, 4706. (b) Felker, P. M.; Zewail, A. H. J . Phys. Chem. 1985, 89, 5402. (5) (a) Amirav, A.; Jortner, J. Chem. Phys. Leu. 1983, 95, 295. (b) Majors, T. J.; Even, U.; Jortner, J. J . Chem. Phys. 1984, 81, 2330. (6) (a) Troe, J. Chem. Phys. Lett. 1985,114,241. (b) S c h r d e r , J.; Troe, J. J . Phys. Chem. 1986, 90, 4216. (7) Saltiel, J.; Sun, Y.-P. J . Phys. Chem. 1989, 93, 6246, and references cited therein. (8) Saltiel, J.; DAgostino, J. T. J . Am. Chem. SOC.1972, 94, 6445.
0022-3654/89/2093-8310$01.50/0
i
= +,
-
or
Accordingly, the friction coefficient for twisting is related to the friction coefficient for translation, TI, by the e x p r e ~ s i o n ' ~ ~ ' ~
(9) Maneke, G.; S c h r d e r , J.; Troe, J.; Voss, F. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 896. (10) SundstrBm, V.; Gillbro, T. Ber. Bunsen-Ges. Phys. Chem. 1985.89, 222. (1 1) Hochstrasser, R. M. Pure Appl. Chem. 1980, 52, 2683. (12) Kramers, H. A. Physica (Amsterdam) 1940, 7 , 284. (13) Chandrasekhar, S. Rev. Mod. Phys. 1943, 15, 1 .
0 1989 American Chemical Society
Stilbene Photoisomerization in +Alkanes
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8311 As with the original Kramers equation, these approaches require the assumption of the hydrodynamic model for solvent f r i c t i ~ n . ’ ~ s ~A~third J ~ approach that assumes that Eovaries from alkane to alkane9~l0has been shown to be in conflict with the data.’ Most promising are approaches that assume that the macroscopic parameter vs is not a good measure of the effective viscosity experienced by the twisting molecule and thus fault the StokesEinstein approximations in the hydrodynamic model for the failure of the Kramers equation. Specifically, rotational reorientation times, 70r,17J1.30,31 and solvent self-diffusion coefficients3’ have been employed as measures of tr,and hence 8, in eq 5 , or an effective viscosity, v*, based on the empirical relationship
L
75
0
30
60
90
I20
8Figure 1. Piecewise parabolic representations of the SIexcited-state potential function for twisting about the central bond, 6 (deg), eq 28 (see text for details). Curve B is corrected for the symmetry factor of 2 for equivalent clockwise and counterclockwise twisting.
and since I = m r t , where m is the mass of the twisting molecular group, it follows that the reduced friction coefficient for rotation is taken to be equal to the reduced friction coefficient for translation of a molecule of radius r and mass m. Br =
( t r / I ) = G / m ) = Bt
(7)
Early fits of koM to eq 5 were based on the assumption14 that the friction coefficient is proportional to the shear viscosity of the solvent, os, as predicted by the Stokes-Einstein hydrodynamic equation for t,
tr = arvsrr2
(8)
where CY = 4 for the slip-boundary condition and a = 6 for the stick-boundary condition.15 Substitution of eq 8 into eq 5 gives
where A = (w/2r) and B = ( 2 1 w ’ / ~ ~ r r r=~(2mw’/arr). ) Applications of eq 9 generally assume the slip condition, a = 4,and that A and B are temperature independent so that the preexponential term is a function of vs only. Accordingly, koM in a solvent series obtained at different T values in order to maintain a constant qSvalue, have been treated by Arrhenius-type isoviscosity plots to yield Eo = 3.5 f 0.1 kcal/mol.10Jb21 The preexponential term F(7J = koM exp(Eo/R7‘) is then fitted with use of eq 9 to obtain parameters A and B. This procedure has yielded unsatisfactory fits. In particular, A and B parameters that account for the rate constants in the lower viscosity alkanes underestimate observed rate constants for the higher viscosity alkanes.I4 Several approaches have been employed in order to explain the failure of the Kramers equation in its hydrodynamic limit and obtain better fits for the kobd values. Two of these are based on refinements of the Kramers model that predict a frequency-dependent friction coefficient22-26 or extend the model to the multidimensional (14) Rothenberger, G.; Negus, D. K.; Hochstrasser, R. M. J. Chem. Phys. 1983, 79, 5360.
(15) McCaskill, J.; Gilbert, R. G. Chem. Phys. 1979, 44, 389. (16) Velsko, S. P.; Fleming, G. R. Chem. Phys. 1982, 65, 59. (17) Velsko, S.P.; Waldeck, D. H.; Fleming, G. R. J. Chem. Phys. 1983, 78, 249. (18) Sundstrom, V.; Gillbro, T. Chem. Phys. Lett. 1984, 109, 538. (19) Courtney, S. H.; Fleming, G. R. J . Chem. Phys. 1985, 83, 215. (20) Hicks, J. M.; Vandersall, M. T.; Sitzmann, E. V.; Eisenthal, E. B. Chem. Phys. Lett. 1987, 135,413. (21) Lee, M.; Bain, A. J.; McCarthy, P. J.; Han, C. H.; Haseltine, J. N.; Smith, A. B.; Hochstrasser, R. M. J . Chem. Phys. 1986, 85, 4341. (22) Grote, R. F.; Hynes, J. T.J . Chem. Phys. 1980, 73, 2715. (23) Grote, R. F.; Hynes, J. T. J . Chem. Phys. 1982, 77, 3736. (24) Hynes, J. T. In The Theory of Chemical Reaction Dynamics; Baer, M., Ed.; Chemical Rubber: Boca Raton, FL, 1985. (25) Carmeli, B.; Nitzan, A. J . Chem. Phys. 1983, 79, 393.
