3330
J. Phys. Chem. 1991, 95, 3330-3334
by the profiles of the alcohol solubility as a function of the surfactant concentration. In these profiles no anomalies were found to correspond to the micellar transitions.6 To explain the breakdown in the plot of C*,Rvs mR,we recall that some alcohols (butoxyethanol, 2-propanol and tert-butyl alcohol) form microheterogeneities in water."' Medium chain length primary alcohols possess hydrophilic and hydrophobic moieties, which would enable them to form these microstructures. The fact that this does not occur can be ascribed to the rigidity of the hydrogen bonds between the OH groups, which leads the associated alcohol molecules to form a separate phase rather than to form microstructures. Simple amines behave like primary alcohols, whereas it is known that their chlorides and bromides form micelles. On the other hand, the ionic head group shifts the cmc toward higher values as is shown by comparing the cmc values of dodecyldimethylamine oxide and the corresponding hydrochloride; in fact, for the former the cmc is ca. 0.0018 mol kg-' whereas for the latter the cmc is ca. 0.0052mol kg-l, Le., 3 times greater.M According to these data, one should expect for amines a cmc of ca. one-third of that of the corresponding hydrochlorides; for example, since the cmc of octylamine hydrochloride3' is ca. 0.15 mol kg-I, the expected cmc for octylamine is ca. 0.05 mol kg-I, which is higher than its solubility in water, the value we have determined to be 0.006mol kg-I. On the other hand, amphiphiles with a poly(oxyethy1ene) head group and butoxyethanol do form micelles and microheterogeneities, respectively, because of the presence of the ether oxygen atoms. In conclusion, it seems that medium chain alcohols and amines do not form micellelike microstructures in water because of the hydrogen bonds between their hydrophilic groups. As a matter of fact, amphiphilic molecules added to alcoholic solutions can promote these alcoholic microstructures by breaking down the hydrogen bonds. This is similar to what happens in micelle formation, where an equilibrium is attained between the apolar groups attraction and the polar groups repulsion. Returning to the AC*3R/bmRvalues, according to the literature2* they can be ascribed to the alcohol distribution between the aqueous and the micellar phases before the break point and to the formation of alcoholic microstructures stabilized by the surfactant above the break point. According to this idea, first we recall that micelles behave like a hydrophobic solvent medium; for example, for PentOH AC*,R/AmRis ca. 300 J K-' mol-l in octane2 and in 0.8 mol kg-I DTAB2 whereas it is ca -60 J K-' (30) Benjamin, L. J. J . Phys. Chem. 1964,68, 3575. (31) Klevens, H. B. J . Am. Oil Chem. Soc. 1953, 30, 74.
mol-' in water.2 So, the initial increase of C,,R with mR and the evolution of the slope with both the surfactant concentration and the alcohol tail account for the distribution constant of alcohol between the aqueous and the micellar phases: at a given surfactant concentration by increasing the alcohol concentration and the alcohol tail, the amount of alcohol solubilized in the micellar phase increases more quickly than that in the aqueous phase. The negative values of AC*,R/AmRat higher alcohol concentrations might be tentatively explained by assuming that the alcoholic microstructures possess a less hydrophobic character than that of micelles so that beyond the break point the alcohol and surfactant solubilization into these microstructures implies a decrease of C*,R. In other words, the alcohol Concentration a t which the break in AC*,R/hR is detected possesses the feature of a critical concentration that depends, like the surfactant micellization, slightly on the additive concentration and highly on its alkyl chain tail (see Figures 6 and 7). One could take of a critical microstructure concentration ( c q c ) . From this point of view, the surfactant could assume the role of an additive, which, by enhancing the alcohol solubility, leads to attaining the cm,c. In conclusion, by progressively adding a medium chain length alcohol to a micellar solution, a t low concentrations the alcohol is distributed between the aqueous and the micellar phases forming mixed micelles. As the composition becomes richer and richer in alcohol, the micelles lose progressively their identity. When the cm,c is attained, the alcoholic microstructure is formed. According to a small angles neutron scattering investigation of the water-hexadecyltrimethylammonium bromidebutoxyethano1 system,)2these alcoholic microaggregates contain only u n d a t e d surfactant molecules. Therefore, it seems that moderately soluble alcohols can form mixed microaggregates when the surfactant and alcohol concentrations are higher than a certain value. In the formation of alcoholic microstructures the surfactant plays the role of an additive which, at the same time, increases the alcohol solubility so that the c q c value can be reached and makes possible, by breaking down the hydrogen bond network, the formation of mixed microstructures.
