J. Phys. Chem. 1980, 84, 2381-2390
Acknowledgment. We thank Dr. H. Muto for his cooperation in the supplemental experiments. References a n d Notes Iwasaki, M; Toriyarna, K.; Muto, H.; Nunome, K. Chem. Phys. Lett. 1978, 56, 494. Toriyama, K.; Iwasaki, M.; Nunome, K. J . Chem. Phys. 1979 71, 1698. Bell, R. P. Proc. R . SOC.London, Ser. A 1935, 148, 241. Le Roy, R. J.; Sprague, E. D.;Williams, F. J . Phys. Chem. 1972, 76, 546. Toriyama, K.; Iwasaki, M. J. Phys. Chem. 1978, 82, 2056. (a) Sprague, E. D.; \Nilliams, F. J. Am. Chem. SOC.1971, 93, 787. fb) Hudson. R. L.: Shiotani. M.: Williams. F. Chem. Phvs. Lett. 1977, 48, 193 arid references cited therein. (c) Sprague,-E. D.J. Phys. Chem. 1977, 81, 516. Torivama. K.: Nunome, K.; Iwasaki, M. J. Am. Chem. SOC.1977, 99,-5823. Toriyama, M.; Iwaseki, M. J . Am. Chem. SOC.1979, 101, 2516. Iwasaki, M.; Toriyama, K.; Muto, H.; Nunome, K. J. Chem. Phys. 1976, 65, 596; Chem. Phys. Lett. 1976, 39, 90. Iwasaki, M.; Toriyarna, K.; Nunome, K.; Fukaya, M.; Muto, H. J. Phys. Chem. 1977. 81. ‘1410. Iwasaki, M.; Muto,”.; Toriyama, K.; Fukaya, M.; Nunome, K. J. phys. Chem. 1979, 83, ‘1590. Iwasaki. M.; Toriyama, K. J . Phys. Chem. 1979, 83, 1596. Foner, S. M.; Cokran. E. L.: Bowers, V. A.: Jen, C . K. J. Chem. Phys. 1960, 32, 963.
.,
2381
(14) The annealing temperatures 26-35 K given in ref 1 for the results obtained from Xe matrices are lower by 10-15 K as compared with those in the present work. The detrapping temperature of hydrogen atoms may depend upon the conditions of the sample preparation of the matrices. However, we have never experienced such a low detrapping temperature as 26 K for Xe matrices slnce these preiiminary experiments. So, the involvement of accidental errors In measurincl the annealina temwature in the Drellminarv - exDeriment . might b e b o r e probabk. ’ (15) Pacansky, J.; Coufal, H. J . Chem. Phys., 1979, 71, 2811. (16) (a) Aditya, S.; Wilky, D.D.;Wang, H. Y.; Willard, J. E. J. phys. Chem. 1979, 83, 599. (b) Wang, H. Y.; Willard, J. E. IbM. 1979, 83, 2585. (17) (a)Toriyama, K.; Iwasaki, M., unpublished wok. (b) Muto, H.; Nunome, K.; Iwasaki, M.,submitted to J. Phys. Chem. (16) Yang, K. J. Am. Chem. SOC.1962, 84, 3795. (19) Toriyama, K.; Iwasaki, M.; Nunome, K. Int. Congr. Radiat. Res., 6th, 1979 1979, abstract p 176. (20) Unpublished work. (21) Sullivan, J. H. J . Chem. Phys. 1959, 30, 1292; 1962, 36, 1925. (22) Morton, J. R.; Preston, K. F.; Strach, S. J.; Adrian, F. J.; Jette, A. N. J. Chem. Phys. 1979, 70, 2889. (23) Miyazaki, T.; Kasugai, J.; Wada, M.; Kinugawa, K. Bull. Chem. Soc. Jpn. 1978, 5 1 , 1676. (24) Iwasaki, M.; Toriyama, K.; Muto, H. J. Chem. phys. 1979, 71,2853. (25) Kinugawa, K.; Miyazaki, T.; Hase, H. J. phys. Chem. 1978, 82, 1697. (26) Bouldin, W. V.; Gordy, W. Phys. Rev. A 1964, 135, 806. (27) Kinugawa, K.; Miyazaki, T.; Hase, H. Radiat. Phys. Chem. 1977, 10,341.
Influence of Hydrophobic Solutes on the Dynamic Behavior of Water K. Hellenga,+J. R. Grlgera,’ and H. J. C. Berendsen’ DepaHment of Physical Chemistty, University of Groningen, Nyenborgh 16, 9747 AG Groningen, The Netherlands (Received: March 12, 1979; In Final Form: April 30, 1980)
From dielectric relaxation measurements it is concluded that thermodynamic quantities which characterize hydrophobic interactions are directly related to changes in the dynamic properties of water. Very sensitive dielectric difference measurements in the range 8-25 GHz have shown that the dielectric relaxation rate of water is slowed down by the dissolution of largely hydrophobic solutes like aliphatic monohydroxy alcohols and monocarboxylic acids at concentrations between 0.001 and 0.1 M. The amplitude of the water relaxation decreases in agreement with the volume fraction occupied by the solute, which excludes the occurrence of strongly bound water molecules. No spread in relaxation times of the solvent was found. Comparison with relaxation times from nuclear magnetic resonance indicates that the rotation of higher alcohols does not contribute separately to the dielectric relaxation; the hydroxyl groups probably participate in the hydration layer. Experimental results are analyzed in terms of a model in which water molecules are exchanging between the bulk solvent and the first hydration layer around the solute molecules. Depending on the number of water molecules in the hydration layer, their rotational motions are found to be between two and three times slower than those of the bulk solvent. The results are not sensitive to the value of the exchange rate between the two species. A linear relation between the dynamic effect and the entropy of solvation is demonstrated for alcohols up to C6isomers, corresponding to an entropy decrease of 1 eu per mole of water for 25% reduction of the rate of rotation. The experiments suggest that the entropy of solvation due to the solute is positive but overcompensated by a strongly negative contribution of the water molecules.
