J. Phys. Chem. 1991, 95, 10336-10344
10336
2-naphthol (2.05 g, 14.2 mmol), and 40% dimethylamine (2.03 mL, 16 mmol) was added 0.469 g (14.9 mmol) of paraformaldehyde. The mixture was allowed to stir for 1 h at room temperature. The solvent was removed by rotary evaporation, and the resulting oily solid was dissolved in dichloromethane, washed with water, and extracted with 2 X 30 mL portions of dichloromethane. The extracts were combined, dried over magnesium sulfate, filtered, concentrated, and dried under reduced pressure for 1 h to give l-[(dimethylamino)methyl]-2-naphthol(1 DMN2) (2.55 g, 12.7 mmol, 85%) as a pink solid (mp 75-76 "C): N M R (CDCl3, 300 MHz) b 2.41 (s, 6 H), 4.09 (s, 2 H), 4.80 (s, 1 H), 7.09-7.12 (d, 1 H, J = 8.85 Hz), 7.25-7.31 (dd, 1 H, J = 7.76, 7.14 Hz), 7.40-7.46 (dd, 1 H, J = 7.14, 7.02 Hz), 7.67-7.70 (d,
1 H, J = 8.86 Hz), 7.75-7.77 (d, 1 H, J = 8.24 Hz), 7.80-7.83 (d, 1 H, J = 8.67 Hz). The material was further purified by high-pressure liquid chromatography on a silica gel column using acetone for 15 min followed by 50% acetonepetroleum ether. The product was obtained as a colorless crystalline solid, mp 75-76 OC.
Acknowledgment. Support of this research by the National Science Foundation through Grant CHE-8805577 is gratefully acknowledged. Registry No. DMBA, 103-83-3; N2, 135-19-3; N2M, 93-04-9; 1 DMN2, 5419-02-3; dimethylamine, 124-40-3; paraformaldehyde, 30525-89-4.
Application of the Medium-Enhanced Barrier Model to the Photoisomerization Dynamics of Substituted Stilbenes in n-Alkane Solvents Ya-Ping Sun,*.+Jack Saltiel,*,t N. S. Park,! E. A. Hoburg,§ and David H. Waldeck*?s Department of Chemistry, The University of Texas a t Austin, Austin, Texas 78712-1 167, Department of Chemistry, The Florida State University, Tallahassee, Florida 32306-3006, and Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (Received: March 15, 1991)
Photoisomerization rate constants for trunr-4,4'-dimethylstilkne, trum-4,4'dimethoxystilbene,and truns-4,4'-di(tcrt-butyl)stilbene in n-alkane solvents are treated with the transition state theory equation (Eyring equation) and the medium-enhanced barrier = AH',+ aE,,. The a value, a measure of the effect of medium viscosity, increases in the order of stilbene, model, AH dimethylstilbene, dimethoxystilbene, in contrast to results obtained by applying the power law kobsd= k,B/q;. The medium-enhanced activation enthalpies and entropies for photoisomerization of the stilbenes obey the isokinetic relationship, and so do the corresponding viscous flow parameters, in the n-alkane solvent series. These relationships show that the apparent slope from the logarithmic form of the power law plot for n-alkane solvents is not equal to -a but is a function of -a. As previously concluded, microviscosity should be used in the power law because when shear viscosity is used the n-alkane series does not behave as a continuous medium with respect to the twisting groups. The results presented further support the conclusion that the intrinsic energy barriers for photoisomerization of stilbene and its derivatives are constant in the n-alkane solvent series. The relationship between the potential energy barrier height, the Arrhenius activation energy, and the activation enthalpy is discussed.
Introduction The photoisomerization of the stilbenes has been studied extensively for several decades.l It has been well established that photoisomerization of tram-stilbene by direct excitation proceeds through its excited singlet states.* The initial step of the isomerization is subject to an intrinsic potential energy barrier and corresponds to twisting of planar excited trans-stilbene, It*, to a perpendicular state, Ip*. In supersonic jet expansions, the barrier height for the isolated molecule was determined to be 3.4 f 0.3 kcal/mol from the excess excitation threshold energy required for radiationless decay (i~merization).~Alternative values of 2.64a5b and 3.7 f 0.2 kcal/mo14"sdhave been based on the optimal fitting of the RRK and RRKM theories to the excess energy dependence of fluorescence quantum yields and decay rate constants, respectively. The isomerization of stilbene can be viewed as a large-amplitude twisting motion about the central bond. In solution, this motion depends strongly on medium friction. Therefore, tram-stilbene has been used ~ i d e l y ' ' * ~as- ~the prototypic system with a high intrinsic energy barrier ( > k T ) for the study of medium viscosity effects on reaction rate constants, a subject that has been attracting great attention. A generally applied theoretical model in accounting for medium effects is based on the concept of a Brownian particle escaping over a one-dimensional potential energy barrier that exists in a 'The University of Texas at Austin. $The Florida State University. 8 University of Pittsburgh.
0022-3654/91/2095-10336$02.50/0
potential energy curve that is piecewise parabolic. It gives rise to the Kramers e q u a t i ~ n which * ~ ~ ~is derived as an approximate solution to the Langevin equation. The Kramers equation is given by koM = (w/27r)(p/2w')([l ( 2 ~ ' / p ) ~ ] '-/ ~lJe-Eo/RT (1)
+
(1) (a) Saltiel, J.; D'Agostino, 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 Photochromism, Molecules and Systems; Durr, H., Bouas-Laurent, H., Eds.; Elsevier: Amsterdam, 1990; p 64 and references therein. (d) Waldeck, D. H. Chem. Rev. 1991, 91, 415. (2) (a) Saltiel, J. J. Am. Chem. SOC.1967,89, 1036; 1968, 90,6394. (b) Orlandi, G.; Siebrand, W. Chem. Phys. Lett. 1975.30, 352. (c) Hochstrasser, R. M. Pure Appl. Chem. 1980, 52, 2683. (3) (a) Syage, J. A,; Lambert, W. R.; Felker, P. M.; Zewail, A. H.; Hochstrasser, R. M. Chem. Phys. Lett. 1982,88, 266. (b) Syage, J. A,; Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1984, 81, 4706. (c) Felker, P. M.; Zewail, A. H. J. Phys. Chem. 1985, 89, 5402. (4) (a) Amirav, A.; Jortner, J. Chem. Phys. Lett. 1983, 95, 295. (b) Majors, T. J.; Even, U.: Jortner, J. J. Chem. Phys. 1984,81, 2330. (c) Troe, J. Chem. Phys. Lett. 1985, 114, 241. (d) Schroeder, J.; Troe, J. J. Phys. Chem. 1986, 90, 4215. ( 5 ) (a) Saltiel, J.; Sun, Y.-P. J . Phys. Chem. 1989, 93, 6246. (b) Sun, Y.-P.; Saltiel, J. J. Phys. Chem. 1989, 93, 8310. (6) Park, N. S.; Waldeck, D. H. J. Chem. Phys. 1989, 91, 943. (7) Schroeder, J.; Schwarzer, D.; Troe, J.; Voss, F. J . Chem. Phys. 1990, 93, 2393. (8) Kim, S . K.; Courtney, S. H.; Fleming, G. R. Chem. Phys. Leu. 1989, 159, 543. (9) Kramers, H. A. Physica 1940, 7, 284.