Acknowledgment. We are grateful to the National Research Council of Italy (CNR, Progetto Finalizzato Chimica Fine 11) and to the Ministry of University and of the Scientific and Technological Research (MURST) for financial support. Registry No. DTAB, 11 19-94-4; PrOH, 71-23-8; BuOH, 71-36-3; PentOH, 71-41-0; HexOH, 111-27-3; HeptOH, 111-70-6. (32) Quirion, F.; Magid, L. J . Phys. Chem. 1986, 90,5193.
Stoichlometry and Formation Constants of Pyrene Inclusion Complexes wlth @- and y-Cyclodextrin A. Muiioz de la Peiia? T. Ndou, J. B. Zung, and I. M. Wamer* Department of Chemistry, Emory University, Atlanta, Georgia 30322 (Received: June 18, 1990) This study focuses on the inclusion complexes of pyrene with (3- and y-cyclodextrin (CD) in aqueous solutions. The stoichiometries and association constants of these complexes have been investigated by monitoring the I/III vibronic band intensity ratio for pyrene in the presence of CDs. The data are analyzed by both linear and nonlinear regression methods, which indicate 1/1 y-CDlpyrene and 2/1 (3-CDlpyrene complexes. The estimated association constants are 250 M-' and 8.53 X los M-2 for y- and &CD, respectively. A discussion of the stoichiometries of both systems using molecular modeling is included.
Introduction Cyclodextrins (CDs) are cyclic oligosaccharides that are well-known for their unique ability to form inclusion complexes
with a variety of guest Owing to their inherent usefulness in the areas of analytical separations and pharmaceuti=, numerous studies have been undertaken to evaluate the complexing
Corresponding author. 'On leave from the Department of Analytical Chemistry, University of Extremadura, Badajoz 06071. Spain.
( I ) Szejtli, J. Cyclodextrins and their Inclusion Complexes; Academiai Kiado: Budapest, 1982. (2) Saenger, W . Angew. Chem., Int. Ed. Engl. 1980, 19, 344-362.
0022-3654/91/2095-3330%02.50/0 0 1991 American Chemical Society
Pyrene Inclusion Complexes with ,9- and y-Cyclodextrin ability of CDs. In particular, many studies have focused on the ability of CDs to include guests of varying sizes in different stoichiometric ratio^.^-^ It is well documented that, depending upon the host CD (i.e. a-,b-, or y-CD) and the size of the guest, different host/guest stoichiometries are possible. Obviously, stoichiometry is an important consideration when examining a host/guest complex. The correct evaluation of the stoichiometry of a complex is crucial for an accurate determination of the formation constant. For analysis based on complexation, a knowledge of the formation constants for CD complexes is necessary in order to predict and better understand the interactions between CDs and guest in solution. The formation constant provides a measure of complex stability, and an examination of these values can give a clearer understanding of the factors affecting complexation. The results of such studies can then be implemented in the design of specific analytical or pharmaceutical methodologies. Recently, much attention has focused on the ability of pyrene to form complexes with both P- and y-CD.&'l Unfortunately, some of these previous studies have lacked consistency, and different conclusions have been derived. Most notably, low aqueous solubility (approximately 5 X IOp7 M) has not always been considered in sample preparations of pyrene and CDs, and hence different results have been reported. In our previous studies using fluorescence lifetime measurements, we reported that pyrene forms a 1:l complex with B-CD.I2 Our data also suggested the possibility of a 2/1 O-CDlpyrene complex at higher C D concentrations, but the presence of a second complex was not resolvable by using fluorescence lifetime measurements. Pyrene is an especially suitable fluorescence probe because of its vibronic fine structure and sensitivity to microenvironmental changes.s*6 The polarity of the microenvironment surrounding pyrene can be estimated from the ratio of intensities of band I (373 nm) and band 111 (385 nm). This sensitivity to microenvironmental changes allowed Edwards and ThomasS to investigate the interaction of 8-CD with pyrene. Nakajima6 and other^^.