I. Introduction Model building of hydration cages with structures similar to those of gas hydrates and other clathrates have The interaction between water and hydrophobic solutes evoked pictures of rather static hydration structures. is characterized by lairge negative entropy effects, ASosol, However, NMR experiments by Hertz and Zeidlers indifor the transfer of hydrophobic substances from an apolar cated that the rotational motions of water molecules in the environment into wa1;er.l The origin of the large ASosol first hydration layer around tetraalkylammonium salts in values is usually ascribed to an increase of water aqueous solutions are only about two times slower than “structure” in the hydration spheres around hydrophobic in bulk water. In addition, cold neutron scattering ex~ o l u t e s ,although ~,~ a contribution from the decrease of rotational freedom of solute particles cannot be e ~ c l u d e d . ~ , ~ periments’ on 2-methyl-2-propanol solutions have shown that also translational motions of water molecules are somewhat slowed down. Both effects have been borne out Vrije Universiteit Brussel, Faculteit Wetenschappen, Organische by NMR experiments of Goldammer et alS8f9Microwave Chemie, Brussels, Belgium. *Departmentof Biophysics, I.M.B.I.C.E., 1900 la Plata, Argentina. dielectric measurements on aqueous solutions of soluble 0022-3654/80/2084-2381$01 .OO/O
0 1980 American Chemical Society
2382
The Journal of Physical Chemistry, Voi. 84,No. 19, 1980
alcohols1° and tetraalkylammonium salts'l have indicated an increase in rotational correlation time of the water molecules in the entire solution. Until now solutions with solute concentrations below about 1 M could not be studied for reasons of limited instrumental sensitivity. This has prevented the investigation of the dynamic behavior of water in solutions containing largely hydrophobic solutes. The importance of hydrophobic interactions, in particular in biochemical systems as a major factor determining the stability of macromolecular conformation, justifies a further investigation into the molecular basis of this interaction. The purpose of this article is twofold: (a) to present experimental data from microwave dielectric and NMR measurements giving quantitative information about rotational motions of water molecules in dilute (0.1 M or below) aqueous solutions of alcohols and carboxylic acids up to C8isomers and (b) to investigate the assumption that the thermodynamic effects can be ascribed to an "increase in water structure" which should involve a slowing down of the rotational (and translational) motions of the solvent molecules. This is done by correlating the shift in dielectric relaxation frequency of water resulting from the introduction of the solutes with the corresponding entropy of transfer, ASosol.
Hallenga et al.
a
'9.50
10.00
10.50
11.00
Ib50
12.00
12.50
LOG W
b N I
0
.".A
I
11. Theory A. Phenomenological Description. Earlier dielectric measurements on pure waterI2-l4and aqueous solutions1619 have given a large amount of information on the phenomenology of the dielectric relaxation of water as well as on the changes induced therein by solutes of different nature. It has been c ~ n c l u d e d ' ~that J ~ water behaves as an almost perfect Debye dielectric with a very small spread in relaxation times. Such a behavior is quite well described by the empirical Cole-Cole equation:20 E* =
E,
(to
- €-)/[I
(iW70)~-~]
0-t
(1)
9.50
LOG W
Here E* is the complex dielectric constant with low- and high-frequency limits eo and ,e, respectively, w is the angular frequency of an applied electric field, T~ is the dielectric relaxation time (in this article often replaced by wo 1 / ~ and ~ ) ,a is a parameter indicating the spread in relaxation times. For water a is smaller than 0.02; with a = 0 eq 1 reduces to the Debye equation. The addition of solutes such as inorganic salts, alcohols, amines, carboxylic acids, tetraalkylammonium salts, and heterocyclic organic compounds tends to decrease eo and to shift wo either upward or down.lg Both parameters depend linearly on solute concentration up to about 1.5 M, while a increases slightly to values of 0.1-0.2. The decrease of eo with solute concentration is directly related to the replacement of water by solute molecules with a lower permittivity in a rather straightforward way. The drops in eo may also reflect changes in the average relative orientation of solvent molecules around solute particles (e.g., ions). The change in relaxation frequency has been interpreted as a structure-breaking or structure-making effect of the solute molecules on the structure of liquid water. According to the results obtained on aqueous solutions, we assume a linear dependence of eo, E,, and wo on solute concentration c, expressed in molarity (eq 2). The paE-(c) = t,(l - bc) t0(c) = e O ( l - aoc) (2) a&) = wo(1 + dc) rameters ao, b, and d can be interpreted as the relative change in eo, E,, and wo per unit concentration of solute
Flgure 1. Dielectric difference plots, he' in part a, A€'' In part b, between water and five hypotheticalsolutions with lower eo values (a, = 5%), lower e, values ( b = lo%), and relaxation frequencies changed by, respectively, -10 (I),-5 (+), 0 (XI, 5 (Y), and 10% ("). Cole-Cole parameters of water (a)and of the solutions (p) are equal: a = P = 0.014. eo = 78.4, e, = 5.3, and w, = 1.22 X 10" s-'; solute concentration is 0.1 M.