has been used instead of qs in eq 9.33 Actually, the failure of the Stokes-Einstein version of the Kramers equation, eq 9, is not surprising when one considers that Stokes-Einstein approximations (stick or slip) fail to account quantitatively for the qs dependence of translational diffusion coefficients or of rate constants for bimolecular diffusion-controlled reactions. On the other hand, the rate constants for diffusioncontrolled reactions are accounted for quantitatively when experimental diffusion coefficients are employed in the Smoluchowski relations hi^.^^-^^ The premise of the present paper is that the microfriction experienced by a diffusing toluene molecule is very similar to the microfriction experienced by the rotating portion of the excited stilbene molecule. With use of diffusion coefficients of toluene as the probe, a solute/solvent-specific property, the microviscosity, v,,, is defined so that, upon substitution for vs in eq 9, it gives an excellent fit for koM. Though our approach was initially intuitive, the equivalence of 6, and Pt in eq 5 shows that it is a logical consequence of the theory. Previous approaches for treating medium effects will be evaluated in comparison with our results. As the writing of this paper neared completion, a related publication3’ appeared in which experimental translation diffusion coefficients for naphthalene in hydrocarbons were shown to be equivalent measures of the friction experienced in rotational reorientation of 1,l’-binaphthyl and therefore of the friction experienced by the torsional motion leading to radiationless decay of l , l ’ - b i n a ~ h t h y l . ~It~ can be concluded that the approach described below will succeed in accounting for the medium dependence of the decay rate constants of 1,l’-binaphthyl in n-alkanes. Results and Discussion Diffusion coefficients for toluene in n-alkane solvents were calculated with the semiempirical formula of Spemol and W i r t ~ . ’ ~ This formula relates experimental diffusion coefficients, Dcxptl, to the Stokes-Einstein (stick limit) values, DSE
f = (0.16 + 0.4r/rL)(0.9 + 0.4TrL - 0.25Tr)
(12)
where f is the microfriction factor for translation, r and rL are (26) Zawadzki, A. G.; Hynes, .I. T. Chem. Phys. Lett. 1985, 113, 476. (27) Grote, R. F.; Hynes, J. T. J . Chem. Phys. 1981, 74, 4465. J. (28) van der Zwan, G.; Hynes, T. J . Chem. Phys. 1982, 77, 1295. (29) Bagchi, B.; Oxtoby, D. W. J . Chem. Phys. 1983, 78, 2735. (30) Courtney, S. H.; Kim, S.K.;Canonica, S.; Fleming, G. R. J . Chem. Sac., Faraday Trans. 2 1986, 82, 2065. (31) Kim, S.K.;Fleming, G. R. J . Phys. Chem. 1988, 92, 2168. (32) Zeglinski, D. M.; Waldeck, D. H. J . Phys. Chem. 1988, 92, 692. (33) Lee, J.; Zhu, S . B.; Robinson, G. W. J . Phys. Chem. 1987,91,4273. (34) Schuh, H. H.; Fischer, H. Helu. Chim. Acta 1978, 61, 2130. (35) Saltiel, J.; Atwater, B. W. Adu. Photochem. 1988, 14, 1, and references cited therein. (36) Fischer, H.; Paul, H. A&. Chem. Res. 1987, 20, 200. (37) Bowman, R. M.; Eisenthal, K. B. Chem. Phys. Lett. 1989, 155.99. (38) Bowman, R.M.; Eisenthal, K. B.; Millar, D. P. J. Chem. Phys. 1988, 89, 162. (39) Spernol, A,; Wirtz, K. 2.Naturforsch, A 1953, 8, 522.
8312 The Journal of Physical Chemistry, Vol. 93, No. 26, 1989
TABLE I: Selected Microfriction Factors for Toluene Diffusion in
Sun and Saltiel __
19.2,
n-Alkanes
solvent n-C1
n-C, n-Cs
n-C9
n-Go
f”
T, K 223.2 294.5 303.2 220.0 294.5 304.0 294.5 272.7 294.5 304.0 294.5 247.0 294.5 349.0 294.5 282.6 294.5 327.8 294.5 280.3 294.5 346.1 294.5 291.2 294.5 322.7
0.570 0.617 0.623 0.510 0.557 0.563 0.517 0.461 0.472 0.477 0.448 0.401 0.418 0.438 0.403 0.380 0.383 0.388 0.372 0.356 0.358 0.368 0.350 0.339 0.340 0.344
fb 0.586 (0.842) 0.529 (0.770) 0.491 (0.721) 0.457 (0.682) 0.432 (0.647)
15.21
2.4
0.402 (0.608) 0.395 (0.601) 0.383 (0.586) 0.336 (0.5 15) 0.286 (0.442) 0.277 (0.428)
I
I
3.0
T-I
3.6 x
I
4.2
I
4.8
io3 (K%--+
Figure 2. Isomicroviscosity transition-state plots of the photoisomerization rate constants of tram-stilbenein n-alkanes. Values next to the lines are q,’s (cP). The upper line represents extrapolation of all other points in the figure to qr = 0.