^ have reported the sole existence of a 1/1 8-CDlpyrene complex. Recently, however, Kusumoto7 has reported that in addition to a 1 / 1 O-CDlpyrene complex, a 2/ 1 host/guest complex is also formed at higher CD concentrations. The association constant for the latter was determined on the basis of the variation of the I/III vibronic band intensity ratio for pyrene. Similarly, different reports have also appeared in the literature describing the stoichiometry and the association constants of y-CDlpyrene comple~es.~*~J In this paper, we report the formation of a 2/ 1 0-CDlpyrene complex and a 1 / 1 y-CDlpyrene complex using both linear and nonlinear regression analysis. Formation constants for the resulting inclusion complexes are also presented. Experimental Section Apparatus. Steady-state fluorescence measurements were acquired with a modified Perkin-Elmer 650- 10s fluorescence spectrophotometer equipped with a thermostat4 cell housing." Fluorescence emission spectra were taken at an excitation (3) Herhtrotter, W. G.;Martic, P. A.; Evans, T. R.; Farid, G.J . Am. Chem. Soc. 1986, 108, 3273-3280. (4) Kobayashi, N.; Saito, R.; Hino, H.; Hino, Y . ;Ueno, A.; Osa, T. J . Chem. Soc., Perkin Trans.2 1983, 1031-1035. ( 5 ) Edwards, H. E.; Thomas, J. K. Carbohydr. Res. 1978,65, 173-182. (6) Nakajima, A. Spectrochim. Acra 1983, 3 9 4 913-915. (7) Kusumoto, Y . Chem. Phys. k r r . 1987, 136, 535-538. (8) Kano. K.; Takenoshita, 1.; Ogawa, T. J . Phys. Chem. 1982, 86,
1833-1838. ... . .... (9) Patonay, G.; Shapira, A.; Diamond, P.; Warner, 1. M. J . Phys. Chem. 1986, 90,1963-1966. (IO) Hashimoto, S.; Thomas, J. K. J . Am. Chem. Soc. 1985, 107, 4655-4662. (1 1) Hamai, S. J . Phys. Chem. 1989, 93,6527-6529. (12) Nelson, G.;Patonay, G.;Warner, 1. M. J . Inclusion Phenom. 1988, 6, 277-289. (13) Zung, J . B.;Nelson, G.;Warner, 1. M. tab. Microcompur. 1989, 9, 145-150. ~
The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3331 wavelength of 335 nm, with excitation and emission bandwidths of 5 and 1.8 nm, respectively. All measurements were performed at 21 0.1 OC. Materials. Pyrene (>99%) was obtained from Aldrich and used as received. The j3- and y-cyclodextrins were obtained from American Maize Products (Hammond, IN). The 8-cyclodextrin was recrystallized twice from water before use. Method. A 5.0 X le M stock solution of pyrene was prepared in cyclohexane. Aqueous pyrene solutions were prepared by pipeting an aliquot of the stock solution into a 100-mL flask. Cyclohexane was then evaporated under dry nitrogen, and the remaining flask contents were diluted with deionized water (Continental Water Systems, Atlanta, GA) to give a 2.0 X lV7 M solution. To prepare a pyrene solution in aqueous CD, a 5-mL aliquot of pyrene in water was transferred into a IO-mL flask. A weighed quantity of 8- or y-CD was added, and the flask was filled to the mark with deionized water. Concentrations of 8-CD ranged from 2.0 X lo4 to 0.01 M, and those of y-CD, from 2.0 X lo4 to 0.016 M. The pyrene concentration was held constant at 1.O X M in all experiments. All solutions were allowed to equilibrate for 18 h prior to analysis.
