J. Phys. Chem. 1981, 85, 733-739
b C cj
2 d2
D F
H R V
(u) W X
1
term defined by eq 29 solute concentration term defined by eq 30 solute concentration at the wall of accumulation integration constant in eq 38 integration constant in eq 39 diffusion coefficient term defined by eq 23 plate height retention ratio local velocity cross-sectional average velocity channel thickness distance from wall of accumulation mean altitude of the solute zone from wall of accumulation
X
x
P Y
7t3
Langevin function defined by eq 44 solute migration velocity nonequilibrium parameter defined by eq 16 l/W
reduced velocity defined by eq 19 independent third-degree parameter defined by eq 48
term defined by eq 20 nonequilibrium parameter defined by eq 15 term defined by eq 34 xll Supplementary Material Available: Appendix 11, a calculation of F,and Appendix 111, a final expression for F (7 pages). Ordering information is available on any current masthead page.
Ultrasonic Absorption in Aqueous Binary Mixtures. 3. Tetrahydrofuran-Water at 25 OC G. Atkinson,” S. Ralagopalan,t and B. L. Atklnson Department of Chemistry and Department of Physlcs, Universlty of Oklahoma. Norman, Oklahoma 730 19 (Received: September 3, 1980)
Ultrasonic absorption has been measured in the tetrahydrofuran/water system at 25 “Cat nine concentrations over the frequency range 0.3-630 MHz. It is shown that the frequency dependence can be fitted quantitatively by the “fluctuation” theory of Romanov and Solov’ev if NMR-determined diffusion data are available. The experimentally determined absorption amplitudes agree very well with those calculated from experimental thermodynamic data by use of the RS theory. The discussion puts the thermodynamic and ultrasonic properties of such systems in the context of the latest picture of water generated by “molecular dynamics” experiments.
Introduction The interaction of simple hydrocarbon derivatives such as alcohols and ethers with water has been studied by almost every physical technique. The most extensive review has been given by Franks* in his monumental series of monographs. Although difficult to explain on a molecular level, the thermodynamic properties of such systems show systematic trends. However, the kinetics of the interactions have been very difficult to grasp. The inherently high rates of the interactions limit our choice of experimental methods. In many attractive methods a complex and approximate theory stands between the experimental data and kinetic results. However, methods such as NMR have given us some very useful information. From the definitive work of Hertz and his colleague^,^^^ it is clear that no long-lived “structures” or “complexes” exist in solutions of simple alcohols and ethers in water. The introduction of monofunctional molecules into water has measurable effects on the water easily seen in such parameters as self-diffusion coefficients and rotational correlation times. The magnitude of the effects are related in subtle ways to the hydrophobic/hydrophilic balance in the molecule. In general, the larger the hydrophobic moiety for a given hydrophilic group, the larger the effects on the water. Molecules with more than one hydrophilic group such as polyols or polyamines usually give very small effects on the water and seem otherwise close to ideal. Ultrasonic absorption has seemed to be a promising approach for the study of dynamics in such systems. The ‘Department of Physics, Nagpur University, Nagpur, India. 0022-365418112085-0733$01.25/0
time scale was in a range appropriate for intermolecular rearrangements. Furthermore, systems that showed pronounced deviations from ideality in their thermodynamic properties also showed enhanced ultrasonic absorption in an easily accessible frequency range. Blandamer4is a good source of chemical references in this area while Litovitz5 and Lambs set a more general background. The traditional interpretation of the excess ultrasonic absorption in alcohol-water and similar systems was the H-bond complex equilibrium originally suggested by and extensively developed by Andreae and his collaborator^.^^ That is, the ultrasonic relaxation was attributed to an equilibrium of the kind
characterized by definite rates. The equilibrium coupled to the ultrasonic wave through a nonzero AV and/or AH. The limited frequency range of much of the early data (1) F. Franks, Ed,“Water-A Comprehensive Treatise”,Plenum, New York Vol. 2, 1973; Vol. 6, 1979. (2) E. v. Goldammer and H. G. Hertz, J.Phys. Chern., 74,3734 (1970). (3) E.v. Goldammer and M. D. Zeidler, Ber. Bunsenges. Phys. Chern., 73, 4 (1969). (4)M.J. Blandamer, “Introduction to Chemical Ultrasonics”, Academic Press, New York, 1973. (5) K. F.Herzfeld and T. A. Litovitz, “Absorption and Dispersion of Ultrasonic Waves”, Academic Press, New York, 1959. (6) J. Lamb in ‘‘Physical Acoustics”, Vol. IIA, W. P. Mason, Ed., Academic Press, New York, 1965, Chapter 4. (7) (a) C. J. Burton, J . Acoust. SOC.Am., 20, 186 (1948); (b) J. H. Andreae, P. D. Edmonds, and J. F. McKellar, Acustica, 16, 74 (1965).