which is a measure of the solvent’s resistance to the motion of a macroscopic object, to microviscosity, which is a measure of the solvent’s resistance to the diffusion of a solute molecule. Accordingly, we define microviscosity as
0.266 (0.41 2)
‘Calculated with eq 9-12. bAdjusted by multiplying values from the preceding column by Dcmp/Daw for isobutane taken from ref 34. Values in parentheses are for trans-stilbene as the solute; Dcmp/Dsw values for n-C, and n-C6 were obtained by extrapolation and those for n-C9, n-C,,, n-C,,, and n-C,, were interpolated with the data for even alkanes. solute and solvent molecular radii, respectively, and the parameters TrLand Tr are reduced solvent and solute temperatures, respectively
where Tis the experimental temperature and Tbpand Tmpare the boiling and melting points of solvent for TrLand solute for T,. Molecular radii (cm) are estimated from molar volumes, V (cm3), with
L
where x = 0.74 is the space-filling factor for closest packed spheres and N is Avogadro’s number. Comprehensive studies by Fischer and co-workers on the effects of solvent and T on radical self-termination rate constant^^^-^^ have provided ample verification of the agreement between Spemol and Wirtz diffusion coefficient^,^^ D,, and Dcmpvalues based on strictly empirical equations.@ Dswvalues for isobutane diffusion in the C7 to CI2n-alkanes are generally within 5% of D,, in the 295-365 K range and increase systematically for CI4and C,, from Dsw/Dcmp = 0.8 to Dsw/Demp= 1.0 as the temperature is raised from 300 to 365 K.34,35 DSwvalues for toluene in toluene or cyclohexane deviate by no more than 16% from Dexptlin the 248-330 K range.4’ Selected microfriction factorsf calculated for toluene in the n-alkane solvents by eq 11-14 are listed in Table I. Also listed in Table I is a set offthat is obtained by adjusting the Spernol and Wirtz set by the factors Demp/Dsw for isobutane. The latter procedure assumes that D,, values for toluene will deviate from Dclptlby the factors that they deviate for isobutane. Since the factor fcorrects the macroscopic friction coefficient to a microscopic one, it can also be used to convert the shear viscosity, (40) (a) Van G e t , A. L.; Adamson, A. W. I d . Eng. Chem. 1%5,57,62. (b) Ertl, H.; Dullien, F. A. L. AIChE J. 1973, 19, 1215. (41) Lehni, M.; Schuh, H.; Fischer, H. Int. J. Chem. Kinet.1979,ll. 705.
Theoretical justification for this approach is provided by the microfriction theory of Gierer and W i r t and ~ ~ Dote ~ et al.43 It should be noted that the microviscosity in eq 15 is a solute/solvent-specific parameter whose definition does not depend on koM. In this respect, it differs from the effective viscosity q* defined in eq 10, for which adjustable parameters a and b must be based on the fitting of kohd values. Use of solvent self-diffusion coefficients (based on light-scattering relaxation times) as a measure of the microfriction experienced in the twisting process of the singlet excited state of trans-4,4’-dimethoxystilbeneresults in no significant improvement over the use of q,.32 This is not surprising since self-diffusion coefficients are similar to shear viscosity in that they reflect solvent-solvent molecular interactions and not solute-solvent interaction~.~~ Microviscosity and the Kramers Equation. Substitution of q p for 4, in eq 9 is based on the assumption that the effective viscosity experienced by the rotating portion of the excited singlet state of a stilbene molecule is equivalent to the microviscosity experienced by a toluene molecule as it undergoes translational diffusion. Since as qr or vir approaches zero eq 5 and 9 reduce to kobsd= (w/2*)@0/RT = Ae-EO/RT (16) which should be equivalent to the transition-state-theory expression for the medium-unimpeded rate constant, i.e., eq 3 with a, = 1 and AH, = 0, the parameter A is proportional to T. For the n-alkane solvent series, kow values as a function of T from the literature were tabulated in ref 7. The appropriate microviscosity form of the Kramers equation for the fitting of these data is (kobd/T) = (A’q,,/B)([l + ( B / 7 , ) 2 ] ’ / 2- l)e-Eo/RT (17) where A’ = A / T . In ref 7, Eo = 2.85 i 0.04 kcal/mol was obtained by assuming it equals AHt in the medium-enhanced twisting barrier model. The value for Eo used in previous applications of the Kramers equation is higher mainly by RT since it was based on Arrhenius plots instead of transition-state-theory plots.’ It is also questionable because Arrhenius plots were performed at constant qs despite the failure of qs to fit the kOM values by eq 9. As a first test of eq 17 and of the concept of microviscosity, the value of Eo was redetermined by performing (42) Gierer, A.; Wirtz, K.Z . Naturforsch, A 1953, 8, 532. (43) Dote, J. L.; Kivelson, D.; Schwartz, R. N. J . Phys. Chem. 1981,85, 2169. (44) Gainer, J. L.; Metzner, A. B. Transport Phenomena, Proceedings of the Symposium, London, June 16, 1965; AIChE-I. Chem. E. Symposium Series No. 6; Institute of Chemical Engineers: London, 1965; p 74.