*
Results and Discussion Most complexation studies assume a 1/ 1 stoichiometry between CD and the guest molecule of interest; however, this is not always the case and some researchers have reported that two or more cyclodextrins can encapsulate a single m ~ l e c u l e . ' ~ - ~This ' is important because interpretation of the results in studies involving complexation may vary significantly depending on the stoichiometry. The equilibrium constant (K,)for a 1/1 association between pyrene (P) and cyclodextrin (CD) is given by P + C D + PCD
where [PCD] is the equilibrium concentration of the inclusion complex for a given CD concentration. The classical method for the determination of K l is the preparation of a double-reciprocal plot,22 derived from the following equilibrium equation:
where [PIo and [CD], are the initial analytical concentrations of P and CD. In the study reported here, the concentration of cyclodextrin is large with respect to that of the complex, Le. [CD] >> [PCD]. Thus, eq 3 becomes (4)
The general quantum yield expression for the fluorescent complex takes the form (14) Harata, K. Bull. Chem. Soc. Jpn. 1976, 49, 1493-1501. (15) Gelb, R. 1.; Schwartz, L. M.; Murray, C. T.; Laufer, D. A. J . Am. Chem. Soc. 1978,100,3553-3559. (16) Gelb, R. 1.: Schwartz, L. M.: Laufer. D. A. J. Am. Chem. Soc. 1979. 101, 1869-1874. (17) Rosanske, T. W.; Connors, K. A. J . Pharm. Sci. 1980,69,564-567. (18) Wong. A. B.; Lin, S. F.; Connors. K. A. J . Pharm. Sei. 1983, 72, 388-390. (19) Connors, K. A.; Pendergast, D. D. J . Am. Chem. Soc. 1984, 106, 7607-7614. (20) Pendergast, D. D.; Connors, K. A. J . Pharm. Sci. 1984, 73. 1779-1783. (21) Armstrong. D. W.; Nome, F.; Spino. L. A.: Golden. T. D. J . Am. Chem. Soc. 1986, 108, 1418-1421. (22) Benesi, H. A.; Hildebrand. J. H. J . Am. Chem. Soc. 1949. 71. 2703-2707.
Muiioz de la Peiia et al.
3332 The Journal of Physical Chemistry, Vol. 95, No. 8,I991 QPCD
=
I
2.0 I
FPCD
G[PCDl
where QPCD is the quantum yield of the complex, FPCD is the fluorescence signal from this complex, and G is a constant characteristic of the fluorophore and instrumental parameters. Combining eqs 4 and 5 and then rearranging, we obtain 0.6
Thus, a reasonable estimate of KI can be obtained from a plot of [P]O/FxD versus I/[CD],. To determine F p p , the fluorescence signal of a solution containing only pyrene (Fp)IS subtracted from the total fluorescence signal (F,).However, since the quantum yield of pyrene (Qp) is not small relative to QED and since the concentration of free P decreases as CD increases, the use of this equation can introduce a significant negative error in FED. This is true because the addition of CD causes an enhancement in the fluorescence intensity of pyrene and a slight spectral shift upon forming the inclusion complex. Consequently, it is not possible to measure the individual contributions of the two species to Ft. In addition, this method requires that the concentration of P be constant and exactly known for all solutions. This can be a difficult task due to the low solubilty of pyrene in water and the tendency of pyrene to adsorb onto the walls of the glassware. Patonay et al.23have reported that special precautions should be taken with aqueous sample preparations of cyclodextrin solutions. A feasible approach for determining the association constants of pyrene and CDs is to use the variation of the I/III intensity ratio of the pyrene monomer fluorescence in the presence of varying CD concentrations.' Almgren et al.24have used this I/III ratio to determine the stability constant of the molecular complex between pyrene and micellar tetraethylammonium bromide. In this method, it is assumed that the observed I/III ratio is the weighted average from the complexed and free pyrene. Thus [PCD] Ro- R -(7) [PI, Ro - Rl where the parameters Roand R I denote the ratios for pyrene in water and in the complex, respectively, and R is the measured ratio at a given C D concentration. In the case of a 1 / 1 complex, the following equation is applicable
