J. phys. Chem. 1982, 86, 1711-1721
limited to the surface of the micelles.s7 (Furthermore, this s t u d p nicely demonstrated the liquidlike interior of micelles in contrast to a recent proposalmto the contrary.) Halle and Carlstrijm,44from water 170relaxation data, could demonstrate that the watel-hydrocarbon contact in several ionic micelles is equivalent to less than two fully exposed methylene groups. ‘gF relaxation in H20 and DzO of fluorinated chains is also inconsistent with a deep water penetration into micelles.6s At low amphiphile concentrationsceions stay away from the micelles, while a t higher concentration the effective excluded volume decreases markedly. This is in qualitative (67) In contrant to Menger et al.,BzCabane did not introduce a pol= probe into the alkyl chain. A polar probe may perturb the micelle interior and/or not be confiied to the interior of the micelle. (68) P. Fromherz, Chem. Phye. Lett., 77, 460 (1981). (69) J. Ulmiua and B. Lindman, J. Phys. Chem., 85, 4131 (1981).
1711
agreement (and should be amenable to quantitative tests) with results of theoretical calculations on the basis of the Poisson-Boltzmann equation and can be interpreted as a combination of two effects: (a) with increasing freecounterion concentration (and thus total free-ion concentration), the range of the electric interactions is reduced and the co-ions are repelled to a smaller extent; (b) with decreasing distance between the micelles, the depths of the potential minima between micelles become smaller, which might be called an electric double layer repulsion.
Acknowledgment. Valuable comments on this work from our colleagues, especially Professor Hakan Wennerstrijm and Dr. Bertil Halle, are gratefully acknowledged. The work has been supported by the Swedish Natural Sciences Research Council and the stays of Bjijrn Lindman in Montpellier also by Centre National de la Recherche’ Scientifique.
Isotope Effects in Aqueous Systems. 13. The Hydrophobic Interaction. Some Thermodynamic Properties of Benzene/Water and Toluene/Water Solutions and Their Isotope Effects Malay K. Dutta-Choudhury, Nada Ylljevlc,t and W. Alexander Van Hook’ Chemlsby Department, Unlversny of Tennessee at Knoxville, Knoxvilk, Tennessee 37SS6 (Received: September 25, lS81; In Final Form: December 1, lS81)
Henry’s law constants, KH= KO + KIX, have been measured as a function of concentration for the water-rich and benzene-rich solutions C6&/Hz0 and C6&/D20 and for the water-rich solutions C&/HzO and C6D6/Dz0 at several temperatures. The constants K O and K1are sensitive to temperature and to isotopic label. The vapor pressure results have been supplemented with measurements of the apparent molar volumes of the solutions listed above, as well as for HzO-and DzO-richsolutions of toluene and deuteriotoluene, and with determinations of the solubilities and solubility isotope effects of the toluene solutions. The data have been interpreted in the context of the theory of isotope effeds in condensed-phase systems. That analysis indicates that a significant dynamical vibrational coupling between solute and solvent normal modes occurs in these solutions. The result is of interest particularly as it pertains to models of the hydrophobic interaction.
Introduction A number of the solute-solute and solute-solvent interactions which occur in aqueous solutions of hydrocarbons or solutions of molecules containing hydrocarbon groups are broadly described, rather labeled, as the “hydrophobic effect”.l A proper and quantitative understanding of solutions exhibiting the hydrophobic interaction is of importance in the application of physicochemical theory to some important problems, including the determination of conformation of biopolymers in aqueous solution, the thermodynamics of hydrocarbon/ detergent/water systems found in oil recovery, and others. Tucker and Christian2*(TC) have pointed out that the benzene/ water system forms an excellent prototype model for the hydrophobic interaction. These authors reported vapor pressure measurements for the C6H6/Hz0system at 35 “Ctogether with the Henry’s law constants derived from them, finding KH = 3.527 X lo5- (3.07X 107)XB in pressure units of torr. The contribution of the term describing the first-order deviation from Henry’s law, f B = N.M. is responsible for the major part of the work on toluene/ water systems. 0022-3654/82/2086-1711$01.25/0
KHXB,amounts to nearly 4 % at saturation (4.4 x IO4 = XB). The result is mildly surprising in view of earlier claims that Henry’s law is obeyed reasonably well as far as the solubility limit.m Roasky and Friedman7 (RF) have published an interpretation of the observations reported by TC and concluded that the experimental resulta are in reasonable accord with model calculations earlier reported? In view of widespread interest in the hydrophobic effect, and because of the conflict between the results of TC and the earlier claims>+ we concluded that independent con(1) Kauzmann, W. Adu. Protein Chem. 1959,14,1. Franks, F. Water: Compr. Treatise 1973,4, 1-94. (2) (a) Tucker, E. E.; Christian, S. D. J. Phys. Chem. 1979,83,426. (b) Tucker, E. E.; Lane, E. H.; Christian, S. D. J. Solution Chem. 1981, 10, 1.