Stilbene Photoisomerization in n-Alkanes
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8313
8’07
t
A
-
2.8,0
0.2
0.4
0.6
0.8
0
n
P
4t
- 2 0.0
0.2
0.4
0.6
0.0
1.0
1.0
Tp kP)-
Kramers fit of the photoisomerization rate constants of trans-stilbene in n-alkanes at different temperatures using toluene v,, values based on Spernol and Wirtz in eq 17: F = (kobsd/T)exp(Eo/RT). Figure 3.
a series of transition-state-theory isomicroviscosity plots, Le., plots of In (k,w/T)vs T1at constant q,,. Rate constants used in the isomicroviscosity plots are interpolated values obtained by using microfriction factors f to convert each selected q,, to qs for each solvent, calculating the corresponding T with the applicable Andrade equation, and then calculating kobd with the transition-state parameters in Table I1 of ref 7. A series of essentially parallel lines are obtained for q,, values of 0.21, 0.32, 0.47, 0.69, and 1.00 cP, having an average slope of E,, = 2.89 f 0.07 kcal/mol, Figure 2, in excellent agreement with the value obtained independently in ref 7. A satisfactory fit of the set of kobd values collected in ref 7 (excluded were values in n-CI4for T < 300 K and values in for T < 310 K due to the less satisfactory performance of the Spernol and Wirtz equation in these solvent^^^^^') to eq 17 is obtained by fixing Eo at 2.89 kcal/mol, Figure 3, with best fit parameters A ’ = 7.37 X lo9 s-l K-’ and B = 7.14 X lo-’ g cm-l s-l. Possible causes for the dispersion of the points about the calculated line are as follows: (1) small errors in T leading to significant uncertainties in q,, values and (2) uncertainties in q,, associated with the use of the Spernol and Wirtz equation to calculate the microfriction coefficient f. To reduce the impact of these difficulties, the kOMvalues reported in ref 21 at constant T, 21-23 OC, were fitted separately to eq 17 with q,,’s derived by adjusting the Spernol and Wirtz fvalues with D,,/D, ratios for isobutane as the guide (last column in Table I). An excellent fit is obtained with best fit parameters A ’ = 7.31 X lo9 s-l K-l and B = 8.21 X g cm-I s-l, Figure 4. The A’values for Figures 3 and 4 give w = 1.36 X loi3 and 1.35 X loi3 s-l, respectively, at 294.5 K, which are indistinguishable from w = 1.36 X loi3and 1.34 X lo1’ s-l calculated independently in ref 21 and 30, respectively, by fitting k d values to the Kramers-Hubbard equation with stilbene reorientation times as the measure of microfriction. Since w/2r corresponds to (KkT/h) exp(AS*,/R), the transition-state preexponential term for medium-unimpeded barrier crossing, the derived parameter w was used to fix the intercept for the q,, = 0 line in Figure 2. Prior to the intercept adjustment, the points on each qr # 0 line were merged into a common line by shifting them vertically by the separation of the lines at 283 K, roughly, the median-applicable T. The slope of the resulting q,, = 0 line gives Eo = 2.89 f 0.02 kcal/mol. On the basis of the assumptions made to derive eq 17, w’ = 1.39 X lOI3 and 1.59 X lo1’ s-l follow from 3rrBlm with the B values from Figures 3 and 4, respectively, with r = 3.15 X 10“ cm from g for toluene. Since the definition eq 14, and m = 1.53 X offin eq 11 is based on the stick boundary condition, cy = 6 had to be used in the calculation of w’. To compare our w’ values with the corresponding parameter based on the Kramers-Hubbard equationF1-Mwe must consider the definition of the reduced friction coefficient @ employed in the latter treatment. Derivation of the Kramers-Hubbard equation is based on the assumption that the
Kramers fit of the photoisomerization rate constants of trans-stilbene in n-alkanes at constant temperature*’ with s7; for toluene given in the last column of Table I, see text. Figure 4.