0 1991 American Chemical Society
Photoisomerization Dynamics of Substituted Stilbenes
The Journal of Physical Chemistry, Vol. 95, No. 25, 1991
where Eo is the barrier height, w and w’ are the frequency of the initial well and the frequency corresponding to the curvature at the top of the barrier, respectively, and /3, the reduced friction coefficient, is equal to the ratio of [,, the friction coefficient for the twisting portion of the stilbene molecule, to I , its moment of inertia. Since the friction coefficient f , is not an experimentally observable quantity, a “bridge” is required to connect it to an observable quantity, usually the viscosity of the medium, q. In many studies this is accomplished by assuming f r 0: q. in accord with the Stokes-Einstein hydrodynamic equation. When shear viscosity, qs,is used in the approximation, the Kramers equation fails, almost invariably, in accounting for medium effects on twisting rate constants of trans-stilbeneI0 and other moleculesIcJl in the n-alkane solvent series; rate constants predicted by the equation decrease with increasing viscosity faster than experimental rate constants. This failure has resulted in a wide range of discussions concerning both the Kramers equation and the “bridge”, the Stokes-Einstein equation. One of several modifications to the Kramers-Stokes-Einstein approach is the Kramers-Hubbard in which the effective reduced friction coefficient, [,, experienced by the twisting portion of the molecule is assumed to be proportional to the effective reduced friction coefficient, tor, experienced by the entire molecule as it undergoes rotational reorientation. Use of experimental rotational reorientation times, T~,,obtained under the same conditions as photoisomerization rate constants, leads to generally improved fits of experimental twisting rate constants of trans-stilbene and other molecules to the Kramers-Hubbard e q ~ a t i o n . l ’ ~ ~ The ~ ~ ’ im~-’~ plication is that the failure of the Kramers-Stokes-Einstein approach is primarily due to the use of shear viscosity in the Stokes-Einstein equation as was also recognized in a series of insightful papers by Robinson and co-workers.I6 They defined an effective viscosity, q* = [ ( a / q s ) b ] - ’ , where a and 6 are specific parameters for each solute in each solvent series.16* Substitution of q* for qs in the Kramers-Stokes-Einstein equation introduces a third adjustable parameter and gives very good fits for the medium dependence of rotational isomerization rate constants of excited molecules in homologous solvent series.16a However, this approach does not explicitly account for differences in specific solute-solvent interactions as the solvent is changed and requires the isomerization rate constants in order to define q* through fitting.Ic The inadequacy of using the Stokes-Einstein hydrodynamic equation as a “bridge” for the Kramers equation in the n-alkane solvent series was addressed in our (SS) recent study of transstilbene photoisomerization in n - a l k a n e ~ in , ~ which ~ the isomerization process in stilbene was approximated as translational diffusion, D,, of a toluene molecule along a curved pathway. The relationship of the T ~ and , D, approaches for the measurement of microfriction was also considered, and it was concluded that the two are essentially equivalent provided that the volumes of the diffusing moieties are identical.5b It was shown that the failure of the Kramers-Stokes-Einstein equation in the n-alkane solvent series is inevitable due to the fact that although translational diffusion coefficients and rotational reorientation times in many cases are proportional to viscosity in each n-alkane solvent, the proportionality constants differ from solvent to solvent in the
+
(IO) Rothenberger, G.;Negus, D. K.; Hochstrasser, R. M. J . Chem. Phys. 1983, 79, 5360. (11) (a) McCaskill, J.; Gilbert, R. G.Chem. Phys. 1979, 44, 389. (b) Velsko, S. P.; Waldeck, D. H.; Fleming, G. R. J . Chem. Phys. 1983, 78, 249. (c) Zeglinski, D. M.; Waldeck, D. H. J . Phys. Chem. 1988, 92, 692. (12) Hubbard, P. S. Phys. Rev. 1963, 131, 1155. (13) Courtney, S. H.; Kim, S. K.; Canonica, S.; Fleming, G.R. J . Chem. Soc., Faraday Trans. 2 1986, 82, 2065. (14) 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. (15) Bowman, R. M.; Eisenthal, K. B.; Millar, D. P. J . Chem. Phys. 1988, 89, 762. (16) (a) Lee, J.; Zhu, S.-B.; Robinson, G. W. J . Phys. Chem. 1987, 91, 4273. (b) Zhu, S.-B.; Lee, J.; Robinson, G. W. J . Chem. Phys. 1988,88,7088. (c) Robinson, G. W.; Singh, S.; Krishnan, R.; Zhu, S.-B.; Lee, J. J . Phys. Chem. 1990.94.4. (d) Zhu, S.-B.; Lee, J.; Robinson, G . W. J . Phys. Chem. 1991, 95, 1865.