introduced. We further suppose that the spread in relaxation times, p, for a solution will be different at least in principle from a,ita value in pure water. Combination of eq 1 and 2 leads to a theoretical expression for Ae(c) ("CHzO
- esoln):
At(c) = e,bc
+
€0 -
1
Em
+ (i(J/Wo)l-a
- eO(l - aoc) - ~ , ( 1- bc) 1 + [iw/wo(l
+ dc)]'"
(3) Equation 3 has been used to calculate Ae for various values of the parameters over the frequency range 5 x 108-5 X 10'l Hz, which covers the water relaxation completely. The results have been visualized in plots of A€' and Ad', the real part and the negative imaginary part of AE(c), respectively, against the logarithm of angular frequency. The parameters eo, e,, and wo were fixed during all computer runs at the respective values 78.4,5.3,and 1.22 X 10" rad/s according to results of Pottel et al.14 at 25 "C. Difference curves as shown in Figures 1 and 2 provide a very useful sensitivity analysis for the dielectric difference method. The curves is Figure 1show the large influence of the relaxation frequency shift, being -10, -5,0, +5, and +lo%, respectively. Parameters a. and b were fixed for
Dynamics of Hydrophobic Solvation
a 3
li
8
The Journal of Physlcal Chemistty, Vol. 84, No. 19, 1980 2383
We consider the following model as adequate for the influence of a solute on the water relaxation. The water molecules are classified into two species: the bulk water A and the hydration shell B. The intrinsic relaxation times are TA and ~ B respectively. J In order to d o w for exchange of molecules between A and B in which the polarization is at least partly preserved, we introduce exchange terms with a rate constant k in the time-dependent equations for the polarization of both species. This model leads to the following coupled rate equations: dmA/dt - ( ~ / T A + kp)mA + k(l - p ) m ~ (4)
1 dmB/dt = kpmA - [ 1 / + ~ k(1 - p ) ] m ~
(5)
Here mA and mB are the total polarizations of species A and B, respectively, p is the fraction of solvent molecules in species B, and kp and k ( 1 - p) are the rate constants for the transfer from A to B and from B to A, respectively. The general solution has the form:
+ bl eXp(+zt) mB(t)/(mA" + mB") = a2 exp(-Xlt) + b2 exp(-X2t) mA(t)/(mA" + m~') = a1 eXp(-Xlt)
(6)
(7)
where mAo = mA(0) and mgo = mB(0). The boundary condition at t = 0 gives a1 bl = 1 - p a2 bz = p
+
+
For the total fraction x of the polarization relaxing with time constant Xl one can derive X ai + a2 = 1 x 2 - P / ~ B (8) - (1 - p)/7A]/(b - X i ) with h2,l = + '/2(1/7A + 1/7B + k) f 1/2[(1/7A - 1/'d2 2 k ( 1 / 7 ~- 1 / 7 ~ ) 4kp(l/~A- 1 / 7 ~ ) k2]'/' (9)
+
+
LOG W
F@re 2. Dlelectrlc difference plots between water and h e hypothetical solutions. Parameters as in Figure 1 except for a larger spread in relaxation times of the solutions: p = 0.024.
this run at 0.05 and 0.10, respectively, while cy = P = 0.014. Clearly, small shifts in relaxation frequency can be determined accurately from At measurements around w". Figure 2 shows the influence of an increased spread in relaxation times. Parameters a,,, b, d, and a have the same values as in Figure 1, while P = a + 0.01 = 0.024. This small increase of spread in relaxation times is reflected quite strongly in both Ad and Ad', while the influence of shifts in oobecomes relatively smaller. All curves shown demonstrate the possibility to determine shifts of wo as well as changes in the spread in relaxation frequencies from accurate A6 measurements over a relatively limited frequency range. B. Relaxation Model with Molecular Exchange. The change in macroscopic dielectric relaxation frequency of the solvent has been interpreted until recently as an overall and homogeneous effect of the solute molecules on the bulk solvent. GieseZ1has given an interpretation on the molecular level by using a model in which water molecules exchange with a finite rate between sites in a hydration layer around solute molecules and positions in the bulk solvent. The jump frequency between sites and the dielectric relaxation frequencies in the two different environments as well as a possible difference in dipolar correlation of the solvent (Kirkwood g factor) at the two sites appear to be important factors that need further consideration.
Here the first subscript refers to the + sign. The time dependence of the total solvent polarization can then be described as mA
+ mB = (mA" t mgo)[xe-hlt + (I - ~ ) e - ~ a t(IO) ]
We will consider this solution for four different values of the exchange rate parameter k: (a) Infinitely fast exchange (k = a). Here we have A1 = p / +~(1 - p)/7A A2 = k with x = 1 (1la) There is only one observable relaxation process with an average relaxation rate X1. (b) The exchange process is as fast as the relaxation in the bulk solvent ( K = 1/7A). Here one gets X2,l
= 1/2(2/7A + 1/7B) '/2(1/7A)[(7A/7B)' + 4P(l - 7A/7B)I1"
x = / X 2 7 A - pTA/TB - (1 - p)]/[(TA/TB)' + 4p(l -
1
TA/~B)
(1Ib)
(c) The exchange occurs at the same rate as the relaxation in the hydration layer (k = 1/7B), This gives X2,l
= 1/2(1/7A + 2/7B)
f
1/2(1/7A)[1 + 4(1 - P ) ( T A / T B - 1)7A/7B11'2
x=[
k - P~~
A / ~ -B (1 - p ) l / [ l
+ 4(1 - p) X
(7A/TB
- 1)7A/7BI1/' (1lc)
(d) No exchange at all (k = 0). This is simply 1/7B x 2 = 1/?A x =p (1W It is reasonable to assume that k will be of the order of rA-I or TB-~, because the diffusion time between the bulk A and
2384
The Journal of Physical Chemistry, Vol. 84, No. 79, 7980
the monolayer B is expected to be comparable with the dielectric relaxation time. In the following analysis we show that the experimental observations are not very sensitive to the value of k . In our difference measurements we determine the difference between the polarization of a solution as described in eq 10 with the pure solvent polarization given by eq 12.