@ 1981 American Chemical Society
734
The Journal of Physical Chemistry, Vol. 85, No. 6, 1981
Atkinson et ai.
made fitting to a single classical relaxation possible. However, as experimental equipment improved it became clear that the relaxation process in most systems was too broad to be fitted to a single relaxation. Therefore, two relaxation fits were introduced (see, for example, ref 8 and 9). That is, it was assumed that two equilibria of type (1) were occurring in some coupled way. For example
Although the data could be fitted quantitatively to such schemes, the resulting analyses had certain very serious problems and raised more questions than they answered. Why were relatively large values of n (or m and n) needed to fit the data and why did the simple 1-1 complex never appear? T o have eight or nine molecules reacting in a single, simple kinetic process seemed unlikely. Why were the rate constants obtained, typically on the order of lo9 s-l, so low? They seemed unrelated to the H-bond formation rates observed in dielectric relaxation. In fact, they seemed slow compared to expected translational diffusion-controlled rates. And this was in concentrated mixtures where molecules only had to rotate to find a H-bond reaction partner! Why were the thermodynamic parameters (AV, AH) obtained not relatable to the known thermodynamics of H-bond formation or the measured thermodynamics of the systems? In an attempt to clarify these questions and put the technique on a sounder basis, we have initiated a systematic measurement and interpretation program on such systems. In the first paper in this series,'O we examined tert-butylamine in H20. We showed that a combination of amine hydrolysis and a fluctuation thermodynamic approach could fit the data quantitatively. However, the amine hydrolysis and lack of standard thermodynamic data on the system made a complete analysis impractical. In a second paper'l we reexamined the p-dioxane-water system that had been previously analyzed by Hammes and Knoche12using a two-step equilibrium model. We showed that the Romanov-Solov'ev fluctuation theory13 fitted the frequency dependence of the broad relaxation very well. Moreover, the amplitude of the absorption was fitted quantitatively by using the known classical thermodynamics of the system. No unlikely kinetic processes needed to be assumed and no peculiar rate constants or thermodynamic parameters were deduced. In an attempt to further demonstrate the utility of the fluctuation approach, we decided to examine the tetrahydrofuran (THF)-water system. Since T H F has only a single oxygen, it would be expected to H bond to water only half as well as p-dioxane. Yet the known thermodynamic parameters show that it is more nonideal than p-dioxane in water.14 And preliminary measurements showed that the THF-water systems gave rise to much (8) M. J. Blandamer, N. J. Hidden, and M. C. R. Symons, Trans. Faraday SOC.,66,316 (1970). (9)K. Tamura, M.Maekawa, and T. Yasunaga, J. Phys. Chem., 81, 2122 (1977). (10)G. Atkinson, M.M. Emara, H. Endo, and B. L. Atkinson, J. Phys. Chem., 84,259 (1980). (11)G. Atkinson, S. Rajagopalan,and B. L. Atkinson, J.Chem. Phys., 72,3511 (1980). (12)G. G. Hammes and W. Knoche, J. Chern. Phys., 45,4041(1966). (13)V. A. Solov'ev and V. P. Romanov, Sou. Phys.-Acoustics, 11, 68 (1965). (14)K.W.Morcom and R. W. Smith, Trans. Faraday SOC.,66,1073 (1970).
70 01
, , ' , ! , ' $
1
I
'
1
,
,
I
,
,
!