effective reduced friction coefficient, &, experienced by the rotating portion of the stilbene molecule is proportional to the effective reduced friction coefficient, Po,, experienced by the entire stilbene molecule as it undergoes rotational reorientation, where poris the proportionality constant. As shown in eq 18, Po, is related to experimentally measured stilbene, rotational reorientation times, T,,, by the Hubbard relati0n,2l*~~ where I,, = 3.4 X g cmz was used as the moment of inertia of the rotating stilbene m o l e ~ u l e . Two ~ ~ *independent ~~ treatments of k,w values at constant T i n n-alkanes using the Kramers-Hubbard relationship have yielded remarkably similar (w’/p,,) values of 3.42 X 10” SKI(ref 21) and 3.43 X 1013s-l (ref 30). The ratio (w’/p,,) from ref 21 and the 0’ value from our treatment of the Same data gives por= 2.2, which probably reflects mainly the difference in the moments of inertia for rotational reorientation of the entire molecule and for twisting of only half of it. Microviscosity and Rotational Reorientation Times. Since both stilbene rotational reorientation times and toluene translational diffusion coefficients have been demonstrated to be satisfactory probes of the microscopic environment experienced by the twisting portion of the excited stilbene molecule, we are encouraged to examine the relationship of 7,, and q,,. Pertinent literature concerning the dependence of T,, on shear viscosity has been reviewed in ref 21. It is given by the modified Stokes-Einstein-Debye equation where V,, is the hydrodynamic volume of the molecule,fand C are solute- and solvent-specific correction factors, and 7,:, the free-rotor correlation time, is given by 1,;
= (Io,/kT)l~z
(20)
Plots of T,, vs qs at constant T i n a series of solvents show pronounced deviation from linearity,2’-30reflecting in part the dependence of C on the solvent. This is also evident in plots of T , ~ vs ( q s / T ) that, though linear in a single solvent, exhibit solvent-specific slopes,31in much the same way as do plots of D,’s vs (T/q,).34 Since the factors fC in eq 19 have the same purpose as the Spernol and Wirtz microfriction factorf, the assumption that rotational and translational diffusion are similarly impeded by the medium allows substitution of q,, for qJC in eq 19. Spernol and Wirtzfvalues for trans-stilbene were calculated with eq 12-14 cm based on the density of trans-stilbene with r = 3.79 X liquid at its mp.46 The plot of T ~ in, n-alkanesZ1vs q,,, Figure 5, is linear, 9 = 0.996, with intercept = 2.65 f 1.22 ps and slope (45) Hubbard, P. S. Phys. Rev. 1963, 131, 1155. (46) Saltiel, J.; Shannon, P. T.; Zafiriou, 0. C.; Uriarte, A. K. J . Am. Chem. SOC.1980, 102, 6799.
8314 The Journal of Physical Chemistry, Vol. 93, No. 26, 1989
Sun and Saltiel
l - - - - - r 6 ' 32 f
2 3 . 3 1 1
22.41 -2.3
I
-1.8
I
-1.3
I
-0.8
I
-0.3
I
0.2
In ~ ~ ( c P ) d Figure 5. Dependence of trans-stilbene rotational reorientation times2' on qM based on D,, for trans-stilbene. Dashed line is based on qr for toluene.
= 55.7 f 1.5 ps/cP. These parameters are in remarkable agreement with :,1 = 2.89 ps calculated from eq 20 with I,, = 3.4 X cm2 g from ref 21, and Vh/kT = 56.1 ps/cP at 294.5 K based on the hydrodynamic radius of stilbene. The linear dependence of rotational reorientation times on microviscosities based on empirical translational diffusion coefficients is consistent with the premise of this paper, namely, that rotational and translational molecular motions experience similar microfriction in a specific medium. Moreover, the quantitative adherence of T , ~ to eq 19 provides strong evidence that applicable microviscosities for these motions are not just proportional but are equivalent. Extrapolation of this conclusion to the twisting motion of an isomerizing molecule implies that the proportionality constant p,,, would be unity provided that the volume of the twisting portion of the molecule is equivalent to the volume of the probe molecule whose translational or rotational diffusion is employed as an empirical measure of microviscosity. We expect that T,, values for toluene would lead to a more satisfactory application of the Kramers-Hubbard equation to stilbene isomerization. To illustrate the volume equivalence requirement, a plot of T , ~for tram-stilbene vs v,, based on translational diffusion coefficients of toluene is also shown in Figure 5 . Though the linearity of this plot is just as acceptable 9 = 0.996, its intercept and slope, 1.64 f 1.25 ps and 87.4 f 2.4 ps/cP, respectively, are not in as good agreement with the values obtained from eq 19 and 20 for trans-stilbene. Microviscosity and the Medium-Enhanced Barrier Model. Fischer and co-workers first proposed a linear dependence of In koM on In vs in accounting for the medium dependence of 6,for trans-stilbene and related compound^.^^^^* Their approach was based on the free-volume theory for viscosity that relates qs to V,, the mean free volume of the s o l ~ e n t ~ ~ * ~ ~ qs =
qoeVdvf
(21)
where Vo, the critical free volume for molecular translational diffusion, and the preexponential factor vo are solvent-specific properties. The additional free volume required to allow rearrangement of a solute was assumed to be a fraction a -< 1 of Vo, and the rate constant for a medium-dependent rearrangement was written as koM = k,e-aVo/vr (22) where k, is the limiting rate constant at zero ~iscosity4~ Combining eq 21 and 22 gives In kobsd= In k, + a In qo - a In
qS
(23)
(47) Gegiou, D.; Muszkat, K. A.; Fischer, E. J. Am. Chem. Soc. 1968, 90, 12. (48) Sharafi, S.; Muszkat, K. A. J . Am. Chem. Sor. 1971, 93, 4119. (49) Doolittle, A. K. J. Appl. Phys. 1951, 22, 1471. (50) Cohen, M. H.; Turnbull, D. J . Chem. Phys. 1959, 31, 1164.
Figure 6. Adherence of kowvalues to eq 2 5 .