Hailenga et al.
a 1.or
mHzO= moH20e-WOt (12) In fact the steady-state response to a single excitation frequency w is determined. This corresponds to the Fourier transform of mHz0- (mA+ mB)written as eq 13.
Here the factor f is the total volume fraction of water in solution, as it contributes to the dielectric polarization. We now want to compare eq 13 with the phenomenological description of the dielectric differences as given in the previous paragraph (eq 3). The function At(o) is equivalent in both equations if h = 0, CY = p = 0, and the trivial difference in em is neglected. The Cole-Cole equation cannot be Fourier transformed in an analytical closed form, but the small values of CY and p in the plots of Figure 1 have almost no influence on the A€ plots when a = p. The parameters of eq 13 now relate to those of eq 3 as h1 = p/TB
+ (1- p)/TA = W O ( 1 + dc) f = (1 - aoc)
Since wo =
1/TA,
b
(14a) (14b)
we see that
04c) p[TA/TB - 11 = dc Hence, at least in the case of infinitely fast exchange ( k = a),the hydration fraction and the relaxation rate of the hydration layer cannot be determined independently. Assuming a faction p = 0.05 for a 0.1 M solution (e.g., a C4alcohol having a hydration number of about 25), one sees that the d values -0.10 through +0.10 (used as parameter in Figure l) correspond to T A / T B values of 0.8 through 1.2. In Figure 3 At is plotted according to eq 13 with parameters f = 0.995 (corresponding to a 0.1 M solution with a. = 0.05 as in Figure l),p = 0.05, and T A / T B appearing as parameter. Four exchange conditions are shown. Although h = 0 and h = are clearly distinguishable, there is virtually no change in the At curves with k for any reasonable value around 1/TA, 1/TB. This demonstrates that the T A / T B ratio, or rather the value of p . [ ( T A / T B ) - 11, to be determined from the experimental A6 values, is almost independent of the actual exchange rate, while the latter cannot be derived from experimental data.
~
01
w TA
1
10
Figure 3. Dielectric difference plots, A€' in part a, At" in part b, between water and five hypothetical solutions using the exchange model described in section IIB, for various values 7A/7Bof the ratio of relaxation tlmes of pure water and solvation layer: (solid line) infinitely fast exchange (k= m); (broken line) no exchange (k= 0); (dots) intermediate exchange (k = 7 A - l and k = TB-' are indistinguishable).
00
111. Experimental Section 1. Instrumentation. A. Dielectric Measurements. The microwave system operating in the frequency range 8-26.5 GHz was designed to detect extremely small changes in the real and imaginary part of 6. It has a sensitivity at least one order of magnitude better than other systems described in the literature except for the microwave bridge reported by Van Casimir et aL2' Both his system and ours have a sensitivity of 0.005 in Ad and At". The system used for our experiments is based on the principle of cavity resonance perturbations. A block diagram is shown in Figure 4. The features of the system contributing to the enhanced sensitivity are the following: (1)the combination of a high-frequency stability of the microwave source (1 part in los over a period of seconds) with a phase-sensitive
i
'
MICROWAVE GENERATOR
'* Isvncsnq I , gy ~
I
~
I
COliROl
REFEREHCE 1
1
Flgure 4. Block diagram of the microwave system used for the dielectric difference measurements. A double frequency modulation, 1 kHz through the frequency control and 2 MHz directly, allows the simultaneous registration of the first and second derivative of the frequency response of the cavity.