,
I
, , , , I 8
100
FREQUENCY lMHrl
Flgure 1. Representative ultrasonic absorptions fitted to single and double relaxations.
larger excess sound absorption a t the given concentration than did p-dioxane-water. Experimental Section The T H F was purified by a standard method.16 Solutions were made up by weight with doubly distilled water and stored in the dark until use. Sound Absorption Measurements. High Frequency. In the range 5-630 MHZ, the absorption coefficient a,defined by I , = IOe-2a+in terms of intensities and distance, was measured by a standard two-crystal send-receive pulse apparatus.16 Three cells of different dimensions with quartz crystals of fundamentals 1, 5, and 30 MHz were used. The electronic equipment was a Matec 6600 mainframe and 700 series plug-ins with a 1235B amplitude monitor. Frequencies were measured by the heterodyne beat method with Hewlett-Packard signal generators and a HP5237C digital frequency meter. The cells were jacketed and maintained a t 25 f 0.05 "C by using a Lauda K2R bath. Low Frequency. In the range 0.3-10 MHz absorption was measured by using the cylindrical resonator method of Eggers.17 A complete description of the apparatus will be published later.l8 It used a Hewlett-Packard 3330B frequency synthesizer and HP461A amplifiers with a DVM. The measurement is controlled and the data accumulated by a dedicated HP9815 calculator. Velocity Measurements. The sound velocities were determined by the "sing-around" techniquelg with an NUS 6100 laboratory velocimeter and a Hewlett-Packard 5237C digital frequency meter. The cell temperature was controlled to 25 f 0.02 OC with a Forma bath. A complete set of (a,f ) data is available from one of the authors (G.A.) on request. Results and Analysis Measurements were made at nine THF-water concentrations in the range XTHF= 0.1 to XTHF= 0.95 a t 25 "C. For each concentration measurements were made between 0.30 and 630 MHz. Figure 1 shows typical data a t XTHF = 0.2 and 0.5. The solid lines are computer-fitted singleand double-relaxation fits of the data obtained by using (15)F. Franks, et al., Trans. Faraday Soc., 66,582 (1970). (16)R. Garnsey and D. W. Ebdon, J. Am. Chem. Soc., 91,50 (1969). (17)F. Eggers, Acustica, 19, 323 (1967). (18)G. Atkinson, S. Rajagopalan, and E. Enwall, Chem. Instrum., submitted for publication. (19)R. Garnsey, R. J. Boe, R. Mahoney, and T. A. Litovitz, J. Chem. Phys., 50, 5222 (1969).
Ultrasonic Absorption in Blnary Mixtures
The Journal of Physical Chemistry, Vol. 85, No. 6, 198 1 735
TABLE I: Parameters f r o m Double-Relaxation Fit f o r the THF-Water System f,, MHz 79.09 61.43 56.47 54.68 59.99 45.09 79.94 53.59 122.13
XTHF
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.86 0.95 av std dev, % range, %
10.0 4.3-16.8
5.9 3.1-11.5
a nonlinear least-squares program (NLSQ). The doublerelaxation expression
clearly provides a very adequate fit of the data. It should be clear that eq 3 is a five-parameter fit where the background value, B, is more or less fixed by the high-frequency limit of the data. The other four parameters (Al, Az,frl, frz) are allowed to vary freely to achieve the best fit. There is no question that the double-relaxation fit describes the data well at all concentrations. Table I gives the actual two-relaxation fitting parameters. A t the bottom of the table, the range of standard deviations of fit and the average standard deviation are given in percentages. And there is no question that one could fit the parameters in Table I with a mechanism such as that in eq 2. However, the results of such an analysis would have the same deficiencies that were described in the introduction. Therefore, we decided to fit the data using the fluctuation approach that worked so well for the p-dioxane system. Then (see ref 20)
- = ?! 1(Z,) cy
P
w
+B
(4) (5)
where P is the molar volume of solution, u is the sound velocity in solution, p, is the adiabatic compressibility and C p is the specific heat of solution.
V” 3 (a2v,/aX2z) H”
f,, MHz 4.66 5.69 5.14 5.39 5.47 4.16 3.65 5.08 4.67 18.9 6.9-46.1
1017A,, s z cm-l 89.25 377.62 485.75 367.70 199.14 135.90 38.27 22.05 31.68
(a2HE/dX22)
YEis the excess thermodynamic property, and
2Itan-l [ ( ~ U ) ~ / ~ Z ,+ 11 - tan-1[(2a)1/2Z, - 11) ( 7 ) where a = w/2D12and w = 27rf. DI2is the mutual diffusion coefficient and I , is the minimum interaction length.