(3) Green, W. J.; Frank, H. S. J . Solution Chem. 1979,8, 187. (4) Ben-Naim, A.; Wilf, J., Jr.; Yaacobi, M. J. Phys. Chem. 1973, 77, 95. (5) Ben-Naim, A.; Wilf, J. J. Chem. Phys. 1979, 70,771. (6) Franks, F.; Gent, M.; Johnson, H. H. J. Chem. SOC.1963, 2716. (7) Rossky, P. J.; Friedman, H. L. J.Phys. Chem. 1980,84,587. (8) Friedman, H. L.; Krishnan, C. V. J . Solution Chem. 1973,2, 119; 1974,4,727. Ramanathan, P. S.; Krishnan, C. V.; Friedman, H. L. Ibid. 1972,1,237. Friedman, H. L.; Krishnan, C. V.; Jolicoeur, C. Ann. N. Y. Acad. Sci. 1973,204, 79.
0 1982 American Chemical Society
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Dutta-Choudhury et al.
The Journal of Phplcal Chemlstty, Vol. 86, NO. 9, 1982
Flgura 1. Experlmental apparatus: (A) benzene resevolr, (B) vapor dosing bulb, (C) sokrtkn TBsBvdT, @) TI quartz Bowdon pressure gage, (4 ) valves.
firmatory experiments were demanded. Further, we decided to extend the experiments and make measurements on isotopically labeled solutions (CsH6/D20, C6D6/H20, and Ca6/D2Oin addition to C&/H20) in order to obtain information about the details of the effect of intermolecular forces on the motions of the solute molecule^.^ The benzene/water system is particularly apt for such studies because the thermodynamic isotope effects on the properties of the pure component benzeneslO and waters"J2 and solutions of the isotopic isomers, one in the other, of benzeneI3and waterI4have been measured and thoroughly analyzed in terms of the theory of isotope effects in condensed-phase systems.15 In addition to studies on twocomponent hydrocarbon-rich and water-rich solutions, we report vapor pressure measurements for C6H6 and C6D6 (subsequently symbolyzed as Bh, or Bd6, respectively) dissolved in water/urea (6.931 m) or water/tetramethylurea (3.991 m) solutions. Considerable interest in the effect of urea and substituted ureas on the structure of water and aqueous solutions is to be found in the recent literature,I6 and Jakli and Van Hook" have recently completed a thorough investigation of the thermodynamic properties of these solutions and their isotope effects. Finally, measurements of the apparent molar volumes of water/ benzene and benzene/water solutions and their isotope effects are reported. The molar volume experiments are of interest for two reasons. First, because the deviations from Henry's law for fugacities are small (and for that reason alone their existence is arguable), the observation of a corresponding deviation in another thermodynamic property using a completely independent experimental technique should strengthen the conclusion that Henry's law is not obeyed. Secondly, recent rationalizations of the origin of free energy effects in solutions of isotopic isomers, one in the other, in terms of the partial molal volume and molar volume isotope e f f e ~ t s ' ~ Jsuggested ~J~ that information on the volume effects would be of interest. The extension of experiments where feasible to toluene/water solutions was another natural confirmatory step. (9) Jancso, G.; Van Hook, W. A. Chem. Rev. 1974, 74, 689. (10) Jakli, G.; Tzias, P.; Van Hook, W. A. J. Chem. Phys. 1978,68, 3177. Jancso, G.; Van Hook, W. A. J. Chem. Phys. 1978,68,3191. (11) Van Hook, W. A. J. Phys. Chem. 1968,72,1234. Gellai, B.; Van Hook, W. A. Fluids Fluid Phose Equilib. 1980, 5, 19. (12) Jakli, G.; Van Hook, W. A. J. Chem. Eng. Data 1981, 26, 243. (13) Janceo, G.; Van Hook, W. A. Physica A (Amsterdam) 1978, 91, 619. (14) Van Hook, W. A. J. Phys. Chem. 1972, 76,3040. Dutta-Choudhury, M. K.; Van Hook, W. A. Ibid. 1980,84,2735. Phutela, R. C.; Fenby, D. V. Aust. J. Chem. 1979,32,197. Fenby, D. V.; Chand, A. Ibid. 1978, 31, 241. Jancso, G.; Jakli, G.; Illy, H. Ibid. 1980, 33, 2357. (15) Bigeleisen, J. J. Chem. Phys. 1961, 34, 1485. (16) Franks, F. Water: Compr. Treatise 1973, 2, 355. Franks, F.; Pedley, M. D. J. Chem. SOC., Faraday Ram. 1 1981, 77, 1341. (17) Jakli, G.; Van Hook, W. A. J. Phys. Chem., 1981,85, 3480.