detection based on frequency modulation of the stabilized source and (2) careful construction of the microwave cavities in particular with respect to mechanical stability
Dynamics of Hydrophobic Solvation of the sample tubes and temperature control to within 0.002 OC of cavities and samples. Details of the microwave system, the cavity constructions, and the derivation of formulas for the cavity resonance perturbation have been given e l ~ e w h e r e . ~ ~ ? ~ ~ It is important to realize that Ad and Ad’ cannot be determined independently from changes in the resonance frequency Aw, and resonance width Awl, respectively. For cylindrical cavities operating in a TEolnmode as well as for rectangular cavities excited in a TElh mode, the change in resonance written as a complex quantity A o = iAwl + Aw2 is related to the complex quantity AE = A d - iAt” as (C1 C2d’)Ad - C2Ad’ = -C4A02
+
CZLL‘IAE’ 4- (C,
+ C~E’)AC”= C ~ A W ~
The C‘s are constants expressed in terms of dimensions of the cavity and the sample tube. The equations are a second-order series expansion of the perturbation formula derived in a general form by SlaterZsand more specifically by Collie et for cylindrical cavities. A derivation of these equations is given in ref 23 and 24. For rectangular cavities with length 1, width b, in which a capillary tube is inserted with radius r at a position of minimal electric field, the constants are C1 = wr3r4n2(1- 5r2n2rZ/24l2)/bl3
C2 = w3.rr3fln2/12b13c2
c, = 1 B. Magnetic Relaxation, Deuterium spin-spin relaxation times of dilute solutions of some deuterated alcohols in H20 were determined with a sensitive NMR pulse spectrometer equipped with a signal averager using the modified Carr -Purcell pulse sequence.27v28Density measurements of alcohol and acid solutions were performed with an Anton Parr IIMA 02/10 vibrating sample digital instrument at 25 “C. 2. Materials. All1 alcohols and acids were obtained commercially in pa grade quality. Their purities, tested by gas chromatographly,were found to be better than 99%. Concentrations were 0.1 M except for l-hexanol, 2,2-dimethyl-l-butanol, and some higher heptanols and octanols, which had solubilities below 0.1 M. 2-Methyl-2-propanol solutions were prepared over the range 1mM to 0.4 M in order to study concentration dependence of Ad and AE”. IV. Results 1. Dielectric Measurements. The assumption of linear dependence of A€ on riolute concentration was first tested carefully on 2-methyl-2-propanol solutions at 18.1 GHz over the concentration range 0.001-0.4 M. The results, shown in Figure 5, demonstrate almost perfect linearity over the whole conceintration range studied. This result is not a trivial one. I t might very well be that different water molecules will have a different dielectric environment when the distance between solute particles increases upon dilution. For 0.1, 0.01, and 0.001 M solutions the distances between solute molecules on the basis of a cubic arrangement are 25,54, and 117 A, respectively. It will be clear from this that averaging of differences in dielectric relaxation times within a time 7” becomes questionable upon dilution from 0.1 to 0.001 M. Translational diffusion with a root mean square displacement of 3.5 A in s will be too slow to achieve the averaging. Either the internal field and the orientational correlation of neighbouring molecules arc! effective factors in averaging the dielectric relaxation time or the rotational motions of water molecules close to a solute molecule are not very different
The Journal of Physical Chemistry, Vol. 84, No.
19, 1980 2385
0 7,
T BUTANOL 25’C,l&3 G
ON 0 61
,
M
,
I
A€’ W
i iI
I
4
0
000
002 003 004 CONCENTRATION(MOLES/LITRES)
001
005
Flgure 5. Dielectric difference measurements (At’ and Ae“) on 2methyl-Ppropanolsolutions in the concentration range 0.0008-0.405 M. The points with error limits (I) correspond to the range 0-0.05 M as indicated by the axls labels. The open circles correspond to the range 0.05-0.5M; for these points the units on the axes are 1OX larger than those indicated. from those at distances several tens of angstroms away from a solute particle. Difference measurements on alcohol and acid solutions were performed at 8.12, 10.95, 14.05, 18.35, and 24.53 GHz by using rectangular cavities. All of the raw data can be found in ref 23. Data fitting to eq 3 with variable parameters ao,d, and /3 was done first, using Pottels’s valued4 of 78.4, 5.3, and 1.22 X 10l1rad/s for, respectively, cot e,, and wg. Variation of t, gave no improvement, and therefore the parameter b was kept equal to zero. The complete results are given in Table IA of the supplementary material and in ref 23. (See paragraph at end of text regarding supplementary material.) Differences between experimental and calculated At’ and AE” values are typically 0.05. The standard deviations of parameters a. and d are about 10 and 5%, respectively. An important point is that the spread in relaxation times in the solutions, given by /3, is within experimental error equal to its value in pure water. In addition, least-squares fits to eq 13 using all four exchange velocity conditions ( k = m, k = TA-’, k = TB-’, and k = 0) were carried out. The case k = ~0 is practically equal to the one described above with (Y = /3 = 0 and gave almost identical results. In the exchange conditions k = 7A-l and 7B-l a threeparameter fit using ao,TA/TB, and p as adjustable parameters was not successful. This is not surprising since in eq l l b and l l c p and 7A/7B are not independent (see section IIB). We therefore tried to make reasonable estimates of p from hydration models29and NMR studies3” that give as hydration numbers n for propanol and 2methyl-2-propanol 22 and 25, respectively. Then the least-squares fits with k = 7A-l and k = 7B-l were repeated with fiied values of p (p = 0.04,0.05,0.06, and 0.07). For all measurements we now obtained good fits. Using p = 0.04 (n = 22) for the propanols, p = 0.05 (n = 26.5) for the butanols, p = 0.06 (n = 33) for the pentanols, p = 0.07 (n = 37.5) for the hexanols, and corresponding p values for the carboxylic acids, we could then calculate the most probable values for T A / ~ Band d. The least-squares fits with 12 = 0 did’not give satisfactory results and will not be considered further. The results for k = 7B-1for the alcohols and carboxylic acids up to c6 isomers are given in Table I. Results for tZ = and k = 7A-l are given in Tables IB and IC (supplementary material). Plots of a. vs. d for all substances in Table I (Figure 6) show a rather close correlation between the shifts in relaxation frequency and the changes
The Journal of Physical Chemstry, Vol. 84, No. 19, 7980
2386
Hallenga et al.