Using the medium-enhanced barrier model, Saltiel and D'ago st in^^^^ derived an equation having the same form as eq 23 by combining eq 3 and 4 In koM = In k, + In a, + (AH,/E,,) In A , -(AH,/E,) In q, (24) In eq 24, a = AH,/E,, is explicitly defined as the fraction of the activation energy of viscous flow that gives AHv, the enthalpy increment by which the medium augments the intrinsic barrier for solute rearrangement. Saltiel and DAgostino emphasized that, provided that the temperature dependence of vsobeys the Andrade relationship, eq 24 should give the dependence of kW on vs when kohd values are determined in a single solvent as a function of temperature.8 However, different solvents have different A,, values, and eq 24 cannot be expected a priori to account for the dependence of koM on q, when their variation is achieved at constant temperature by changing the solvent. Nonetheless, for small solvent changes and constant T, the tangential use of In k, vs In q, plots was proposed and demonstrated for the glycerol/water solvent system.s Since vo also varies from solvent to solvent,49 Velsko and Fleming similarly recognized that eq 23 could not properly be applied to data obtained in different s~lvents.~l Empirically a good linear relationship between In koM and In q, was obtained by Fleming and co-workers for trans-stilbene in the n-alkane solvent series at constant T.52353 However, the significance of this relationship is unclear since it gives a = 0.32, which is smaller than a = 0.39 f 0.02 obtained independently with the medium-enhanced barrier model by plotting AH*obsdfrom transition-state plots vs E,.7 In view of the success of 1 , in the Kramers equation fits of kobd,we were encouraged to examine the possibility that it might serve as the proper measure of viscosity in eq 23 and 24. The plot of In kobsd2'vs In q,,, Figure 6, is reasonably linear, giving a = 0.40 f 0.02 in excellent agreement with the expected value.' In principle, interpretation of this plot could be based on an Andrade treatment of vMvalues, which would provide E , and A , parameters for substitution in eq 24. Since the temperature dependence of experimental D,'s for toluene in the n-alkanes is not known, this approach could be based on calculated Spernol and Wirtz microfriction factorsf. However, the T dependence off predicted by eq 12 is weak and is within the acceptable uncertainty limits ofJ suggesting that the following procedure may be considered more reliable. Addition and subtraction of a In f to eq 24 gives In kobd = In k,
+ In a, + a In (fA,) - a In vfi
(25)
The adherence of kOMvalues to eq 25 implies that changes in a In VA,,) are nearly exactly compensated by changes in In a,. (51) Velsko, S. P.; Fleming, G. R. J . Chem. Phys. 1982, 76, 3553. (52) Courtney, S. H.; Fleming, G. R. J . Chem. Phys. 1985, 83, 215. (53) Fleming, G. R.; Courtney, S. H.; Balk, M. W. J . Stat. Phys. 1986, 42. 83.
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8315
Stilbene Photoisomerization in n-Alkanes
where w1 = w and w2 are the well frequencies of It* and Ip*, respectively, 8 , and 82 are the dihedral angles at the barrier and at Ip*, Eo is the barrier height for It*+lp* twisting, Etpis the energy difference between the two minima, and 4, and are dihedral angles at the connecting points of the three parabolic segments. Since the potential energy function is continuous at the joints, it follows that at 8 = +1, VI = V2and aVl/a8 = aV2/a8 and, at 8 = 42, V2 = V, and dVz/a8 = (aV3/a6). These relationships give
I
,
= W’2/(WI2
I
0.31 0.3
-
’O
I
0.5
I
0.7
I
I 1.1
0.9
-aln(A,f/c)
+ w’2)
(29a)
I
1.3
Figure 7. Compensation between entropy terms in eq 25.
Eyring’s approximate equation for qs, based on reaction rate theory, can be used to show that the plot in Figure 7 represents a linear entropy r e l a t i ~ n s h i p . Replacing ~~ the ratio of partition functions in Eyring’s equation with exp(-AS*,,/R) and the height of the potential energy barrier with AH*:$ and multiplying both sides of the equation by f give ‘lr = W h / V exp(-G*,/R)
exp(AH*,/RT)
(26)
where V is the molar volume of the solvent. Since f is nearly independent of T, it can be assumed to correct primarily the preexponential and entropy parts of Eyring’s equation. It follows that E, = E,,, = AH*,, and that In (fA,) = In (flvh/V) - AS*,/R. In the absence of information that allows partitioning off into corrections for changes in V and in AS, separately, we assume that
fA, N c exp(-AS’,/R)
(27)
where c is roughly a constant for the n-alkane solvent series. It follows that In (fA,) - In c = -AS*,/ R where AS is the activation entropy for microviscous flow. Since In a, = d V / Rwhere AS, is the medium-imposed entropy increment for motion of the solute, it can be seen that the plot in Figure 7 is equivalent to hs, = UAS,,~. The value of c was obtained as the intercept of the In a, vs -a In ($A,,) plot with a fixed slope of unity. The linear relationship between In k,, and In q,, for It* twisting in n-alkanes, Figure 6, has thus been traced to the existence of a true linear free energy relationship, AGv = aAG*,, , where AGv is the medium-imposed free energy change on sofute motion and AG*,,#is the free energy of activation for microviscous flow. It can now be seen that eq 23 cannot account for the relationship in Figure 6 because it lacks a term analogous to In a, in eq 24 and 25. This situation can readily be remedied by including a medium-specific proportionality constant m in eq 22: kobd = k,me-aVo/vf. Potential Energy Curve for It* Ip* Twisting. The approximate potential energy curves in Fig 1 were based on the equations used by Millar and Eisenthal in deriving potential energy curves for the SIstate of l,l’-binaphthyl.s6 As assumed in the derivation of the Kramers equation, the potential surface is taken as harmonic in the well, It*, and barrier regions. In addition, we assume that the surface is harmonic in the ‘p*-well region. The potential energy of the surface, V(8),is calculated relative to the energy of It* that for this purpose is set equal to zero