Experimental Section The vapor pressure apparatus is schematized in Figure 1. Liquid benzene is vaporized from container A thermostated at the experimental temperature, T,, into a dosing bulb, B, of known volume, vb,and temperature Tb > T,. Therefore, nB,the number of moles of benzene to be injected per dose, can be accurately calculated from the benzene vapor pressure, POB ( T = T,),'O the bulb temthe bulb volume, vb,and the virial coefficient perature, Tb, of benzene vapor.'* Degassed water or previously dosed solution contained in bulb C is then cooled to liquid-nitrogen temperature and the vapor slug quantitatively transferred from bulb B to bulb C. The system is warmed back to the experimental temperature, mixed, and allowed to come to thermal equilibrium while the total pressure is monitored with a Texas Instruments quartz Bourdon pressure gage, D. Bulb C has a volume of -48 cm3. The number of moles of benzene added per injection varies with temperature but at 25 "C was approximately 2.4 X so the mole fraction increase per slug (again at 25 "C) was about 7 X giving six data points to define the leastsquares constants which describe the Henry's law line. The system is designed with a m i n i u m of vapor space in bulb C and in the lines connecting bulb C to the gage. The ratio of liquid to vapor volume, R, = VEq/Vvap,was typically 3. The point under discussion is important because in the experiment only nB and P are measured. One then calculates the partitioning of benzene into the liquid and vapor phases from R,, the measured total pressure, and assumptions concerning the PVT properties of the vapor phase. The uncertainty in nB(liq),the number of moles of benzene which dissolve, is therefore sensitive to R, and is minimized at large values of R,. The effort to increase R, is limited by the fact that, during the first step in the process, the liquid is frozen so that sufficient vapor volume to allow for the expansion on freezing must be included. The shape of container C is also important. We found that cylindrical or spherical bulbs were likely to break during freezing. In spite of these various difficulties, the present value, R, = 3, is significantly larger than that selected by TC. On the other hand, our laborious techniques imply fewer data points, so the present statistical uncertainty is larger. The generally fine agreement of the Henry's law constants reported here or by TC is therefore gratifying (see Table I). Agreement at lower temperatures is within the quoted experimental uncertainties. A t higher temperatures (compare 30, 33, and 35 "C data, Table I) the present values for the ratio Kl/Ko compare favorably with TC's earlier report,& not as well with their more recent values.2b Further discussion on this point is given in a later section. In the present work benzene solubilities were determined as the intersection of the experimental lines for the vapor pressures of the unsaturated and saturated solutions. They are in good agreement with the solubilities as determined by other techniques (vide infra). Apparent molar volumes of benzene and toluene solutions were measured with a Mettler-Paar DMA f30vibrating densitometer using well-established technique^.'^ Solutions were gravimetrically prepared in sealed containers with minimal vapor volume. Solubilities of toluene solutions were measured spectrophotometricallyby using a Cary-17 spectrophotometer and techniques found in the literatureSw The method was calibrated with measurements on benzene/ water solutions. (18) Dymond, J. H.; Smith, E. B. "The Virial Coefficients of Gases"; Clarendon Press: Oxford, 1969. (19) Dutta-Choudhury,M. K.; Van Hook, W. A. J.Phys. Chem. 1980, 84, 2735.