10337
series.”-I9 A common proportionality constant for the series can be obtained if microviscosity, qq
is used, where f = DSE/De, I is the microfriction factor, and DSE and De, I are diffusion coefhcients calculated from the stick limit of the itokes-Einstein equation and based on experiment, respecti~ely.l’.~~ Twisting rate constants of trans-stilbene in the n-alkane solvent series were fit very well by the Kramers-microviscosity approach using reasonable parameters.5b Although the Kramers-microfriction approach, which accounts for specific solvent-solute interaction by using either rotational reorientation times or microviscosities based on translational diffusion coefficients as a measure of medium friction, fits experimental rate constants quite well, it can reasonably be argued that a successful least-squares fit does not prove the validity of the Kramers equation. Nor does a good fit diminish the significance of other improvements on the Kramers model, such as the frequency-dependent friction approach20,21or its extension to a multidimensional model.229z3 Actually, the equation based on the frequency-dependent friction also fits experimental twisting rate constants of trans-stilbene in n-alkane solvents successfully.I0 However, until the multidimensional and the frequency-dependent friction models are evaluated based on microscopic measures of friction, the physical meaning(s) of the fitting parameters will remain problematic. It is likely that the success of the microfriction approach rests, in part, on the proximity of the time scales of the microfriction probe and the isomerization process.lsc Discussions of the failure of the Kramers-Stokes-Einstein equation also renewed interest in semiempirical equations which were used in the earlier studies of medium effects on stilbene photoisomerization.z4-z6 A frequently discussed equation is the power l a ~ ~ ~ 3 ~ ~ (3) where kt is the limiting rate constant at zero viscosity, and B and a are assumed to be constants with 0 I a I l.5b Equation 3, initially proposed as a semiempirical expression based on the free volume was also derived based on the medium-enhanced barrier modelz5and was recently reintroduced26 as an empirical modification of the Smoluchowski limit of the Kramers equation. Applications of the power law to the twisting rate constants of trans-stilbene and other molecules in n-alkane solvents appeared to be generally successful with plots of In kow vs In qsbeing linear at constant temperature. In the free volume theory, the additional free volume required to allow rearrangement of a solute was assumed to be a fraction a of the critical free volume for molecular translational diffusion.24 Obviously, a larger a value corresponds to the requirement of a larger additional free volume for solute rearrangement and consequently to a larger medium viscosity effect. Within the context of the Kramers equation when the value of a is large (closer to unity), the viscosity effect approaches the Smoluchowski or the high friction limit of eq 1. In an attempt to examine the role of twisting group size on the effect of medium viscosity, we (PW) carried out a comparative study of the photoisomerization dynamics of trans-stilbene, trans-4,4’-dimethylstilbene,and trans4,4’-dimethoxystilbene.6 Since in this series of isomerizing solutes (17) Schuh, H. H.; Fischer, H. Helu. Chim. Acfa 1978, 61, 2130. (18) Saltiel, J.; Atwater, B. W. Adu. Photochem. 1988, 14. 1 and refer-
ences therein. (19) Kim, M. S.; Fleming, G.R. J . Phys. Chem. 1988, 92, 2168. (20) Grote, R. F.; Hynes, J. T. J . Chem. Phys. 1980, 73, 2715; 1982, 77,
_-.
1176 -.
(21) Hynes, J. T. In The Theory of Chemical Reaction Dynamics; Baer, M., Ed.; Chemical Rubber: Boca Raton, FL, 1985. (22) Carmeli, B.; Nitzan, A. J . Chem. Phys. 1983, 79, 393. (23) Agmon, N.; Kosloff, R . J . Phys. Chem. 1987, 91, 1988. (24) (a) Gegiou, D.; Muszkat, K. A.; Fischer, E. J . Am. Chem. SOC.1968, 90, 12. (b) Sharafi, S.; Muszkat, K. A. J . Am. Chem. SOC.1971,93,4119. (25) Saltiel, J.; D’Agostino, J. T. J . Am. Chem. SOC.1972, 94, 6445. (26) Velsko, S. P.; Fleming, G.R. J . Chem. Phys. 1982, 76, 3553.
10338 The Journal of Physical Chemistry, Vol. 95, No. 25, 1991
the size of the twisting group progressively increases in the order stilbene, dimethylstilbene, dimethoxystilbene, the amount of volume displaced upon isomerization is expected to increase in the same order. Intuitively, a larger volume displacement should correspond to a stronger viscosity effect and, consequently, to a larger a value. However, a values obtained from eq 3,0.33,0.26, and 0.22 for stilbene, dimethylstilbene, and dimethoxystilbene, respectively, in the n-alkane solvent series: exhibited the opposite trend. This unexpected trend was tentatively interpreted to suggest multidimensional twisting motions in these molecules.6 We shall show below that, although the possible need to consider multidimensionality in photoisomerization dynamics is a valid concern whenever the simpler one-dimensional theoretical model is used, a multidimensional model need not be invoked to explain the viscosity effect on trans-stilbene isomerization. Since the discussion of multidimensional effects in substituted stilbenes relied heavily on the assumptions6that there should be a correspondence between the displacement volume of the twisting group and the viscosity effect and that the apparent a values obtained from the slopes of the In koMvs In qsplots (eq 3) serve as measures of the viscosity effect, a closer examination of the meaning of these apparent a values is needed. In our (SS) earlier discussion on the failure of the Kramers-Stokes-Einstein equation, we showed that it is necessary to use microviscosity or an equivalent microscopic probe of medium friction in a solvent series in order to maintain the physical meanings of parameters obtained from the best leastsquares fits.5b In fact, the a value for trans-stilbene in n-alkanes obtained from the microviscosity version of eq 3 is significantly larger than the value quoted above and is in excellent agreement with the a value obtained using the medium-enhanced barrier modeL5 The medium-enhanced barrier modellc*5a~25 is based on the assumption that solvent organization around the solute, ‘t, is random with respect to the geometric requirements of the twisting motion and that, upon excitation, a distinct population of It* is produced in solvent cages, S’, which do not restrict rotation while the remaining population occupies solvent cages, S, which are unfavorable to isomerization. The two populations interconvert rapidly compared to k, and are related by the equilibrium constant K,. The overall twisting process can be represented by (’t*S)