1017A,,s z cm-’ 69.57 346.03 484.47 375.63 391.28 124.73 12.76 7.18 31.99 8.6 3.0-21.4
10”B, s z cm-’ 40.57 50.71 58.09 64.41 77.23 74.03 71.20 58.88 79.98 6.1 0.9-15.7
TABLE 11: Diffusion Coefficients f o r THF-Water at 25 “ C 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.85 0.95 ~
1.580 1.409 1.505 1.773 2.124 2.273 2.555 2.866 2.926
0.873 0.965 1.133 1.363 1.644 1.962 2.304 2.834 3.180
6.857 5.377 4.278 3.330 2.523 2.153 3.353 10.812 21.220
Note: In our work on dioxane-HzO eq 5 was used in the analysis but an incorrect form is given in the paper. In addition, it should be pointed out that eq 6 and 7 are commonly given incorrectly in the original papers. The forms given show great resistance to normal curve fitting and, at times, even contain mathematical discontinuities in the desired frequency range. The concentration fluctuation approach was first suggested for such noncritical mixtures by NomotoZ1but has been most extensively explicated by the Russian In the work it is assumed that the introduction of the hydrocarbon derivative into the water causes fluctuations in the thermodynamic properties of the water in the immediate vicinity of the hydrocarbon. The fluctuations are treated by using the classical methods.25 A Debye distribution of relaxation times (and fluctuation lengths) is assumed. It is also assumed that the fluctuations dissipate by diffusion. The horrendous problems created by longrange correlation in real critical ~ y s t e m s do ~ ~not ~ ~bother ’ us here. Equation 4 is a four parameter fit (Q, ,I DI2,B). B can be estimated from the high-frequency limit of the experimental data. This is not ideal but usually works well. Dlz can be obtained from mutual diffusion experiments or estimated by using NMR self-diffusiondata. Q is directly related to the observed thermodynamics of the system. Only, , Z the minimum fluctuation length cutoff, is not clearly related to other observables. The approach has the virtue that no kinetic model such as eq 1 or 2 need be assumed for data analysis. Obviously, then, no molecular model information can be derived. Equation 4 does not work well as a four-parameter fitting equation for data. Even when sophisticated NLSQ strategies are used, no unique set of parameters can be (21) 0. Nomoto, J.Phys. SOC. Jpn. 11, 818,827, 1146 (1956). (22) V. P. Romanov and V. A. Solov’ev in “Water in Biological Systems”, Vol. 2 Consultants Bureau, New York, 1971. (23) P. Kruus, L. K. Kudryashova, I. G. Mikhailov, and V. P. Romanov, Sou. Phys.-Acoustics, 19, 82 (1973). (24) Yu. S. Manucharovand I. G. Mikhailov, Sou. Phys.-Acoustics, 23, 522 (1977). (25) L. D. Landau and E. M. Lifshitz, “Statistical Phvsics”.. Pereamon Press, London, 1958. (26) M. Fixman, Adu. Chem. Phys., 6, 175 (1964). (27) K. Kawasaki and M. Tanaka, Proc. Phys. Soc., 90, 791 (1967). ~
(20) M. J. Blandamer and D. Waddington, Adu. Mol. Relaxation Processes, 2, 1 (1970).
The Journal of Physical Chemistry, Vol. 85, No. 6, 1981
738
Atkinson et ai.
TABLE 111: Fluctuation Parameters for THF-Water System at 25 " C XTHF 0.1 0.2 0.3 0.4 0.5 0.6 0. I 0.85 0.95
1O60,2, cm2 s-' 6.86 5.38 4.28 3.33 2.52 2.15 3.35 10.81 21.22
lO"Q, em2 s 1.44 i 5.47 f 5.75 i 3.12 t 2.01 f 0.60 f 0.13 f 0.51 f 2.87
0.05 0.13 0.13 0.07 0.09 0.03 0.006 0.02 0.20
lO'l,, 1.25 t 1.30 f 1.27 2 1.15 t 1.21 t 0.96 + 0.71 f 1.45 t 2.06 t
cm
10' 'Qca~cc~, cmz s
1 0 1 7 ~ s* , cm-1
0.03 0.02 0.02 0.02 0.04 0.03 0.03 0.05 0.1
40.57 50.71 58.09 64.41 77.23 74.03 71.20 58.88 79.98
i 2
t t t 2 i 2 2
1.69 4.85 9.12 3.81 6.19 4.92 1.11 0.51 1.73
0.26 3.47 5.55 4.58 3.13 0.52 0.05 0 0.004
0.a
%*3.0 "
THF-H,O
A
0.4
ai
FREQUENCY IMHsI
Flgure 3. Representatlve ultrasonic absorptions In THF-water fitted with fluctuation theory.