Isotope Effects In Aqueous Systems
The Journal of Physlcal Chemistty, Vol. 86, No. 9, 1982
TABLE I: Henry's Law Constants, KH = K" t K,X, for Benzene/Water Solutions, Where fB(t0rr) = KHXB no. of temp, data soln C pointsC 10-5~"
ref 2a ref 2b
Bd,/H,O Bd, /D ,0 H,O/Bh,d D,O/Bh,d Bh, in 6.931 m urea in H,O Bd, in 6.931 m urea in H,O Bh, in 3.991 m TMUb in H,O Bd, in 3.991 m TMUb in H,O
8 15 25 33 35 35 30 15 8 25 33 33 33 33 33 33
11 5 5 4
33
3.98 3.90 4.01 4.37
7.853 7.868 7.833 7.775
0.003 0.004 0.003 0.003 0.005 0.0006 0.0004 0.003
3.64 3.62 3.80 4.64 4.18 41.5" 34.Eia 7.26
7.930 7.929 7.863 7.687 7.796 5.823 5.968 7.218
0.006
-11 c 12
7.81
7.165
0.006
0.0003
-49c 3
> 30
5.722
0.011
0.0010
-35+ 3
> 30
5.660
-0.54 -0.63 -1.03 -3.59 -3.07 -2.11 -1.61 -0.63 -0.39 -0.66 +0.26 c -1.51 -1.88 f -0.0861 -0.0969 +0.14 f
0.18 0.31 0.39 0.32
0.005 0.007 0.011 0.005
0.09 0.19 0.16 0.19 0.36 + 0.0048 * 0.0038 0.14
4
1.808 f 0.010
-0.20
0.22
33
5
0.4158 + 0.0007
-0.205
33
5
0.4013
-0.142
f
0.0012
f
t
In
-51 + 18 - 4 0 + 21 -43 1 6 -110 f 1 2 - 87 - 61 - 56 -42 8 -34 -25+ 7 t7+5 -5Of 6 -55 f 11 -68 f 10 -75 f 10 t7+8
1.055 c 0.004 1.537 c 0.007 2.402 c 0.010 3.238 c 0.008 3.527 3.486 2.890 1.484 1.139 f 0.002 2.644 c 0.004 3.538 c 0.004 3.047 c 0.005 3.398 f 0.008 0.1275 + 0.0008 0.1291 c 0.0005 1.856 t 0.007
10 5 4 4 4 9 8 4
1713
Solubility not determined in present experiment. Value cited is from ref 21. Present data extend only to ~ ~ 1X. 2 TMU = tetramethylurea. Number of data points below solubility limit. In colums 4, 5, and 7 read 'K or K , instead of K" or K , . e Solubility. a
TABLE 11: Apparent Molar Volumes of BenzenelWater and TolueneIWater Solutions at 25 O c a soln
no. of data points
X(") or Y(max)
Bh/H,O Bh/D,O Bd/H,O Bd/D,O H,O/Bh D,O/Bh H,O/Bd D,O/Bd Th/H,O Th/D,O Td/II,O Td/D,O
5 6 7 5 4 5 2 1 15 14 21 6
3.3 x 101.8x 10-4 3.6 x 10-4 2.4 x 10-4 24 x 10-4 22 x 10-4 20 x 10-4 23 x 10-4 1.1 x 10-4 1.1 x 10-4 1.1 x 10-4 1.1 x 10-4
o * ~ cm3/mol ,
82.55 * 82.79 t 82.87 + 84.38 t 24.37 c 25.05 + b
0.07 0.22 0.16 0.23 0.41 0.49
10-JA
[a'/(n -
l)] l i 2
0.77 t 0.36 -11.8 1.5 1.19 c 0.65 -5.6 2 1.2 -1.03 t 0.26 -1.14 0.30
0.04 0.06 0.06 0.07 0.22 0.20
13.5 c -12.0 t 13.9 c -2.5
0.07 0.13 0.05 0.04
C
97.47 f 0.22 98.36 c 0.36 98.11 f 0.15 98.49 c 0.10
3.1 4.6 1.8 1.2
a @v= @ * v + AX or @ ,, = @*v t AY. V',(C,H,) = 89.432, V',(C,D,) = 89.234, V',(C,H,CH,) = 106.879. V',(C,D,CH,) = 106.715, V',(H,O) = 18.068, V',(D,O) = 18.137. @(H,Oin C,H,) - @(H,Oin C,D,) = 1.0 f 0.2 at x = 1.9 X C @ ( D , Oin C,H,) - @(D,Oin C,D,) = 0.8 e 0.2 at x = 2.4 x
*
Results Least-squarea parameters defining Henry's law constants for various isotopically substituted benzene/water solutions and benzene/urea/water and benzene/ tetramethylurea/water solutions are reported in Table I together with the number of data points (below saturation), the root mean square error, the errors in the derived constants, and the measured solubilities as determined from the intersection of the two branches of the pressure/mole fraction graph. Throughout the paper Henry's law is written for water-rich solutions as fB = KHXB = (KO + K1XB)XB (la) and for hydrocarbon-rich solutions f, = KHY, = (KO + KlY,)Y, Ob) where fB and f, are the benzene or water fugacities, respectively, as calculated from the vapor pressure measurements, the virial coefficients'* and condensed-phase partial molar volume measurements (vide infra) of the solute and solvent components, and well-established relations.20 The corrections applied to measured pressures