--
and a plot of l/(Ro - R ) versus l/[CD], should give a straight line. Similarly for a 2/1 complex, the overall equilibrium constant (K2) is given by P
+ 2CD
P(CD)2
and under the assumptions [CD], obtains
--1
Ro - R
-
I
(9)
>> (P(CD)2] >> [PCD] one
1 +KZ(R0 - R2)[CD]02 RO- R2
(11)
where R2 denotes the I/III ratio for pyrene in the 2/1 complex. In this case, a straight line should be obtained when l/(Ro - R ) is plotted against 1 /[CDIo2. The application of this method (use of the I/III vibronic band ratio for pyrene) allows one to graphically determine the stoichiometry of the system under study. For example, consider the variation of this ratio in water, as monitored with increasing (23) Patonay, G.; Rollie, M. E.; Warner, 1. M.Anal. Chem. 1985, 57, 569-57 1. (24) Almgren, M.;Gricsser, F.; Thomas, J. K. J . Am. Chem. Soc. 1979, 101, 219-291.
3
0
6
13
10
16
CD CONCENTRATION (mM)
Figure 1. Plots of the I/III vibronic band intensity ratio for pyrene versus cyclodextrin concentration with [PI = IO-' M and A,, = 335 nm: A, y-CD; A, fl-CD; solid lines, computed from eqs 12 and 13 for y-CD/
pyrene and fl-CDlpyrene complexes, respectively. 18.0
4 r
1
1
7.2
- -I:
18.0 1
.I
1
J 0
10
$
1
i
/'
721
,'
, i
36
I I
- ~20 30 40 50
00
0
4
7
11
IllCDI2 x
I l I C D I x lo-' W1
14
18
M-2
Figure 2. Double-reciprocal plots for the @-CDlpyrenecomplex: (a) l/(Ro- R) versus l/[fl-CD]; (b) I/(Ro - R ) versus 1/[fl-CDI2
,
22 1 0
7
6
v
I
$
~
lj// b'
132
g
zl/~
I I I
44
44
00
00 0
12
24 IllCD1 x
36
48 h4-l
60
0
5
10 [1/CD12 x
16
21
26
M-'
Figure 3. Double-reciprocal plots for the y-CDlpyrene complex: (a) I/(& - R ) versus l/[y-CD]; (b) l/(Ro - R ) versus l/[y-CDI2.
concentrations of 8- and y C D (Figure 1). The I/III ratio decreases with increasing CD concentration because of a change in the microenvironment surrounding the pyrene. In this figure, it is seen that the I/III ratio decreases significantly with increasing 0-CD concentration. At approximately 5.0 X M 8-CD (I/III = 0.62), the curve levels off, indicating that effectively all of the pyrene molecules are included in the cavity. The low value of the ratio also suggests that pyrene is located in a very nonpolar environment or more specifically inside the CD cavity. Pyrene evidently seeks out the inner hydrophobic region, and is far removed from the bulk aqueous environment. In the case of */-CD, the I/III ratio also decreases as the CD concentration increases, but the effect is less pronounced. The fact that less CD is needed to reach the constant low level for 8-CD than for y-CD suggests that the complexation equilibrium is more favorable for the 8-CD complex. A typical double-reciprocal plot for a 8-CDlpyrene complex is shown in Figure 2. An upward concave curavature is obtained when l/(Ro - R) is plotted against I/[CDIo (Figure 2a). This suggests that the stoichiometry of the complex is not simply 1/ 1. In contrast, a linear relationship (for the entire range of concentrations tested) is obtained (Figure 2b) when these data are plotted according to eq 11. It can then be inferred that a 2/1 @-CD/pyrene complex exists, and this appears to be the predominant form. It is clear that a 1/1 P-CDlpyrene complex should also be formed at extremely low p-CD concentrations, and the association constant should be very small. Hence, one can conclude that the fraction of pyrene present in 1 / 1 complexes is not large enough to influence the graphical analysis of the B-CD complexation under these experimental conditions. The linearity
Pyrene Inclusion Complexes with 8- and y-Cyclodextrin
The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3333
I a
I b
f
l
Figure 4. CPK models for the inclusion of pyrene in 8- and y-CD. Part a: ( I ) pyrene inside 8-CD as viewed from the secondary hydroxyl side; (11) side view showing only part of the pyrene inside the 8-CD cavity. Part b: ( I ) pyrene inside y-CD as viewed from the secondary hydroxyl side; (11) side view showing pyrene inside the y-CD cavity.