XTHF
0.3
Figure 2. Diffusion coefficients for THF-water at 25 OC.
derived. This is caused by the close coupling of D12and 1, in the equation. We have avoided this problem by calculating Dl2 from self-diffusion data. Hertz's group has measured D1, the self-diffusion coefficient of HzO in THF, and D2, the self-diffusion coefficient of T H F in H 2 0 by NMR methods using appropriate isotopic labeling. We fitted the (D1, X,)and (Dz,X,) data with simple polynomials and interpolated the proper D values for our concentrations. DlZ is then calculated by using a Darken expression
THF-HzO 10.74 X
Intercept
I y*
**
.i= 0.1
0.0
0
1
lo+
k
* IO
b
I1
14
10
*; la
x165iMHzl''~
f"l
Figure 4. Low-frequency llmit of excess absorption in THF-water.
Treiner's dataz8was used to calculate the thermodynamic term. The calculated Dl2's and other values are given in Table I1 and shown in Figure 2. Using these Dl2 values, we could fit all the data to eq 4 allowing Q,,,Z and B to vary arbitrarily. In all cases a unique "best fit" could be obtained. The derived values are given in Table 111. Figure 3 shows a fit of three of the T H F concentrations to the fluctuation equation. Despite the fact that we are fitting to only three arbitrary parameters rather than the five used for the double relaxation, the fit is statistically superior a t all concentrations. ~
~~
(28) C. Treiner, J.-F. Bocquet, and M. Chemla, J. Chim. Phys., 70,68 (1973).
Another instructive exercise is to compare the low-frequency limit of the relaxation data. A t frequencies below the relaxation range, eq 3 predicts +
AI + A2
+ B = constant
(9)
On the other hand, eq 4 predicts
This is, a / f approaches a square-root dependency on frequency. Figure 4 shows such a plot. Clearly, the data does approach linearity with f1I2 at low frequencies. All
The Journal of Physical Ch8miStty, Vol. 85, No. 6, 198 7
Ultrasonic Absorption in Binary Mixtures
737
TABLE IV: Comparison of Fluctuation Fitting Parameters for THF-Water at 25 " C 1 0 1 5 ~ 1
intercepta
XTHF
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.85 0.95 a x lOI5 from ( a / f fitting parameters.
(DMb
1.74 7.99 10.74 8.25 6.82 2.90 0.55 0.32 0.69 - B ) vs. f 1 ' 2 plot.
1.68 7.83 10.58 8.15 6.58 2.90 0.55 0.33 0.66 Calculated from
" .l.O
XT",
Flgure 6. Excess functions of THF-water at 25
OC. I
I
0 n
Y
n 1.101
0 ,
0.3
0.4
0.5
06
0 7
01
09
4.3
10
Y
XTHF
Figure 5. Thermodynamlc parameters of THF-water at 25
OC.
concentrations show the same behavior. This enables us to calculate the fitting parameters (8, 1,) in an independent fashion. Table IV compares the fitting parameters found in these two ways. It is clear that there is substantial internal consistency in the fitting. We now must compare the Q values derived experimentally with those calculated from the thermodynamics of the system. Equation 5 can be usefully rewritten
(z)(&,,)'(
8=4 IzT P
-
g)p
(11)
The density data needed for T' and a were obtained from was the work of B e n ~ o nu, ~ ~ from our own work and, together, with the density data mentioned above was used to calculate p,. Cpwas calculated from the data of Morcom and Smith.14 a was calculated from the temperature-dependent VE data of Morcom and Smith." The interpolated values of the quantities are given in Table V and illustrated in Figure 5. The Y"calcu1ations present a very difficult problem as they involve the _secondderivatives of the measured excess quantities. For G E we used the data of Treiner et al.% For VEwe used the data of Benson and K i y ~ h a r a , checking '~ it against the data of Signer et al.30 and Morcom and Smith.I4 The more extensive data of Benson and Kiyohara are much more useful when second derivatives must be obtained. The RE data used was also from Benson and (29) 0.Kiyohara and G . C. Benson, Znt. Data Ser. B, Part 2a (1978). (30) R. Signer, H.Arm, and H. Daeniker, Helu. Chirn. Acta, 52, 2347 (1969).