to obtain the fugacities reported in this paper are small at the relatively low pressures employed in the present series of experiments. The mole fraction of hydrocarbon in water-rich solutions will subsequently be represented by X, and that of water in hydrocarbon-rich solutions by Y. Note that the algorithm for extracting the fugacities from the pressure measurements must account for the inteirelation between X (or Y), fB (or f,), the total amount of solute injected, and R,. An iterative technique was therefore employed for the calculations. Benzene solubilities as determined by the intersection of the saturated and unsaturated branches of X, fB curves are also entered in the table. The corresponding solubilities of H20 or D20 in Bb are from Moule.21 Results of apparent molar volume measurements on the various isotopically substituted benzene/water and toluene/water solutions are reported in Table 11using the same format as Table I. Solubility data for toluene/water so(20) McGlashan, M. L.'Chemical Thermodynamiii";Academic Press: New, York, 1979. (21)Moule, D.C. Can. J . Chem. 1966, 44, 3009.
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DuttaGhoudhury et al.
The Journal of physlcal Cbmbtry, Vol. 86, No. 9, 1982
TABLE 111: Solubilities of Some Isotopic Toluenes in Water and Heavy Water
solute
solvent
temp, "C
C,H,
H,O
C,H,
D,O
C,D,
H,O
C,D,
D,O
C,H,
H,O
33 10 33 10 33 10 33 10 25(ref 37) (ref 38 ( 3 7 ) ) (ref 39) (ref 23)
104(mole fraction) 1.16 f 1.08 * 0.85 i 0.82 i 1.07 t 1.08 i 0.99 i 1.02 i 1.21 1.11 1.12 1.23
0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.04
i
t
+
$
+
lutions as determined spectrophotometrically are found in Table 111.
Discussion Benzene/ Water Solubilities. Solubilities determined in the present study (Table I) are compared with results 2. Agreement is from the l i t e r a t ~ r e ~ * * ~ in~ Figure -~' satisfactory. The minimum in the solubility observed by a number of other authors is nicely reflected in the present data points which lie very nearly on top of the values given and Franks, Gent, by Arnold, Plank, Erikson, and and Johnson,G although our point at 33 "C deviates positively from these authors. Other also show the minimum, but displaced positively, except for BenNain and Wilf,6 and Ben-Naim, Wilf, and Yaacobi; whose data depart from the weighted average of other workers, especially at lower temperatures. Heat of Solution. The temperature coefficient of the Henry's law constant, the solubility, and the heat of solution are straightforwardly related. It is interesting to make a test of thermodynamic consistency between the present results and calorimetric data. The benzene free energy in aqueous solution is expressed pB = poB + RT In KHX N p o B + RT In K O + R T K I X / K o + RT In X (2) where KH is given by eq 1 and poBis the partial molar free energy of benzene in the dilute gas reference state and we have recognized that K I X / K o D20). This particular effect can be seen as the difference between lines labeled as "2" and "3" in Figure 4. In the material below we show that one implication of this observation is that a significant dynamical vibrational coupling exists between solute and solvent normal modes. Heretofore a wide variety of condensedphase isotope effects on free energies have been successfully treated9J0J4by application of the theory of isotope effects in condensed phased5 using a pseudoharmonic cell model. In this approach the "solute" species is described in terms of a 3n-dimensional partition function evaluated via an isotope-independent pseudoharmonic F (force constant) matrix. The number of atoms per solute molecule is designated n. The solutesolvent interaction enters only insofar as the condensed-phase F matrix has been perturbed from the vapor-phase value by means of the intermolecular forces operative in the high-density phase. The marked solvent isotope effect on the isotopic free energy difference between Bh,and B& can be rationalized in the context of this approach only if the dimensionality of the model is expanded to include one or more solvent molecules dynamically interacting with the solute species. The approach is outlined in more detail below, but from the outset i t k to be recognized that a further effect arising from the hard-sphere interaction parameters complicates the analysis. In the material which follows, this later effect is separated out and treated approximately by using