-
vl(e)= (i/4)zw,2e2 v2(e)= E , - (i/4)rw’2(e - e# v3(e)= ,tztp+ (i/4)zw;(e - e,)2
Thus, the Kramers fit parameters w 1 = w and a’,together with Eo = 2.89 kcal/mol from the isomicroviscosity plots, allow calculation of & and 81 from eq 29a and b, respectively, and the reasonable assumption that 82 = ~ / would 2 allow calculation of 82 and w z from eq 29c and d if Etpwere known. We note that our eq 29b and d differ by a factor of 2 from the corresponding eq 12a and b in ref 56. The origins of the well at lp* and the maximum in the ground-state surface at Ip have been attributed to an avoided crossing produced by the correlation of the ground-state lA, states with the third excited (A state^.^'-^^ For trans-stilbene, the latter has been assigned58to t i e most intense two-photon band in the thermal lensing spectrum at 122 kcal/m01.~~We consider this value an upper limit since the origin of the 4IA 1‘A transition is not known. Assuming that the same energy difference applies for the cis isomer and taking into account the -4.6 kcal/mol higher energy content of I C (ref 60) allows estimation of the crossing-point energy as 63.4 kcal/mol at 82 = ~ / 2 .If we further assume that the energies ‘p and Ip* are symmetrically disposed about this energy, the known energy of ‘p of -48 kcal/molbO permits the estimation of the energy of Ip* as -78.8 kcal/mol. Since the energy of It* is 88 kcal/mol,61 we take EtP= -9.2 kcal/mol as a rough upper estimate (lower in the absolute sense) of this parameter. Curve A in Figure 1 was based on the average values of the parameters for Figures 3 and 4, i.e., wI = 1.35 X l O I 3 s-I and w’ = 1.49 X lOI3 s-I. It corresponds to 81 = 16.4O, + I = 9.0°, 4 . ~ ~ = 23.3*, and w2 = 4.8 X 10I2 s-I. The use of w1 = 1.35 X 10” s-l assumes that the equilibrium geometry of trans-stilbene in the excited state is not planar62v63so that there is always a bias for either clockwise or counterclockwise rotation, depending on the distortion initially present.I4 This assumption is questionable in view of a series of recent spectroscopic studies that indicate planar equilibrium geometries for trans-stilbene in its ground and lowest excited singlet states.6448 There is also ample experimental
-
(57) Orlandi, G.;Siebrand, W. Chem. Phys. Lett. 1975, 30, 352. (58) Hohlneicher, G.;Dick, B. J . Photochem. 1984, 27, 215. (59) Fuke, K.;Sakamoto, S. A.; Ueda, M.; Itoh, M. Chem. Phys. felt. 1980. 74. 546. (60) Saltiel, J.; Ganapathy, S.;Werking, C. J. Phys. Chem. 1987,91,2755. (61) Dyck, R. H.; McClure, D. S. J . Chem. Phys. 1962, 36, 2336. (62) Traetteberg,M.; Frantsen, E. B.; Mijlhoff, F. C.; Hoekstra, A. J. Mol.
--.
26. -57.-Strurt .. .._ .. 1975. -- 1
0 5 41
(284
dl 5 e I42 (28b) (28~) e 2 b2
(54) Eyring, H. J . Chem. Phys. 1936, 4, 283. (55) Kierstead, H. A,; Turkevich, J. J . Chem. Phys. 1944, 12, 24. (56) Millar, D. P.; Eisenthal, K. B. J . Chem. Phys. 1985, 83, 5076.
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(63) Warshel, A. J . Chem. Phys. 1975,62, 214. (64) Myers, A. B.; Trulson, M. 0.;Mathies, R. A. J . Chem. Phys. 1985, 83, 5000. (65) Suzuki, T.; Mikami, N.; Itoh, M. J . Phys. Chem. 1986, 90, 6431. (66) Baskin, J. S.;Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1987,86, 2483. (67) Spangler, L. H.; van Zee, R. D.; Zwier, T. S.J . Phys. Chem. 1987, 91, 2782. (68) Spangler, L. H.; van Zee, R. D.; Blankenspoor, S. C.; Zwier, T. S. J . Phys. Chem. 1987, 91, 6077.
J . Phys. Chem. 1989, 93, 8316-8323
8316
or counterclockwise rotation, thus introducing a symmetry factor evidence showing that upon excitation the aryl-vinyl bonds in 1,2-diaryl olefins assume more double-bond c h a r a ~ t e r , di~ ~ . ~ ~ of 2 above the usual single barrier transition-state rate constant.'I minishing the amplitude of torsional excursions about these bonds. We have, accordingly, adjusted w , to 6.75 X 10l2s-l and repeated In fact, as drawn in Figure 1, the potential energy curves with the calculations to obtain curve B in Figure 1 with 8, = 26.6O, 41= 22.1°, +2 = 34.7O, and w2 = 5.7 X 10l2s-I. We present these the well minimum at Oo torsional angle about the central bond are consistent with a planar It* geometry. From a planar curves as approximate guides for future quantum mechanical equilibrium geometry, isomerization is equally likely by a clockwise calculations. (69) For reviews, see: (a) Scheck, V. B.; Kovalenko, N. P.; Alfimov, M. V. J . Lumin. 1977, 15, 157. (b) Fischer, E. J . Photochem. 1981, 17, 331. (c) Fischer, G.; Fischer, E. J . Phys. Chem. 1981, 85, 261 1. (d) Mazzucato, U. Pure Appl. Chem. 1982, 54, 1705. (70) Sun, Y.-P.; Sears, D. F., Jr.; Saltiel, J.; Mallory, F. B.; Mallory, C. W.; Buser, C. A. J . Am. Chem. Sor. 1988, 110, 6974.