=(‘t*S’) l+(‘p*S’) k
KB
(4)
where Ip* is the perpendicular state. Since K,is small, it follows that the effective twisting rate constant is kobsd = K,k,. In terms of transition state theory kobd = (KkT/h)K* = (KkT/h) exp(U*obsd/R) exp(-U*obsd/RT) (5)
*,
*,
the model gives AS *obsd = AS + hs, and AH *obsd = AH + AHv, where AS, and AHv are associated with K, and are the medium-imposed entropy and enthalpy, respectively. Our previous work shows that the medium-imposed energy barrier increment, AHv,is proportional to the activation energy for viscous flow, E,,,, with a, 0 I a 5 1, being the proportionality constant. E,, , the viscous flow activation energy, is obtained from the Andrade equation plot In vs = In A s + E J R T (6)
Sun et al. obtained either from global transition state theory plotsSaor from In kobsd vs ln v,, plots at constant t e m p e r a t ~ r e . ~ ~ In this paper, treatment of the twisting rate constants of tran~-4,4’-dimethylstilbene,tran~-4,4’-dimethoxystilbene, and tram-4,4’-di(tert-butyl)stilbene in n-alkane solvents will be based on the medium-enhanced barrier model. Medium effects on the substituted twisting portions of the molecule will be reexamined. Adherence of experimental quantities to extrathermodynamic relationships will be demonstrated in these systems, and the critical importance of correcting for fluid discontinuity, even when a homologous solvent series is used in the study of medium effects, will be discussed. A more complete picture of the medium-enhanced barrier model emerges from this treatment. Experimental Section Experimental details concerning materials, fluorescence decay data acquisition, and analysis for trans-4,4’-dimethylstilbene6and trans-4,4’-dimetho~ystilbene’’~ have been described. trans4,4’-Di(tert-butyl)stilbene was prepared using the Wittig reaction from 4-tert-butylbenzyl bromide (Aldrich) and 4-tert-butylb e n ~ a l d e h y d e . ~The ~ aldehyde was prepared by oxidation of 4-tert-butylbenzyl alcohol (Aldrich) with pyridinium dichromate in methylene chloride and was purified by column chromatography. The stilbene was purified by recrystallization from ethanol and its structure confirmed by N M R and mass spectrometry, mp 179-180 OC, lit. 179-180 0C.28 The complementarity of fluorescence and photoisomerization, established for trans-stilbene experimentally,lb was assumed for the substituted stilbenes so that twisting rate constants, koM, could be calculated from kobd = 7;’ - kf (9) where 7;s and k t s are singlet excited state lifetimes and fluorescence radiative rate constants, respectively. For the substituted stilbenes kf was corrected for changes in index of refraction, n: kf = 3.7n2 X lo8 s-’ for dimethylstilbene: kf = 4.2n2 X lo* SKI for dimethoxystilbene,Ic and kf = 6nZ X lo8 S-I for di(tert-buty1)stilbene. The radiative rate constant for di(tertbuty1)stilbene was obtained by combining fluorescence quantum yields and lifetimes in n-hexane and n-decane. Twisting rate constants for dimethylstilbene, dimethoxystilbene, and di( tertbuty1)stilbene and n-alkane solvents at different temperatures are listed in Table I.
Results and Discussion Andrade Equation and Transition State Theory Plots. Shear viscosity as a function of temperature for 12 n-alkane solvents, n-pentane to n-hexadecane, was obtained from ref 29. Andrade equation plots, In qs vs T’, are linear, yielding In A,,8and E,,, values listed in Table 11. Twisting rate constants for each stilbene in each solvent (Table I) adhere closely to the transition state theory equation
(8)
In (kobsd/T) = Aobsd - u*obsd/RT (10) where AoM = (Kk/h) exp(AS *oM/R);In Aow and AH*oMvalues are listed in Table 111. For each stilbene, there is a clear trend of increasing AH*oM value with increasing n-alkane solvent chain length. Plots of A H * o b s d vs EvBfor dimethoxystilbene and di(tert-buty1)stilbene are close to linear (Figures l a and 2a), yielding AH*,= 3.80 f 0.32 and 3.66 f 0.15 kcal/mol (intercepts) and a = 0.61 f 0.11 and 0.45 f 0.06 (slopes), respectively, but the corresponding plot for dimethylstilbene, AH*,= 2.85 f 0.20 kcal/mol and a = 0.56 f 0.09, shows curvature (Figure 3a). Parallel behavior is observed in the plots of In AoM,the entropy term, vs E,,, (Figures lb, 2b, and 3b). Deviations from the best
In the medium-enhanced barrier model, a = AHv/E,,,has a clear physical meaning since it represents the fraction of the viscous flow activation energy that is imposed by the medium on the twisting motion as an additional barrier. The Smolukowski limit of the Kramers equation, for which a = 1 , can now be seen to represent the high friction limit because it requires full imposition of the viscous flow activation energy. Correct a values can be
(27) Cameron, D. W.; Mingin, M. Aust. J . Chem. 1977, 30, 859. (28) Han, G. Y.;Han,P. F.; Perkins, J.; McBay, H. C. J . Org. Chem. 1981, 46, 4695. (29) American Petroleum Institute. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; Carnegie Press: Pittsburgh, PA, 1953. (30) Weisberg, S.Applied Linear Regression; John Wiley & Sons: New York, 1980.
Combining eqs 5 and 6 gives In kobd = In k, + AS,/R
+ a In A,,, - a In 7,
which is equivalent to eq 3 provided that In B = M , / R + a In A,,,
(7)
Photoisomerization Dynamics of Substituted Stilbenes 00
10
2 0
30
The Journal of Physical Chemistry, Vol. 95, No. 25, 1991 10339 0.0
,
4 0 25 2
r
1 .o
2.0
3.0
4.0 I 24.0
I I
23.4
b
3
22.0
5
ri
22.2
6.8
21.2
3.6 0.0
1 .o
3.0
2.8 0.0
4.0
E,, (kcal/mol)
10
2.0
3.0
5.5
3.5 1 0.0
22.5
1 .o
1 2.0
1
1 .o
2.0
1
3.0
21.6
4.0
EQ (kcal/mol)
Figure 1. Plots of (a, bottom) AHtabsd vs E, and (b, top) In (0) and In A,,, ( 0 )vs E, for truns-4,4’-dimethoxystilbenein n-alkanes. The error bars represent 90% confidence ranges30 (see Table 111). The line in (a) is based on AH, and a values obtained from the global transition state theory plot, and the line in (b) is from the best least-squares fit for In A,,,, = In A, + b E , . 0.0
1
1
I 2.0
5.2
-9
3.0
4.0
E,, (kcallmol) Figure 2. Same as in Figure 1 for rruns-4,4’-di(rert-butyl)stilbene.
straight lines in the two kinds of plots for each stilbene follow the same pattern. This is especially obvious in the case of dimethylstilbene for which both plots exhibit the same curvature. are associated with opposite signs in Since In AoM and eq 10, the observed deviation pattern may reflect compensation between the two parameters due to uncertainties in individual plots (see below and Figure 6). Global Tramition State Theory Plot. Since the transition state theory plots for individual solvents, from which A H t O M and
Figure 3. Same as in Figure 1 for trans-4,4’-dimethylstilbene.