TABLE I: Results of Least-Squares Fits of A e Measurements to Eq 1 4 Using Exchange Condition k = 1 / 7 B (Eq l l c ) for Alcohols and Carboxylic Acidsa soln
substance
concn, M
selected p value in %
00
TA /TB
-d
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 21 22 -23 24 25 26 27 28
1-propanol 2-propanol 1-butanol 2-methyl-1-propanol 2-butanol 2-methyl-2-propanol 1-pentanol 2-pentanol 2-methyl-2-butanol 2,2-dimethyl-l-propanol 1-hexanol 3-hexanol 2-methyl-2-pentanol 4-methyl-2-pentanol 2,2-dimethyl-l-butanol 3-methyl-3-pent an cyclohexanol 1,5-pentanediol 1,6-hexanediol propanoic acid butanoic acid 2-methylpropanoic acid pentanoic acid 3-methylbutanoic acid
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.049 0.1 0.1 0.1 0.055 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
4 4 5 5 5 5 6 6 6 6 4 7 7 7 4 7 7 6 7 4 5 5 6 6
0.0791 0.0852 0.0974 0.1034 0.116 0.128 0.107 0.121 0.127 0.116 0.123 0.132 0.138 0.132 0.121 0.135 0.117 0.106 0.123 0.086 0.105 0.103 0.118 0.128
0.572 0.567 0.592 0.578 0.513 0.467 0.631 0.602 0.576 0.627 0.729 0.632 0.606 0.636 0.681 0.599 0.654 0.638 0.650 0.737 0.725 0.710 0.716 0.720
0.171 0.173 0.204 0.211 0.242 0.267 0.222 0.239 0.254 0.224 0.221 0.257 0.276 0.255 0.232 0.280 0.242 0.217 0.241 0.105 0.139 0.145 0.170 0.168
a Selected hydration fractions are given in column 3, the relative changes of e o per molar in column 4. The relative changes in w o given in column 6 are calculated from the T A / T B ratios using d = ~ [ ( T A / T -Bl)] / c .
TABLE 11: Volume Fractions f s of Alcohol or Acid as Calculated with Eq 15 soh
1o3fS
soh
103fs
soln
103fs
1 2 3 4 5 6 7
7.1 7.3 8.6 8.6 8.9 8.0 10.1’ 10.2
9 10 11 12 13 14 15 16
10.1’ 10.2 5.2 11.2 11.5 11.4 6.0 11.2
21 22 23 24 25 26 27 28
10.3 10.0 12.2 6.7 8.2 8.2 9.4 9.4
8
0.:
02
in to for any solution. The magnitude of a pair of a. and d values for isomers of the same alcohol can be quite different, however. 2. Density Measurements. The decrease in to upon addition of solute to pure water can be related to the corresponding change in water content, i.e., with the volume fraction of the solute f,, which was calculated from density measurements. Assuming densities po both for pure water and for the water fraction in solution and pc for a solution containing a solute with molecular weight M , we can write eq 15. Table I1 contains the f, values f, = 1 - (P, - CWPO (15) calculated from density measurements on most of the alcohol and acid solutions studied. A theoretical relation between a. and f,/c can be derived from the well-known dielectric mixture expression31 (e
- e l ) / ( € + ~ € 1 =) f,(cz
+~€1)
-~J/(Q
/
(16)
In which e, el, and t2 denote permittivities of the solution, the solvent, and the dissolved particles, respectively. The factor x denotes the shape factor which is equal to 2 for spheres. The relative change in eo (defined earlier by co(c) = to(l - aoc))can be directly related to f,/c as
Taking €1 equal to the low-frequency dielectric constant eo of H20, e equal to the low-frequency permittivity of the
/
d
/
fl‘
01
1 0 10
I
0 05
1
0 15
00
Flgure 6. Relative shift In relaxation frequency dvs. relatlve reduction In low-frequency dielectric constant a for a number of alcohol and acld
solutions,as spclfed In Table I. Data were derived from a least-squares flt to eq 13 with Intermediate exchange rate k = iB-’ (eq l l c ) .
solution, c2, the permittivity of the solute, equal to 4, and x equal to 2, we obtain the solid line shown in Figure 7. Here the experimental a. and f,/c values from Tables I and I1 have also been plotted. For ellipsoidal solute particles with t2 = 4 and axial ratio 2, the shape factor x is 1.8 (see ref 19),which would increase the slope of the solid line by about 4%. The dotted line in Figure 7 corresponds to a = 1, which disregards dielectric mixing effects. The values of a. can also be compared with literature values of the low-frequency dielectric decrement for
Dynamics of Hydrophobic Solvation
o'201 1 0.15
The Journal of Physical Chemistry, Voi. 84, No. 19, 1980 2387
/
/
parameter of the field gradient tensor, e& the deuteron quadrupole moment and 7J2) the effective rotational correlation time of the second-order spherical harmonics. When, in addition to rotation of the whole molecule with correlation times TM, intramolecular rotation occurs with correlation time q,the effective correlation time is equal to 7, =
0.ost
/ ,'
Figure 7. Plot of ao,the relative change In E? per molar solute, vs. molar solute volume fraction f,/c. The solid line is drawn according to eq 17. The broken line Indicates a dielectric decrement proportional to volume fraction, disregarding any mlxture effects.