scaled particle theory. The analysis leads to the important conclusion that the modeling of hydrophobic phenomena in terms of an interaction of the solute with a homogeneous and structureless solvent of appropriate dielectric constant, density, etc., is inadequate. It would be interesting to have complementary data on the Bh,/B& solvent isotope effect for the benzene-rich solutions in order to compare the difference In [~"~,o(Bh~)/K"~,o(Bhe)l - In [Ko~,o(Bd6)/ KO~(B&)], with the analogous one for water-rich solutions In [KoBi#20)/KoBd&H20)1 - In [Ko~i&D20)/Ko~(D20)1 treated below in detail. However, the measurement in Bd, solvent would be inordinately expensive in our apparatus because of the large amount of solvent which is required. Volumetric experiments reported in Table I1 address this point, but less directly. Transfer of Benzene from Vapor or Liquid to Aqueous Solution. Consider the free energy of transfer of benzene between any two reference states in terms of a multidimensional and isotope-independent potential energy surface (PES). The free energy is directly calculable from properly established PES'S by the methods of statistical thermodynamics. A one-dimensional schematic of the situation which concerns us is outlined in Figure 5. Within the realm of validity of the Born-Oppenheimer approximation, the value of the potential energy at the position of a given minimum (designated as 2, see Figure 5 ) , as well as the curvature of the surfaces near the minima (designated in terms of an arbitrary shape function, s(Z)),is a function of the intermolecular forces operative between solute and solvent and hence on the particular reference phase. The present approach recognizes that Zo(Bh,) = Zo(B&), T(Bh,) = Z(B&), Z*H (Bb) = z*~&Wj), and Z*,,o(Bh& = Z*D,o(Bd6)but t f a t Zo # z' # Z*Ho # Z*D,O.Furthermore, S(Zo(Bh,)) = S(zo(Bd6)),S(z'(hh6)) = s(Z'(Bd,)), and S ( ~ * Ho(Bh6)) = S(z*~,o(Bd6))= S(Z*D,o(BhJ) = s(Z*D o(B%,))but 42') # ~ ( z# ' )s(Z*). Our purpose is to deduce both the shift in the position of
The Journal of Physical Chemistry, Vol. 86, No. 9, 1982 1717
I
Flgure 5.
sol n.
Schematic representation of benzene potential energy
surfaces in several different reference states.
the minimum (Z* - z', Z* - Zo,or z' - Zo in the figure), and the change in shape of the potential function, both on phase change. The present data including the transfer free energies and their isotope effects allow the two contributions to be assessed in a model calculation. For this purpose we choose an approach which combines aspects of the scaled particle theory as applied to aqueous solution~ and ~ ~the theory of isotope effects in condensed phase^.^.'^ In scaled particle theory the free energy of transfer from the dilute vapor to the dilute aqueous reference state is given in terms of the free energy necessary to create a cavity of suitable size in the solvent, G,, the free energy of interaction of the particle in that cavity with the neighboring solvent species, Gi, the partition functions of the vapor- and solvent-phase solute molecule, and the partial molar volume of the solvent Vl
--
P* - N o
RT
- In KO
= (G,
Q* + In R T + G i ) / R T + In 8"
T
VI
(11)
In many previous applications the effect of intermolecular force on the details of the solute PES were not of interest was set equal to unity. That process and the ratio Q*/T is equivalent to assuming that the shape of the PES is invariant on phase change, unacceptable in the present analysis since it is just those shape changes which determine the solute isotope effects. It is convenient to rewrite eq 11 and reference the free energy to the pure liquid solute standard state. Since p' = p o + R T In P , and because to a sufficiently good approximation In ( P ' * / P ) = In [(s/s?(f'/f")l, then
--- P* - In K o / P = RT P*
(G, + G J / R T + In
Q*
-
+ In R T - In P
Q"
(12)
Vl
Ap* - Ap' s f* Vl = (AG, + A G i ) / R T + In -, - + In 7 (13) RT S P Vl (30) Pierotti, R. A. Chem. Rev. 1976, 76,717.