In Aohdvalues were obtained, consist of different numbers of data points and uncertainties, it is desirable to treat all data points in a global plot based on the relationship AH * o b d = AH * t + aE,,,, as was previously done in the case of stilbene.sa Equation 10 can be rewritten as In kobsd/T -t&,,,/RT = In Aohd - A H * t / R T (1 1) Plots of (In kobsd/T+ aE,,,/RT)vs T’should generate a family of parallel lines with common slope AH*t provided that the correct a value is selected. Therefore, the procedure employed to obtain best-fit parameters consisted of selecting a value of a and then merging the resulting lines into a single line by arbitrarily moving all points by the separation of the individual lines at 298 K. Stepping the value of a to minimize the standard deviation of the global fit yielded the best a and AH * t values listed in Table IV. For dimethoxystilbene and di(tert-butyl)stilbene, the parameters obtained from the global treatment are quite close to those obtained directly (Figures l a and 2a). The a value increases from 0.39 in stilbene5ato 0.51 in dimethylstilbene and to 0.60 in dimethoxystilbene, in the order expected for increasing twisting group size. The actual global plots are shown in Figure 4. As discussed in the Introduction, the power law, eq 3, yields the opposite trend in a values6 The discrepancy between the two treatments is a result of applying eq 3 to rate constants for a solvent series. In eq 3, B is assumed to be a constant independent of solvent. Examination of eq 8 shows that B is a constant only if A S v / R and a In A,,,,both solvent-dependent quantities, were to compensate exactly as the solvent is changed within a series of closely related solvents. It is true that koM’s in n-alkane solvents can be fitted very well to eq 3. For dimethylstilbene, as an example, plots of In koW vs In 7,at T = 273.15, 303.15, and 333.15 K are linear (Figure 5), with slopes of -0.364 f 0.008, -0.283 f 0.009, and -0.251 f 0.01 1, respectively. However, linearity in In k,, vs In Q, plots does not prove that B is a constant because linear plots, with different slopes, would also be obtained if In B were a linear function of In qs. It will be shown below that this is indeed the case so that apparent a values for the stilbenes in the n-alkane solvent series based on eq 3 are systematically smaller than the actual a values. It will also be shown that correct a values are obtained when microviscosity is substituted for shear viscosity in eq 3. The a value of 0.46 for di(tert-butyl)stilbene, which has the largest twisting group in the series, deviates from the expected
10340 The Journal of Physical Chemistry, Vol. 95, No. 25, 1991 18 2
1
I
Sun et al.
I
17 6 oc
+
I-
%
170
%= k
2 n
164
y“
-C
15
8
15 2
2 5
33
29
37
41
1 O O O ~ T(K-’)
Figure 4. Global transition state theory plots for trans-4,4’-dimethylstilbene ( O ) , rrans-4,4’-dimethoxystilbene(A), and trans-4,4’-di(tertbuty1)stilbene (0) in n-alkanes, extrapolated to the E,, = 0 limit.
21.8
22.5
23.2
23.9
24.6
In A obsd Figure 6. Isokinetic relationship, eq 14, for trans-stilbene ( O ) , trans4,4’-dimethylstilbene (O),trans-4,4’-dimethoxystilbene(a), and trans4,4’-di(tert-butyl)stilbene (A) in n-alkanes. The lines are the best least-square fits. where, for the case of rates, 6AH * and 6AS * are activation enthalpy and entropy changes, respectively, resulting from the systematic change of an experimental parameter, and 0 is a constant. For the effect of medium change on the photoisomerization of stilbenes in the n-alkane solvent series, the isokinetic relationship can be written as
22
AH.,= Pm., U*obsd
PR In Aobsd
(13)
+a
(14)
where a = AH*,- j3R In A,, AHtOM= AH*, + AH.,, and AS*,M = AS U.,. The possibility that medium-enhanced activation parameters for trans-stilbene photoisomerization and, more generally, for radiationless decay processes associated with torsional relaxation obey the isokinetic relationship was first suggested by Saltiel and D ’ A g o s t i n ~ . ~More ~ . ~ ~recently, Bagchi recognized that a search for such a relationship might be As shown in Figure 6, plots of AH *&sd vs In Aobd values in Table 111 for the three substituted stilbenes adhere closely to eq 14, yielding a and j3 parameters listed in Table V. The plot for trans-stilbene is not as good, probably due to the smaller range of experimental enthalpy and entropy changes. Since 6AGv = 6AHv - T6ASv,eq 10 can be rewritten as
*,+
20 -1 8
-1
2
-0.6
0.0
0.6
1.2
In r7 (CP) Figure 5. Plots of In /cow vs In vrreq 3, at constant temperatures of 0 OC (0), 30 OC (a),and 60 O C (A), with slopes of -0.364, -0.283, and -0.251, respectively.
order. Since the experimental kfvalues for this stilbene seem anomalously high, the data were also analyzed using k f = 4n2 X lo8 s-I, leading to an even smaller a = 0.36 f 0.03 and AH*, = 3.46 f 0.2 kcal/mol. It is possible that for a molecule with a fairly large nonrigid substituent the expectation that the volume of the twisting group will be the dominant factor in accounting for the medium effect is not realistic. The origin of this anomaly is not clear at this point and will be the subject of future investigations. Isokinetic Relationship. In a variety of systems, over a wide range of reactions, enthalpy and entropy changes associated with rates and equilibria have been shown to obey the isokinetic relationship3’
6AH’ = j36AS’
6AGv = ( 1 - T/j3)6AHv
The plot of E,,, vs In A,,,, shown in Figure 7, yields p,,, = 904.9 f 36.2 K and a,,,= -5.451 f 0.324 kcal/mol. From these isokinetic relationships, it becomes clear that In B is indeed a linear function of In qs a t constant temperature. Substitution of eqs 14 and 16 into eq 8 gives
(12)
( 3 1 ) Lefflcr, J. E.; Grunwald, E. Rates and Equilibria of Organic Reactions;John Wiley and Sons: New York, 1963.
(15)
Examination of eq 15 readily shows that at T = /3 the medium effect on the twisting rate constant vanishes due to exact compensation between medium-induced enthalpy and entropy changes3’ An analogous relationship exists between Andrade equation parameters (Table 11)
(32) Bagchi, B. Int. Rev. Phys. Chem. 1987, 6, 1.