of the double-reciprocal plot, assuming only the existence of a 2/1 complex, supports these views. This is in agreement with the observation by Kusumoto' of a 2/1 &CD/pyrene complex formed at higher CD concentrations. In view of these findings, we feel that the interpretation presented above is more applicable, as determined from the variation of the I/III ratio. The results presented in Figure 3a for the y-CDlpyrene system give a straight line indicating a 1/ 1 stoichiometry. A downward concave curvature (Figure 3b) is obtained when these data are fitted for a 2/1 y-CDlpyrene complex system, confirming that the stoichiometry of the complex is not 2/ 1. Once the stoichiometry of the system is known, the association constant can also be calculated by application of this method. However, in this approach the data are not properly weighted, because when the values of the independent variables are chosen to be equally spaced, the linearization transformation changes this relationship. The points at low abscissa (x) values are widely spaced as compared to high x values, which are closely spaced. Consequently, the slope of the line is very sensitive to they value of the points having the smallest x value. Nonlinear least-squares regression analysis is an alternative approach to the graphical method.2s Nonlinear regression requires (25) Connors, K. A. Binding constanis. The measurement of molecular complex srahility; John Wiley & Sons: New York, 1987.
preliminary parameter estimates, which have been determined from the linear regression approach. By use of a nonlinear regression program,26the data were directly fitted into the relevant equations. In the case of y-CDlpyrene, in which a 1/ 1 complex is suggested, the following equation is used:
For the P-CDlpyrene system in which a 2/1 complex is suggested, eq 12 becomes Ro + R2~z[CD1o2 (13) 1 + K2[CD]O2 The estimated binding constant for the formation of the 1/1 y-CDlpyrene complex is 250 M-I. The value for the y-CD/ pyrene complex is consistent with estimates from previous studies, i.e., 399 M-' (20 "C), 8 1 M-' (30 "C): 270 M-I:' and 300 M-'." Previous results for the association constant of the B-CD/pyrene complex (assuming a 1 / 1 stoichiometry) exhibit a marked difference in the reported values, ranging from 7.6 to 277 M-1.6*7*9J028 R=
(26) SASISTAT, Release 6.03; SAS Institute Inc.: Cary, NC, 1988. (27) Nelson, G.; Patonay, G.; Warner, I. M. Anal. Chem. I=, 60, 274-219. (28) Hamai, S. J . Phys. Chem. 1989, 93, 2074-2078.