XT",
Figure 7. Components of fluctuatlon amplitude for THF-water at 25
OC.
Kiyohara's workS1checked against the data of Signer,30 Erva,32and Nakoyama and ShinodaSS3The more detailed work of Benson again proved more useful. The excess functions are shown in Figure 6. The excess functions were fitted by using the Copp and Everett expression: n
YE = (1- Xz)X2CAi(l - 2Xz)i i=O
(12)
with n = 6. The second derivative of eq 12 gives quite reasonable values over the whole concentration range with only minor odd features at very low and very high XTHF values. The second derivatives of the simpler polynomials described in the Benson paper^^^,^^ did not give as smooth a curve but were reasonably comparable in their general character. Table VI shows the derived Y" values at the concentrations of interest. Figure 7 shows the important components of Q(thermodynamic) as calculated by using the data described (31)0. Kiyohara and G. C. Benson, Znt. Data Ser. B, Part l a (1978). (32) J. Erva, Suom. Kemistil.. B , 28, 131 (1955). (33) H.Nakayama and K. Shinoda, J. Chem. Thermodyn., 3, 401 (1971).
738
Atkinson et al.
The Journal of Physical Chemistry, Vol. 85, No. 6, 1981
TABLE V: Thermodynamic Parameters for THF-Water at 25 “ C d, g.
XTUW
1 0 - 5 u , cm s - ’
cm s g-l
23.97 30.06 36.30 42.65 49.09 55.60 62.13 71.94 78.45
1.581 1.484 1.415 1.375 1.348 1.328 1.308 1.288 1.276
4.094 4.734 5.293 5.688 5.993 6.247 6.498 6.771 6.941
0.9774 0.9 593 0.9435 0.9297 0.9179 0.9078 0.8994 0.8896 0.8847
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.85 0.95
TABLE VI: Second Derivatives of Excess Functions for THF-Water at 25 “ C
XTHF 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.85 0.95
10-IOGE“, 10-lOG”, VE”, cm3 1 0 - “ H ~ ” , mol-’ erg mol-’ erg mol-’ erg mol-’ -20.494 7.035 28.549 55.597 - 14.479 15.458 15.176 1.004 -11.384 8.251 1.222 0.410 - 10.043 5.089 -0.019 0.273 -9.624 4.297 1.638 0.274 -9.628 4.364 0.310 0.675 - 9.893 - 5.94 8 3.948 1.87 5 -11.243 - 19.156 -0.119 8.138 - 13.638 36.453 - 6.546 - 29.435
b.0
0 Fitting Theory 5.0
H
4.0
-I “g
3.0
d 1.0
1.0
0
0
10)
01
03
0.4
0 6
lO”P,,
V,, cm3 mol-’
Ob
07
0.1
09
10
XTHF
Flgure 8. Comparison of experimental and theoretical fluctuation amplitudes for THF-water at 25 OC.