The Journal of Physical Chemistry, Vol. 95, No. 25, 1991
Photoisomerization Dynamics of Substituted Stilbenes
TABLE I: Photoisomerization Rate Constants of Stilbenes in n-Alkanes solvent T, K k&d, ns-’ solvent T, K kobsd, ns-’ solvent
T, K trans-4,4’-Dimethylstilbene
253.2 263.2 273.2 283.2 293.2 303.2 253.2 263.2 273.2 283.2 293.2 303.2 313.2 323.2 333.2 253.2 263.2 273.2 283.2 293.2 303.2 313.2 323.2 333.2 343.2 353.2 363.2 263.2 273.2
1.08 1.45 1.90 2.48 3.16 3.96 0.93 1.26 1.68 2.20 2.84 3.68 4.76 5.83 7.1 1 0.8 1 1.10 1.50 1.98 2.63 3.32 4.38 5.46 6.86 8.37 9.92 1 1.94 0.97 1.32
283.2 293.2 303.2 313.2 323.2 333.2 343.2 353.2 363.2 37 1.8 293.2 303.2 313.2 323.2 333.2 343.2 353.2 363.2 371.8 273.2 283.2 293.2 303.2 3 13.2 323.2 333.2 343.2 353.2 363.2
313.6 313.8 323.0 325.2 332.8 333.2 323.2 332.8 342.2 35 1.5 362.8
0.97 0.98 1.23 1.28 1.64 1.58 1.15 1.52 1.92 2.38 2.98
37 1.6 318.4 327.0 338.8 347.0 358.3 371.8 323.2 332.8 342.2 351.5
298.1 306.1 314.3 322.1 326.6 337.0 306.1 314.3 322.1 326.6 337.0 344.6 351.1 358.3 365.1 314.3 322.1 326.6 337.0 344.6
2.50 3.11 3.81 4.64 5.49 6.48 2.64 3.30 4.02 4.84 5.76 6.78 7.85 8.99 10.10 2.90 3.61 4.32 5.23 6.28
351.1 358.3 365.1 314.3 322.1 326.6 337.0 344.6 351.1 358.3 365.1 314.9 318.8 322.3 329.5 332.9 338.1 339.9 343.8 348.0
1.79 2.38 3.13 4.06 5.12 6.36 7.89 9.69 11.47 13.56 2.14 2.86 3.69 4.73 5.97 7.53 9.16 1 1.03 12.78 1.06 1.48 2.00 2.67 3.51 4.49 5.63 7.07 8.69 10.37
C ,I
c 1 2
CI3
371.8 303.2 313.2 323.2 333.2 343.2 353.2 363.2 371.4 273.2 283.2 293.2 303.2 313.2 323.2 333.2 343.2 353.2 363.2 371.2 273.2 283.2 293.2 303.2 313.2 323.2 333.2 343.2
kobQdr ns-] 12.08 2.56 3.35 4.30 5.46 6.77 8.42 9.89 12.07 0.88 1.26 1.74 2.35 3.1 1 4.01 5.10 6.44 7.87 9.66 11.15 0.80 1.18 1.64 2.24 2.95 3.83 4.96 6.14
solvent
c14
CI5
c16
T, K
kobsd,
10341
nS-’
353.2 363.2 371.4 283.2 293.2 303.2 313.2 323.2 333.2 343.2 353.2 363.2 371.2 293.2 303.2 313.2 323.2 333.2 343.2 363.2 371.4 313.2 323.2 333.2 343.2 353.2 363.2 371.2
7.68 9.24 11.15 1.08 1.55 2.12 2.82 3.68 4.70 5.99 7.37 9.04 10.47 1.48 2.05 2.74 3.58 4.64 5.81 8.85 10.60 2.59 3.40 4.36 5.52 6.98 8.58 9.98
371.6 333.2 342.6 352.4 361.2 371.6 346.9 347.0 358.4 371.8
3.09 1.10 1.42 1.85 2.33 2.99 1.47 1.47 2.07 2.84
322.3 329.5 338.1 343.8 351.1 358.0 365.6 322.8 329.5 338.3 344.1 35 1.O 357.8 365.6 329.5 338.3 344.1 351.0 357.8 365.6
2.59 3.20 4.01 4.59 5.44 6.52 1.62 2.55 3.10 3.93 4.52 5.31 6.23 7.47 2.81 3.62 4.20 4.93 5.80 6.89
trans-4,4’-Dimethoxystilbene 3.67 0.93 1.21 1.60 2.02 2.65 3.57 1.07 1.41 1.74 2.21
362.8 671.6 327.0 337.8 347.0 358.3 371.8 333.2 342.6 352.4 361.3
2.76 3.51 1.06 1.45 1.85 2.46 3.34 1.17 1.53 2.01 2.48
trans-4,4’-Di( tert-buty1)stilbene 7.37 351.0 8.53 351.1 9.64 354.6 2.65 358.0 3.29 362.3 4.01 365.6 4.84 CI I 322.3 5.70 329.5 6.74 338.1 7.82 343.8 8.94 351.1 2.56 358.0 2.86 365.6 3.14 c 1 2 322.3 3.81 329.5 4.14 338.1 4.71 343.8 4.97 351.1 5.38 358.0 5.89 365.6
6.32 6.26 6.86 7.42 8.14 8.70 2.90 3.55 4.44 5.08 5.93 7.05 8.31 2.75 3.35 4.22 4.84 5.68 6.72 7.87
It follows that the power law, eq 3, for the twisting motion of stilbenes in the n-alkane solvent series or in any other solvent series for which the above isokinetic relationships hold can be written
Therefore, the slope of the In kobsd vs In qs plot does not give a, as assumed in eq 3, but is a function of a and temperature. For dimethylstilbene (Figure 5), n values of 0.56, 0.48, and 0.48 can be calculated from the slopes at T = 273.15, 303.15, and 333.15
CIO
c 1 2
C14
c16
c 1 3
C14
c16
K, respectively, based on 8 = 491.7 and &, = 904.9. According to eq 18, the slope of the In k o b d vs In qs plot is equal to -u only when = &,. We show below that substitution of ’I,, for qs in eq 18 leads to a slope of -a because, ideally, the condition = is fulfilled. 8vuMicroviscosity and the Assumption of a Continuous Fluid. We have established that, for trans-stilbene in n-alkane solvents, the a value obtained from the In koM vs In q,, plot is identical, within experimental uncertainty, to the value obtained by applying the medium-enhanced barrier model, eq 11.jb It follows that unlike B,$B is truly a constant independent of solvent. This can be seen
10342 The Journal of Physical Chemistry, Vol. 95, No. 25, 1991
Sun et al. TABLE V Parameters of the Isokinetic Relationship"