TABLE 111: Values of T,-l,r c ( 2 )and , T M ( ' ) Obtained fiom NMR Experiments on Dilute Solutions of D,O and Some Deuterated Alcohols in WateP T - l e a q Q / 10". substance k1' h, kHz r e ( ' ) , s r _ ~_ ( ~s ) , ____--1.8 1.8 248 1.7 HDO 3.5-9 170 1.8 1.8 (CD,),C(D)OH 4.1 170 4.1 1.6 CH,CDOHC,H, 6.7 5.7 2.2 170 C,Dll OH(cyc1) a Quadrupole coupling constants have been used from ref 22 and 23. Concentrations were between 0.1 and 0.5
M. aqueous solutions. From data quoted by Hasted12we can derive a dielectric decrement for 1-propanol, 2-propanol, and 2-methy1-2-propmol of 4.0,4.3, and 6.3 units per molar concentration. This corresponds to values of a,, of 0.051, 0.055, and 0.080, respectively. Our values (Table I) are about 50% higher. It,should be kept in mind that our data are derived from measurements near the relaxation range only. The difference with results a t low frequency is sensitive to the precise behavior of the dielectric difference in the low-frequency wing of the dispersion. 3. NMR Experiments. Measured deuteron spin-spin relaxation times T 2 were submitted to a least-squares analysis that fits the echo signal to a theoretical curve with one or more exponentials as described by Edzes.28 In all cases only one relaxation time was observed. The results of the least-squares fit are summarized in Table I11 (columns 1and 2). Assuming intramolecular nuclear quadrupole interactions to be the only effective relaxation mechanism and molecular motions to be much faster than the Larmor frequency, we can use the general formula32 for nuclear quadrupole relaxation in the limit of extreme motional narrowing to calculate the effective rotational correlation time of the alcohol molecules:
--1 ---1 -
In this formula T I and Tzdenote the spin-lattice and spin-spin relaxation time, while I is the nuclear spin, eq the principal value of the field gradient, 9 the asymmetry
y4(3 cos2 - 1)27M+ {I - 1/(3 cos2 ')' - 1)2)(1/7i+ 1/7M)-l (19)
where y is the angle between the axis of internal rotation and the principal axis of the field gradient tensor. This formula applies for both T , ( ~ )and T , ( ~ ) ,T , ( ~ )being the rotational correlation function of the first-order spherical harmonic (see Zeidler3,). For CD, groups y is equal to 109.5', while 9 can be neglected in this case. Taking quadrupole coupling constants e2qQ/ h from the literat ~ r e we ~ ~calculated v ~ ~ T C ( ~ )as shown in Table I11 (third column). The T , ( ~ )result for HDO agrees reasonably well with literature values as given by Hertz36and by Hindman36ranging from 1.9 to 2.4 ps. The measurements on the monodeuterated butanol and the undecadeuterated cyclohexanol give direct information about the rotational motions of the C-OH group, since there are no additional internal motions involved. The T C ( ~ )values of both alcohols show that their molecular rotations are 2.5-3 times slower than those of HDO. For the heptadeuterated propanol rM@) can be calculated from eq 18. Depending on the value inserted for the correlation time of internal rotation around the C-C axis T ~ @values ) are found in the range 3.5 X s) to 9 x s (for @) = 9 x 10-l~ s (for q@)= 3.5 x s). This is consistent with the conclusion that rotational motions of the alcohols are about three times slower than those of water. 4 . Determination of Thermodynamic Transfer Functions and Their Correlation with the Shift in Relaxation Frequency. The changes in the thermodynamic quantities ASosol and AHosolcan be calculated approximately for many alcohols from the temperature dependence of their solubilities in water. For a quantity of alcohol (phase 2) in equilibrium with an amount of water (phase 1)we can write for the thermodynamic potential of 1mol of alcohol, considering both phases as ideal solutions plo
+ R T In x1 = p Z 0 + RT In x2
(20)
with plo = H10- TSlo and p20 = HZo- TSzobeing the standard thermodynamic potential, enthalpy, and standard entropy per mole in phase 1 and 2, respectively. The standard state refers to alcohol mole fractions of 1in both we can find ASomI from phases. Using dApo/dAT = -AS", the temperature dependence of RT In (x1/x2). Strictly, one does not determine ASos01 between the aqueous solution and the pure alcohol since the alcohol phase contains some water. Because the temperature dependence of x1 is two orders of magnitude larger than that of x2, it is clear that entropy effects in the aqueous phase are in fact determined. Using solubility data from Seidel137we calculated ASomI values as given in Table IV (column 2). All values are calculated at 25 "C. Also AHoaolvalues have been calculated from plots of R In (xl/x2) vs. 1/T (column 3). For comparison AHsol values determined from calorimetric r n e a s ~ r e m e n t are s ~ ~given in column 4. The reasons for the differences between column 3 and 4 will be discussed below. The change in a dynamic property of water, viz., the macroscopic dielectric relaxation frequency, is plotted vs. the change in the equilibrium properties ASoml and AHosol
2388
The Journal of Physical Chemistry, Vol. 84, No. 19, 1980
TABLE IV : AS" sol and A HsolValues for Transfer from the Pure Liquid into an Aqueous Solution for Several Alcoholsa soln
substance
-AS"p,1
1-propanol 2-propanol 1-butanol 4 2-methyl-1-propanol 5 2-butanol 6 2-methyl-2-propanol 7 1-pentanol 8 2-pentanol 9 2-methyl-2-butanol 10 2,2-dimethyl-l-propanol 11 1-hexanol 1 2 3-hexanol 1 3 2-methyl-2-pentanol 1 4 4-methyl-2-pentanol 1 5 2,2-dimethyl-l-butanol 1 6 3-methyl-3-pentanol 1 2 3
-Arsol
AHsol
11.4 11.8 20.6
1.5 1.6 5.1
14.2 18.3 22.9 16.4
1.4 3.0 5.2 2.2
2.419 3.124 2.249 2.226 3.150 4.172 1.868
19.8 22.2 20.0 21.3 24.0
3.8 3.9 2.7 4.6 2.5
IL
i
0
5
8
f
A 15
t I
I
I L
16
9
Pa
2
P = N-E,,, 3kT
0
5
a
I
relaxation. One can now approximate the dielectric difference between pure water and alcohol solution by eq 21,
m
m
9
where A is the contribution to to due to alcohol polarization, T, is the relaxation time of the alcohol, and CY is defined by eq 17. Putting an alcohol molecule in a spherical cavity in water, the cavity field E,, will be roughly 1.5 times the applied field E. The induced polarization AP due to the alcohol dipole moments pa is given by eq 22.