3334
J. Phys. Chem. 1991, 95, 3334-3350
The formation constant for the system under investigation here (2/1 stoichiometry) was found to be 8.5 X los M-2. Kusumoto' has reported a value of about 1.16 X IO4 M-2 for this constant. At a 1 .O X IC2M CD concentration, he noted that the I/III ratio is about 0.90, which is somewhat higher than 0.62 obtained in this work. Our results are in good agreement with that reported by Edwards and Thomass (I/III = 0.61). We think that the differences in the formation constant values reported by Kusumoto and us can be attributed to the differences in the I/III ratios. While our formation constant is relatively large, it should be noted that the formation constant will vary significantly in going from a 1 / 1 to a 2/1 stoichiometry. The large formation constant is indicative of the strong complex formed when a pyrene molecule binds with two j3-CD molecules. In order to further characterize the inclusion complexes of pyrene and 8- or y-CDs, the Corey-Pauling-Kolton (CPK) space-filling models of these systems have been examined. On the basis of molecular size, it is apparent that pyrene (8.2 A wide and 10.4 A long) is too bulky to fit entirely in the j3-CD cavity (7.8 A wide). Examination of CPK models reveals that only part of the pyrene actually enters the CD due to the bulkiness of the former and the small entrance diameter of the latter, as shown in Figure 4a. By viewing the C D from the secondary hydroxyl side, we see that the internal diameter of j3-CD severely limits the ability of pyrene to totally enter the cavity. Furthermore, a side view of the complex reveals that only part of the pyrene is inside the cavity. Since approximately half of the pyrene is located in the cavity, it can be inferred that the other half (outside the cavity) is equally amenable to complex formation with another 8-CD. This is a reasonable assumption since the CD is in large
excess relative to pyrene, and pyrene prefers to be in a more hydrophobic environment. The larger cavity diameter (9.5 A) of y C D may allow more of the pyrene to be included inside (Figure 4b). As in the case of 8-CD, pyrene cannot fit entirely into the cavity, but it will penetrate more deeply. The limiting factor in this case is the length of pyrene and not its width. Since pyrene is too long to be completely included inside y-CD, part of the molecule is located outside the cavity. The part of pyrene exposed to the aqueous environment is very small relative to that included inside the CD, which ultimately reduces the likelihood that more than one y-CD participates in complex formation. Furthermore, we have recently shown that pyrene is less accessible to an external quencher (I-) in the presence of 8-CD than is the case in Y - C D . ~If~ the stoichiometry is 1/1 in both complexes, then better protection should be afforded in y-CD than in 8-CD, because more of the pyrene is inside the cavity. The better protection afforded by j3-CD strongly support the assumption of 2/ 1 stoichiometry. Acknowledgment. This work was supported in part by the National Science Foundation (Grant CHE-9001412) and the National Institutes of Health (Grant G M 39844). A.M.d.1.P. acknowledges support from the DGICYT of the Ministry of Education and Science of Spain for the grant that made possible his research in Professor Warner's laboratory. We are also grateful to G. A. Reed of American Maize Products for providing the CDs used in this study. (29)
Nelson, G.; Warner, I. M. J . Phys. Chem. 1990, 94, 576-581.
Theory on Reaction Rates In Nonthermallzed Steady States during Conformational Fluctuations In Viscous Solventst Hitosbi Sumi Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305, Japan (Received: April 30, 1990)
Rate constants for chemical reactions limited by electron, proton, or atom-group transfer in a large molecular system are formulated for the case where the reaction coordinate is composed not only of intramolecular or atomic vibrational motions but also of conformational fluctuations of the system itself driven by thermal motions of solvent molecules. These fluctuations are diffusive, being describable in terms of Brownian motions. Their relaxation time 7 regarded as proportional to the viscosity 7 of solvents is much longer than periods of the vibrational motions. In this situation it is shown that when reactant populations get to decay single exponentially after a time of order r in exthothermic reactions, rate constant k, tends, irrespective of initial conditions, to the inverse of the first passage time for initially thermalized reactants. This k, can be expressed as l/(k;' + kc'), where k, represents the thermal equilibrium rate constant obtainable by the usual theories such as the transition-state theory, while kf represents a fluctuation-limited rate constant decreasing with increasing T or q. When k, is written as u exp(-AG*/kBT) with the height AG* of the transition state and the frequency factor v at temperature T, we show kf rWdW exp(TAG*/kBn with positive constants a and y both smaller than unity. When k,