TABLE 11: Parameters from the Andrade Equation Plots" n-alkane t range. OC E,? kcal/mol In A / cs -55-35 1.363 (9) -3.790 (16) -3.926 (101 C' -55-65 1.607 (5) -4.055 (12j -25'95 1.854 i 7 j c; -15-105 2.078 (7) -4.166 (10) C8 -4.309 (26) c 9 -15-105 2.321 (15) 2.606 (31) -4.527 (51) CIO -25-105 -4.597 (51) 2.787 (32) C,I -10-105 -4.708 (60) 2.988 (38) c 1 2 -5-105 -4.859 (74) 3.207 (46) c13 -5-105 -4.910 (76) 3.355 (48) CI, 5-105 3.487 (48) -4.949 (74) Cl, 15-105 -4.972 (70) 3.600 (46) c16 25-105
vs
M'obsd
AH*,,IC vs In Acdc a,kcal/mol @,K
&brd
~~
solute stilbeneC dimethyldimethoxydi(tert-butyl)-
4.0
TABLE 111: Activation Parameters for the Twisting of Stilbenes in n -Alkanes AH*obsd,a
AHcaIc,
In Aobsda trun~-4,4'-Dimethylstilbene
kcal/mol 3.419 3.728 3.921 4.122 4.292 4.364 4.357 4.575 4.709 4.717 4.747 4.718
kcal/mol
(37) (63) (43) (22) (59) (52) (98) (57) (81) (86) (84) (56)
3.695 3.820 3.946 4.060 4.184 4.329 4.421 4.524 4.636 4.71 1 4.778 4.836
22.059 22.506 22.745 22.992 23.180 23.227 23.187 23.444 23.607 23.576 23.604 23.514
(68) (109) (72) (36) (90) (82) (146) (90) (128) (133) (130) (83)
In
Acalc
22.561 22.664 22.785 22.892 23.015 23.173 23.283 23.364 23.492 23.567 23.65 1 23.687
truns-4,4'-Dimethoxystilbene 22.548 22.765 23.169 22.887 23.450 23.466 23.697 24.094
(863) (273) (205) (389) (181) (406) (119) (862)
22.630 22.798 22.935 23.131 23.310 23.570 23.823 23.961
trans-4,4'-Di(tert-buty1)stilbene (374) 4.361 23.364 (208) 4.475 23.195 (232) 4.578 23.555 (281) 4.690 23.494 (44) 4.821 23.608 (98) 4.904 23.762 (90) 4.996 23.762 (101) 5.097 23.929 (80) 5.165 23.933 (154) 5.278 23.951
(595) (314) (345) (417) (65) (143) (133) (148) (117) (223)
23.306 23.337 23.405 23.486 23.621 23.683 23.767 23.870 23.942 24.027
4.743 (554) 4.920 (187) 5.237 (139) 5.055 (268) 5.491 (125) 5.551 (283) 5.756 (83) 6.085 (609) 4.397 4.381 4.679 4.695 4.811 4.958 4.993 5.137 5.159 5.226
4.795 4.943 5.078 5.224 5.395 5.624 5.844 5.991
-38.10 -16.21 -14.90 -22.65
(2330) (116) (91) (312)
889.3 (4949) 446.4 (255) 438.3 (196) 584.8 (665)
-28.60 -18.59 -15.32 -23.54
688.0 497.7 447.4 603.9
"Based on eq 14. bValues in parentheses indicate uncertainty ranges in last significant digit(s) shown calculated for the global treatment.30 Data from ref 5a.
Values in parentheses represent 90% confidence ranges for uncertainty in last significant digit(s) shown.30 TABLE IV: Parameters from the Global Treatments"
,
I
I
I
I
"Viscosity values as a function of temperature from ref 29. bValues in parentheses represent 90% confidence ranges for uncertainty in last significant digit(s) shown.30
n-alkane
&K
a,kcal/mol
3.0
.
I
Ab\
m
0
\
\
Lu"
2.0
1
,
I
1
-6.6
-6.0
-5.4
-4.8
1.0
, \ -4.2
-3.6
In A,, or In ('Aq )
Figure 7. Isokinetic relationship for viscous flow (0)and microviscous flow (A, 0)in n-alkane solvent series. The A,, = f A , values for the microviscous flow are based on f values from empirical diffusion coefficients (0)5b and rotational reorientation times (A).I4 The lines are the best least-square fits.
be assumed thatfcorrects primarily the In A, term of the Andrade equation. Applying Eyring's equation for qs, based on reaction rate theory,33 we (SS)had shown earlierSbthat where LS*,1, is the activation entropy for microviscous flow and c is roughly a constant for the n-alkane solvent series. WhenpB is constant, ASv= ahS *,, follows from eqs 8 and 20, requiring the existence of a linear free energy relation~hip,~~ AG, = aAG*,, where AG, is the medium-imposed free energy change on the twisting process and AG*,, is the free energy of activation for microviscous flow. Since it seems reasonable to assume that the isokinetic relationship established for viscous flow (Figure 7) can be extended to microviscous flow with slope 0,,the above linear free energy relationship requires that 0 , = 0.Substitution of q,,, p,,, and a,,, for qs, p,,,, and a,,,, respectively, in eq 18 then gives In kobsd= -a In q,, + aa,,/@R
~
AH*(, solute stilbene' dimethyldimethoxydi(tert-butyl)-
U
0.39 0.51 0.60 0.46
(2) (3) (3) (3)
kcal/mol 2.85 (5) 3.00 (8) 3.83 (7) 3.62 (8)
bb 0.284 0.516 0.675 0.383
In A,b 23.00 21.84 21.55 22.63
Values in parentheses indicate uncertainty ranges in last significant digit(s) shown.30 A,,lc = bE,, + In A,. 'Reference 5a.