1718
The Journal of phvsicel Chemistry, Vol. 80, No. 9, 1982
In these equations the prime represents isotopic substitution, ((s/s?(f*/f )) is the reduced partition function ratio,15 ( Q / Q ? q u a n t u m / ( Q / Q ? ~and , AP = P' - P, etc. In the present paper we compare solution either of Bh or Bd in H20 or (in a separate comparison) in DzO (in either case In (VI/ V i )is zero),or of Bh, in H20 or D20 (or & in H20 or DzO), in which case In (Vl/Vl') = 0.003 and may be neglected. We now employ the data on the free energies of transfer and their isotope effects to sort out the various contributions in eq 12 and 13. Refer to Figure 5 and the potential minimum in pure liquid as the free energy zero, z'. Then the free energy of protio isotope in the liquid phase is P ' B ~ = z' + RT In (QB), and that for the deuterioisotope is = z' + RT In In the vapor phase the potential % k n u m is 2" and the free energy of transfer from vapor to liquid is given as = (2'- 2") + RT In Q * B b / Q " B b (14) h*Bb -
he&.
and the isotope effect (P*Bb -
- (PoBh, - PoB&) = r
follows readily because (2'- 2 " ) ~ = q (2'- 2")- The effective potential surfacea are identica131for both lsotopic isomers. Detailed calculations of the vapor pressure isotope effects of the deuterated benzenes have been reported by Jancso and Van Hook,1° who employed a pseudoharmonic cell model in the liquid state to evaluate the partition function ratios. Agreement with experiment was quantitative and extends to a detailed understanding of the origins of excess free energies of Bh,/B& mixtures in terms of the volume dependence of the normal mode frequencies used to calculate the condensed-phase partition = function ratios. The VPIE is negative (In (PB%/Pw) -0.027 at 33 "C) and the predominant contribution to the effect is accounted for by the red shift observed in the six carbon-hydrogen stretching modes (6.7 cm:') on condensation to the liquid. The other 24 internal modes show much less isotope dependence, or a smaller frequency shift on condensation, or both, and therefore do not make much of a contribution. The external modes are of low frequency and only display a small isotope dependence so their contribution, while not negligible, is small. In fact at 33 "C the C-H stretch contribution amounts to -0.023 out of a total of -0.027 units and is dominant.32 In the absence of detailed spectroscopic information for benzene in the aqueous reference phase, it is unreasonable to proceed with a calculation in the detail reported by Jancso and Van Hook.'O Still some progress can be made. Assuming, as above, that the IE's can be effectively described in terms of perturbations in the six C-H stretching modes, we have for solution of B b or Bds in H 2 0 (Table I) at 33 "C
(31) Except that the force constantsdescribing internal motion in the liquid are volume and hence isotope dependent according to the pseudoharmonictheory. See ref 10 for a thorough discussion of this very small effect. (32) Contributions from the bending modes vary in sign (i.e, there are both red and blue shifta on condensation) but all are small compared 0.023. See ref 10 for details.
Dutta-Choudhury et al.
and using eq 12 and 13
since the SPT solute isotope shift, AG, + AGi, is properly taken as zero because the solvent, H20or DzO, is in either case common to the solutes, Bh, and Bde. Writing an equation for B similar to eq 18 and taking the difference
or 4.8
= GR(HzO) - [GR(B~)][GR(D,O)]= AGR
X
(20) where the GR (G ratio) functions are defined by comparing eq 19 and 20. These relations are derived from eq 18 by assuming that there exists an isotope-independent F matrix for the aqueous reference state, F*,and isotope-dependent G matrices, G*,which are calculable from the geometry (as yet unestablished) and masses of the benzene/water complex.33 We choose to write the matrices, G*,in diagonal form, and such diagonalizationis assumed for the balance of the discussion. (The F*matrix may also be diagonalized.) In that case the ratios of squares of any two frequencies, v*(l) and u*(2), is given as ~ * ( 1 ) ~ / v * ( 2 ) ~ = FG*(l)/FG(2)* = G*(l)/G*(2) since the F matrix is invariant on isotopic substitution. Equation 19 follows directly on substituting (9*)2 = G*. To go from eq 19 to eq 20, we recognize that to sufficient precision v*Bb(HZO) = (3.07 f 0.05) X lo3 cm-', that is, the C-H stretching frequency is not expected to shift more than a few tens of wavenumbers on transfer from the liquid to aqueous solution. Furthermore, GR(Bh,) is available from the measurements, In (KoBb(H20)/KoBb(D20)= -0.088, so GR(Bh,) at 33 "C is 1.002. Equation 20 therefore establishes the point that the experimental difference, A - B # 0, can be rationalized in terms of the conventional formalism (isotope-independent F matrix, isotope-dependent G matrix) only if m ( H 2 0 ) # m ( D 2 0 ) . This in turn is possible only if there is some kind of dynamical coupling between solute and solvent species because it is only in the Dresence of couded solute and solvent modes ,-model . A complete calculation of a benzene solute par-*.