above. Unlike the p-dioxane case, the H” term in Q plays a significant if minor role in the THF-water system. Finally, Figure 8 shows the Q(thermodynamic) plotted with the Q(experimenta1). The agreement is quite close both in overall magnitude and in concentration dependence. The (1/G’q2 term having a maximum at X T H F 0.45 is range by the V” and H” partially offset in the low XTHF terms shifting the Q maximum to around X T H F = 0.3. Considering the calculational problems associated with Q(thermodynamic) the agreement with &(experimental) is very encouraging. Discussion and Conclusion We have shown that the fluctuation theory can be used to quantitatively fit the ultrasonic absorption in the THF-water system over the full range of concentration and frequency. The theory correctly predicts the low-frequency limit of the excess absorption. In addition, the amplitude of the absorption can be calculated from the classical thermodynamic properties of the system. The data fitting is then reduced to one undetermined parameter, l,, the minimum interaction length. The quantitative fit is achieved without the assumption of any kinetic mechanisms or the derivation of any kinetic parameters unrelated
1030r,mol
ern-’ deg-’ 3.967 3.210 2.661 2.251 1.942 1.704 1.519 1.306 1.188
1 0 9 5 , erg, deg- mol0.863 0.961 1.046 1.118 1.177 1.222 1.255 1.279 1.278
to other known properties of the system. Our work on the p-dioxane-water and THF-water systems together with the work of K r ~ u on s ~CH30H~ water and BlandamerZ0on t-BuOH-water have demonstrated the superiority of the fluctuation approach over the hydrogen-bonding equilibrium approach used in the past. It makes the ultrasonic absorption measurements an integral part of the other known properties of these systems. In some ways this seems less esthetically appealing since no molecular picture emerges. We would like to suggest how such a molecular picture could be constructed. A recent popular review by Stillinge~-~~ gives a clear and reasonable picture of liquid water and some simple solutions. The picture presented draws heavily on the recent “molecular dynamics” and “Monte Carlo” calculations. In this picture pure liquid water emerges as a random, three-dimensional network of hydrogen bonds with a large proportion of strained and broken bonds but with a strong local preference for tetrahedral geometry. Very few nonbonded or one-bonded molecules exist. It does contain a reasonable concentration of short-lived polygons and even polyhedra. However, these polygons are just as likely or more likely to be the pentagons characteristic of clathrate hydrates as the hexagons characteristic of ice Ih. As water is cooled and then supercooled, the concentration of relatively unstrained, bulky polyhedra increases. But this polyhedra concentration and consequent increasing longrange correlation seems to be approaching a n order-disthe order transition a t approximately -45 OC (Ts, “Speedy-Angel1 temperature”). The introduction of hydrophobic entities encourages the formation of such bulky polyhedra. In some ways one could say that the hydrophobic entities increase T,. However, because of the disorder present and the very short lifetime of a hydrogen bond, no one polyhedron is preferred. A distribution of polyhedra exist forming around the hydrophobic entity by a cooperative effort, breaking up on a time scale characteristic of mutual diffusion. Therefore, though the polyhedra are related in a fundamental way to the solid clathrate hydrates, the dynamcs of the liquid state enforce a distribution of sizes and shapes, and a short lifetime. It is then clear that most spectroscopic methods (IR-Raman, NMR, X-ray, and neutron diffraction) whose measurement time scale is much longer than the lifetime of an average polyhedra will only see some averaged structure. The ultrasonic measurements tell us that this average polyhedra lifetime is on the order of 10-Bs. However, it is strongly dependent on composition and temperature. And the fact that there is a distribution of polyhedra means that a distribution of lifetimes is inevitable. The average lifetime is substantially longer than a hydrogen-bond lifetime because (34) W.D. T.Dale, P. A. Flavelle, and Peeter Kruus, Can. J. Chem., 64,355 (1976). (35) F. H. Stillinger, Science, 209, 451 (1980).
J. Phys. Chem. 1981, 85,739-744
the formation and break-up of the polyhedra are cooperative processes involving large numbers of water molecules. Since the H-bond network of the polyhedra is unstrained compared to the bulk random network, the thermodynamic properties of the polyhedra can and should be different from that of the bulk. These dynamic differences give rise to the fluctuations in GE, VE, and HE that cause ultrasonic absorption. Clearly fluctuations in VEmust give rise to fluctuations in refractive index and, so, to anomalous light scattering. Dynamic light-scattering measurements carried out with careful attention to the actual microscopic time scale should yield very valuable information. The fact that the unstrained polyhedra and the strained network should have different thermodynamic properties is a good rationalization of why water behaves so often as a two-state fluid. Our problem, then, is not that a two-state model is inappropriate but that our molecular pictures of the two states have been inappropriate. It should then be possible to modify a model such as that proposed by Mikhail0v3~9~’ to calculate the thermodynamic properties of such mixtures. Consider the following set of equilibria H20(n) F! H20(p) (13) HzO(n) e H20W (14) (15) S(P) ?2S(r) Here n stands for network, p for polyhedra, and r for regular solution. That is water can be in the strained (36)V. A. Mikhailov and L. Ponomareva, J. Struct. Chem. USSR, 9, 8 (1968). (37)V. A. Mikhailov, J.Struct. Chem. USSR, 98 332 (1968).