by multiplying both numerator and denominator of the right hand of eq 3 byfh and then substituting eq 2 in the resulting expression: kobsd
= kJhB/(fqs)"
= kJhB/q,"
(19)
Since the microfriction factorfis nearly independent of T, it can
since the fin, terms cancel. Comparison of eqs 19 and 21 shows that the condition thatpB = constant is fulfilled. The expectation that the In kow vs In I],, plot have -a as slope follows naturally from the existence of a free energy relationship between the medium-imposed free energies for translation and rotation of the twisting molecular moiety. While these self-consistent relationships are based on certain approximations, they are justified by the excellent agreement between a values obtained from eq 19 and a values based on the global transition state theory plot for trans-stilbene in n-alkane solvents, eq 11.5 Also shown in Figure 7 is the isokinetic plot for ~
~~
~
(33) (a) Eyring, H. J . Chem. Phys. 1936,4, 283. (b) Kierstead, H. A.; Turkevich. J. J . Chem. Phys. 1944, 12, 24
Photoisomerization Dynamics of Substituted Stilbenes microviscous flow for stilbene, E,, vs In fA,, where f values are obtained based on empirical diffusion coefficients for tolueneSb or on trans-stilbene rotational reorientation times14 adjusted for the difference in the moments of inertia in the rotating moieties.sb The P,,, value of 563.7 K from the slope is significantly smaller than 8 , and is much closer to the experimental 6 value of 688.0 K. Although the plot is based in part on rough approximations, such as eq 20, and does not exactly give P,,, = 0, it still indicates that the above discussion on microviscosity is generally valid. In principle, similar plots can be performed for the substituted stilbenes as well, provided that microfriction factors for those molecules are available. We prefer to defer that treatment until experimental diffusion coefficients for the relevant twisting groups become available. The inadequacy of using shear viscosity in eq 3 as a reliable measure of the friction experienced by the twisting portion of the stilbene molecule in different n-alkane solvents has the same origin as the failure of the Kramers-Stokes-Einstein approach. Since it is based on classical hydrodynamic theory, the Stokes-Einstein equation is applicable to diffusion of particles in a continuous fluid. Hydrodynamic radii of the twisting groups of these stilbenes are in the range 3.15-3.7 A and are, therefore, comparable in size to the n-alkane solvent molecules which have hydrodynamic radii in the range 3.2-4.4 A. Consequently, the solvent series cannot be regarded as a continuous fluid for the twisting groups because the differences in solvent molecular size are significant relative to the size of the twisting portion of the solutes. While the relative size of solvent and solute plays a major role in this regard, the difference in solventsolute interaction for short and long n-alkane solvents contributes as well. On the other hand, the medium can be regarded as a continuous fluid when microviscosity is employed because the microfriction factor f corrects shear viscosity for differences in size and interaction in a solvent-solute specific fashion. According to microfriction t h e ~ r y ,f~can ~ ,actually ~~ be factored into two terms, one of which is a function of the ratio of solute and solvent radii and the other is a correction for specific solute-solvent interaction^.^^ On the Medium-Enhanced Barrier Model. The linear relationship AH = AH + aE,$with a being a constant has been observed directly in n-alkane solvents. However, for systems like dimethylstilbene, where the plot of AH *obsd vs E,,,.is not linear (Figure 3a), it may appear that the relationship is still an assumption that has been forced into the global treatment. The conclusion that the relationship is real and justified can be based on the isokinetic relationships and the linear In kOMvs In vsplots, all of which are inherent properties of the experimental observations. By combining eqs 14 and 16, and by denoting the intercept and slope of the In kobsd vs In 7, plot as In B' and s, respectively, a linear relationship between AH *obsd and E,,, can be obtained
*,
The Journal of Physical Chemistry, Vol. 95, No. 25, 1991 10343
TABLE VI: AH*tand a Parameters for Dimethylstilbene Based on Ea 220
T,K
In B'
S
273.15 303.15 333.15
20.890 (8) 21.658 (7) 22.314 (9)
-0.364 (8) -0.283 (9) -0.251 (11)
AH*,, kcal/mol
a
2.90 3.08 3.06
0.56 0.48 0.48
"In B' and s values from Figure 5; AH, and u values are from the intercept and slope in eq 22, see text. Values in parentheses indicate uncertainty ranges in last significant digit(s) shown calculated for the global treatment.'" linearity, e.g., Figure 3a, are likely to be caused primarily by experimental uncertainties in the temperature dependence of the rate constants. Relationship between E,, AH*,, and Eo. The conclusion that the intrinsic energy barrier for twisting in the lowest excited singlet state of trans-stilbene is essentially constant in the n-alkane solvent seriessa has now been extended to three stilbene derivatives. Therefore, any dependence of the intrinsic energy barrier on the solvent's index of refraction is negligible and cannot be relied upon as the reason for the failure of the Kramers-Stokes-Einstein equation as has been p r o p ~ s e d . ~ The , ~ ~ assumption that the intrinsic barrier is independent of solvent within a homologous solvent series is implicit in the common practice of basing the barrier height on isoviscosity Arrhenius equation plots.6Jlb,c,14,36b,37 However, as was pointed out recently, the failure of the Kramers-Stokes-Einstein equation to account for changes in kohd in n-alkanes requires that the barrier height be based on isomicroviscosity plots.5b Use of the Arrhenius equation in such plots generates estimates of the Arrhenius activation energy, E,, which has been assumed to equal the barrier height energy Eo. We (SS) have claimed that the intrinsic activation enthalpy, AH*,, provides a better approximation for the barrier height.lc*s We now address a recent criticism7of this proposal in order to clarify the differences in the two viewpoints. Using the potential energy barrier Eo, the transition state theory expression for k, is where Q is the reactant partition function and Q' is the transition state partition function with the contribution from the reactive degree of freedom removed. When contributions of all modes, except the mode corresponding to the reaction coordinate v, are the same for reactant and transition state, eq 23 reduces to k, = (KkT/h)[l - exp(-hu/kT)] exp(-Eo/RT) (24) Basing the calculation of AH*,= R p ( d In K */dT) separately on eqs 5 and 24 gives
+
Eo = AH*, Nhu/[exp(hv/kT) - 11
AH*o&d=
(25) At the high frequency limit (hv >> kT), eq 25 reduces to Eo = AH Itand the Arrhenius activation energy is given by E, = AH *t + RT. At the low frequency limit (hv