Acknowledgment. This research was supported by NSF Grant C H E 87-13093. Registry No. trans-Stilbene, 103-30-0;toluene, 108-88-3. (71) Ladanyi, B. M.; Evans, G. T. J . Chem. Phys. 1983, 79, 944.
Shapes of the Electron-Transfer Rate vs Energy Gap Relations in Polar Solutions Akira Yoshimori, Toshiaki Kakitani,* Yoshitaka Enomoto, Department of Physics, Nagoya University, Chikusaku, Nagoya 464-01 Japan
and Noboru Mataga* Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560 Japan (Received: June 22, 1989; I n Final Form: September 1 1 , 1989)
Taking into account the solvent motions classically and adequately defining the reaction coordinate for the electron transfer, we have derived a formula which relates the free energy curve of the initial state to the energy gap law. Using this relation, we have proved the following symmetrical properties of the energy gap law in the case where the role of the electronic polarizability can be neglected. The energy gap law of the charge separation reaction is symmetric with a maximum at the energy gap equivalent to the reorganization energy. The energy gap laws of the charge recombination and charge shift reactions are symmetric when a linear response of the solvent polarization applies but they can be asymmetric when a nonlinear response due to the dielectric saturation applies. It has been demonstrated that even if these symmetric restrictions exist, the overall shape of the electron-transfer rate vs energy gap relation is greatly modified when the dielectric saturation effect is increased. It is shown that the result obtained within this theoretical scheme is not inconsistent with the experimental results by Rehm and Weller for the charge separation reaction. The possibility of realizing a strong dielectric saturation in polar solutions is examined by the Monte Carlo simulation.
Introduction The free energy gap dependence of the electron-transfer rate (energy gap law) in polar solution has been investigated extensively since it is considered as one of the most important mechanisms to regulate the rate. Marcus developed a macroscopic theory within the framework of the linear He predicted a parabolic (bell-shaped) form of the energy gap law: a sharp increase of the rate in the small energy gap region and a sharp decrease of the rate in the large energy gap region. The former and the latter are then called the normal region and the inverted region, respectively. Many experiments have been carried out for the purpose of examining this theoretical prediction. As a result, a remarkable inverted region6,' and bell-shaped energy gap law7 were observed for the charge recombination reaction (A*.-BT A-B), and a considerable inverted region for the charge shift A-.B*).8 However, for the photoinduced reaction (A'-B charge separation reaction (A*-.B A*--BF), no trace of the inverted region was observed within an energy gap as large as 2.5
-
-
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( I ) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. (2) Marcus, R. A. J . Chem. Phys. 1956, 24, 979. (3) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, IS, 155. (4) Marcus, R. A. J . Chem. Phys. 1965, 43, 679. ( 5 ) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acro 1985, 811, 265. (6) Wasielewski, M. A.; Niemczyk, M. P.; Svec, W. A.; Pewitt, E. B. J . Am. Chem. SOC.1985, 107, 1080. (7) Mataga, N.; Asahi, T.; Kanda, Y . ;Okada, T.; Kakitani, T. Chem. Phys. 1988, 127, 249. (8) Miller, J. R.; Calcaterra, L. T.; Closs, G. L.J . Am. Chem. SOC.1984, 106, 3047.
eV which has been examined h i t h e r t ~ . ~The possibilities of participation of the excited state of ions and of formation of a nonfluorescent charge-transfer complex at the large energy gap in the charge separation reaction were examined by the recent elaborate experiment,1° indicating that the nonexistence of the inverted region up to 1.6 eV in the charge separation reaction is an intrinsic property of the electron-transfer reaction in polar solution. The interpretation of the energy gap law in the range 1.6-2.5 eV still remains controversial. It may be possible that the distribution of the distance between the donor and acceptor molecules considerably modifies the energy gap law. Even in such a case, it can be said that the apparent energy gap law changes drastically, depending on the type of the electron-transfer reaction. In accordance with this, we found a similar property in the experimental data of electrochemical reactions. In many of the A*-.M, M representing a metal ionization reactions (A-M electrode), the transfer coefficient becomes smaller than 0.5 while it is larger than 0.5 in most of the neutralization reactions (AZ-M A'-'.-M, z > O)." Under these situations, we showed that all of the qualitative features of the above experimental results can be reproduced if we introduce a new type of potential that the force constant of the reorientation of polar solvent molecules around a neutral solute molecule is much smaller than that around a charged solute In order to obtain a physical basis of a large change
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(9) Rehm, D.; Weller, A. Isr. J . Chem. Phys. 1975, 63, 4358. (10)Mataga, N.; Kanda, Y.; Asahi, T.; Miyasaka, H.; Okada, T.;Kakitani, T. Chem. Phys. 1988, 127, 239.
0022-3654/89/2093-83 16$01.50/0 0 1989 American Chemical Society