739
network or in the polyhedra or at higher concentration of solute ( S ) in a regular solution with the solute. The hydrophobic solute is contained either in the polyhedra or in a regular solution with the water. We are presently attempting to apply such a model to some of the systems of interest to us. It seems also quite possible that the following equilibrium may be necessary: WP) SdP) (16) That is, evidence accumulates that the so-called “hydrophobic b o n d involves two hydrophobic entities in separate polyhedra which are joined on a polygonal face. This seems to confer additional stability on the polyhedra. The large demands for water in such arrangements also rationalize why very large concentration dependences are found in properties at very low solute mole fractions for strongly hydrophobic groups. For such groups, equilibria such as (13) and (14) should be running out of available water in the concentration range X m CT 0.03-0.05. Much beyond, this, the whole system must break down into a more random arrangement of H-bonded fragments with few polyhedra. Extremely valuable insights could be gained if an array of measurements could be made on such non-hydrogenbonding hydrophobic entities as Xe, CHI, and C3He The pressures needed to cover a wide concentration range are easily accessible a t least for some measurements. Acknowledgment. The authors acknowledge the financial support of the NSF under Grant OCE-7814527. One of us (S.R.) thanks the CIES, Washington DC, and USEFI, New Delhi, for a travel grant.
Phosphate Ester-Nucleophile Reactions in Oil-in-Water Microemulsions R. A. Mackay” and C. Hermansky Depaltment of Chemistty, Drexel Universw, Philadelphia, Pennsylvania 19 104 (Received: September 30, 1980)
The reaction of p-nitrophenyl diphenyl, dihexyl, and diethyl phosphate esters with hydroxide and fluoride ions has been studied in oil-in-water (0/W) microemulsions. The hexadecane-in-watermicroemulsion systems are stabilized by a nonionic (Brij 96) or cationic (CTAB)surfactant and 1-butanolas cosurfactant. The rate constants were also determined in the corresponding aqueous micelles and in water-dioxane solutions. The reaction is first order in both ester and nucleophile in all cases, and the second-orderrate constants (kd in the microemulsion have been measured over a phase volume range ($) of approximately 15-85%. To account for the effective nucleophile concentration in the aqueous phase, a corrected rate constant k2,$ = K2(1 - 9) is employed. The effective nucleophile concentration at the reaction surface, in the absence of charge effects, is lower than in the bulk aqueous phase. The more water soluble esters are partitioned between the micelle and aqueous phase, but are all entirely dissolved in the microdroplet. An effective surface potential (9)of 130 mV is obtained for the CTAB micelle, while values of 1c. = 28-58 mV for $ = 85-15% are obtained in the CTAB microemulsion. These lower potentials reflect dilution of the interface by the alcohol cosurfactant.
Introduction Kinetic and thermodynamic dispersions of oil in water (O/W) or water in oil (W/O) which are optically transparent and isotropic are commonly called microemulsions. These systems are normally stabilized by mixtures of surfactant and alcohol, the latter usually of medium chain length. These dispersions form spontaneously upon mixing1i2 and contain spherical droplets having diameters of (1)W. C. Tosch, S. C. Jones, and A. W. Adamson, J.Colloid Interface Sci., 31, 297 (1969). 0022-3654/81/2085-0739$01.25/0
10-60 nm.3-7 The internal structure of the O/W system of the type examined here can be described as a stable collection of “oil” microdroplets in an aqueous continuous (2)S.Levine and K. Robinson, J. Phys. Chem., 76, 876 (1972). (3) W.Stoeckenius, J. H. Schulman and L. M. Prince, Kolloid Z., 169, 170 (1960). (4)J. H.Schulman and D. P. Riley, J . Colloid Sci., 3, 383 (1948). (5)J. H.Schulman and J. A. Friend, J. Colloid Sci., 4, 497 (1949). (6) J. E. L. Bowcott and J. H. Schulman, 2.Electrochem., 59, 283 (1955). (7) J. E. L. Bowcott, Ph.D. Dissertation, Cambridge, 1957.
0 1981 American Chemical Society