A
Pa z
2
N-E 2kT
Here N is the number density of alcohol dipoles. From this we find
A 15
3
7
I
:z
A
Flgure 9. Same data as in Figure 8, plotted vs. the molar enthalpy of solution AHo& (Table IV).
A
rn
A 13
2Ll
-0 20
13
t
A 16
ikcol/Molel
; :1
-0.2L
-O
4.435
Columns 2 and 3 give values derived from the temperature dependence of solubility. Column 4 gives AHmlvalues determined from calorimetric experiments. AS" is given in cal mol-' K-', AH in kcal mol-'. -0.28
-:.r
Hallenga et al.
0
L
I
I
I
where t, is the vacuum permeability (8.854 X 10-l2F m-l). Inserting numerical values for a 0.1 M solution ( N = 6.022 X loz5mol mT3)with ha = 5.87 X C m (1.76 D), we obtain A = 0.0285. The numerical value of af, is about 0.012, corresponding to a At of about one unit. Thus, the effect of water displacement is at least one order of magnitude larger than the contribution of alcohol dipoles. It follows that for any relaxation time of the alcohol molecules the observed dielectric differences can only be due to relaxation shifts in the water. The experiments indicate that all or nearly all water molecules contribute to the dielectric relaxation. The relation between dielectric decrement and volume fraction, as given in Figure 7 and eq 17, shows a discrepancy of 20%. This corresponds to not more than one water molecule per alcohol molecule, as compared to about 20 water molecules in the hydration shell. Moreover, this discrepancy cannot be considered significant in view of the uncertainty of mixture theories. There is another reason to reject a model with a "static" hydration shell. The microscopic dielectric relaxation time of an alcohol molecule rotating together with its hydration shell is in contradiction with the NMR results as has been pointed out by Hertz? From the Stokes-Einstein equation for a spherical particle in a viscous medium, the correlation time of rotational Brownian motion for a hydrated 2methyl-2-propanol molecule can be calculated as 1.3 X lO-'O s. This is at least an order of magnitude larger than the correlation time derived from the deuterium spin-lattice relaxation data.
Dynamlcs of Hydrophobic Solvation
The Journal of Physical Chemistry, Vol. 84, No. 79, 1980 2389
in rotational correlation time. For the latter Eyrings All arguments given above support the conclusion that transition-state theory is used. From this theory, presented “immobilized” hydration layers around hydrophobic soas Appendix A (supplementary material), it is concluded lutes do not exist. A slowing down of rotational motions that a 25% change in relaxation frequency corresponds to in a monomolecular hydration layer by about a factor of less than a 0.05-unit change in entropy of the hindered 2 is sufficient to explain the observed increase in the rotator. Apparently the entropy change of the rotator macroscopic dielectric relaxation time of the bulk solvent. accounts only for a small part of the entropy of solvation. We have shown that this conclusion holds almost indeA number of two-state and other mixture models for pendent of the value k of the exchange rate from k = I/.Q water have been proposed (for review see ref 43 and 44). to k = 03. The results of the computer fits are even virTheir common feature is that a low-density quasi-lattice tually indistinguishable for the most probable range of k structure (state 1)mixes with one or more higher density values (12 = l / r Bto k = 1f T ~ ) .We have also demonstrated states (state 2). The presence of a hydrophobic solute that the two parameters a. and d (as defined by eq 14) favors state 1 and causes a shift towards state 1, thus obtained from the computer fits are also almost irtdelowering enthalpy and entropy. The difference in standard pendent of p , the froction of solvent in the hydration layer. entropy between the two states (So2- So1)depends on the That is to say, different computer fits using different p particular model, two typical examples being 9.8 eu values between 0.04 and 0.07 give less than 5% changes ~ ) . mole fraction (Wada46)and 14.9 eu ( M i k h a i l o P ~ ~The in the a. and d values obtained for the same set of data. in state 1 is in these cases 0.33 and 0.80, respectively, at There is a large uncertainty in the values of AS”,1 and 25 “C. AHoBol determined from solubility as given in Table IV, It is not an unreasonable assumption to identify state which needs some dicicussion. The far more accurate dH,l 1 and values obtained directly from calorimetric e ~ p e r i m e n t s ~ ~ t ~ ~ state 2 with two exchanging dielectric species with intrinsic relaxation frequencies w1 and w2. For fast ex(Table IV,column 4) demonstrate the large deviations that change the bulk dielectric relaxation frequency wo becomes may occur in AH, values obtained from solubility. Systhe weighted average tematic errors may occur because of temperature dependence of activity coefficients in the (not so dilute) saturated wo = xu1 -I-(1- x)w2 alcohol solutions. Other ASosol values than those given in where x is the mole fraction of water in state 1. A shift Table IV obtained with different methods are available in x due to the presence of a solute now perturbs both for only a few alcohols, or they depend on the assumption of a perfect mtropy/enthalpy c o m p e n ~ a t i o n . ~We ~ ? ~ ~ relaxation and entropy, yielding a linear relation therefore have taken the ASosolvalues from solubility. w2 - (JJ1 Although the thermodynamic parameters are rather inAw = AS s o 2-SO1 accurate, we consider the entropy vs. relaxation frequency correlation, as showri in Figure 8, to be real. The correwhere AS is the excess entropy per mole of water. lation between p H s o l and relaxation frequency, shown in The ratio w1f w2 cannot be derived from experimental Figure 9, is much less convincing. The enthalpy of solution data. Considering the limiting case w1