ticle interacting with a large number of neighboring waters is not warranted until some kind of reasonable approximation to the geometry of the complex is available. Even so we have made a beginning study of eq 20 as applied to this problem by consideringthe interaction of an assumed representative CH oscillator with solvent in a number of different geometries. In the simplest set of calculations, we considered a three-particle problem
solving for the four different G matrices obtained from isotope substitution at H (1 or 2) and W (18 or 20) as a function of the C-H-W angle and the H--.W bond distance. For this grossly oversimplified case nonzero and negative values of A(=) from o to -2 x 104 as a function (33) Wilson, E. B.; Decius, J. C.; Cross, P. C. "MolecularVibrations"; McGraw-Hill: New York. 1955.
The Journal of physcal Chemkfty, Vol. 86, No. 9, 1982 1719
Isotope Effects in Aqueous Systems
of angle were obtained. The observed positive value of A(GR) could not be reproduced unless account was taken of the known HzO/DzO isotope effect on thewater size parameter, Aula (vide infra).l9 In that case A(GR) values from 0 to 4 X lo4 were obtained as a function of angle. With more complicated geometries (and more atoms) it is rather simpler to concoct G matrices which obey the criteria given in eq 20. Although we have not made a detailed study of a wide variety of geometries, we have investigated G matrices for C
.(Wi C ' -H"
C'
complex, calculating A(=) as a function of in- and outof-plane C-H-W angle, and have also made calculations on a planar seven-atom configuration
diameter a2 in a hard-sphere solvent of diameter al has been given asm where u12is the radius of the sphere which excludes the centers of solvent molecules; a12= (al + a3/2. The solvent isotope effect on cavity formation is therefore AGc = ATo + alzATl + alzZAT2+ alz3AT3+ TIAalz + 2Tz~lzA~lz + 3T3a1z2A~1z(22) The Ts are tabulated functions of a solvent number density parameter, y 7rpa13 TO R~ = - l n (1- y ) + - - - - (234 i(1:y) 6RT =
RT
We conclude that it is possible to meet the criteria of eq 20 with a variety of plausible geometric configurations describing the benzene-water interaction. It is not the present purpose to deduce the geometry or even a list of possible geometries for benzene-water interaction complexes. Rather the point is to establish that the analysis of thermodynamic measurements leads straightforwardly to the conclusion that a dynamical vibrational coupling between solute and solvent determines certain of the important macroscopic properties of the solutions. It is worth pointing out that detailed analysis of the thermodynamic isotope effects, especially on vapor pressure, of a large number of systems has been possible to date using a simplified pseudoharmonic cell model for the liquid phase? The effect under discussion, while not unexpected, is therefore of a new kind. Solvent Isotope Effect on Z*; S P T Analysis. In the present section we focus attention on the difference Z*(HzO) - Z*(DzO). Refer to Figure 5. The free energy of a given solute species, e.g., Bh, in HzO, may be written The relation p~k(Hzo)/RT= Z* + 1 / z ( h ~ / k 7 ' ) ( C ~ * H ) . follows from the earlier assumption that the solute dynamical contributions are considered in terms of the C-H stretching motions only, and the contribution from these vibrations to the partition function and the free energy is evaluated in the low-temperature (zero-point energy) approximation. The left-hand side is available from the measurements. Its isotopic difference with Bd, in HzO, for example, has been written out as eq 16. In the last section careful consideration of the effect of coupling on the form of the G matrix elements determining frequency ratios led to the conclusion that solute-solvent coupling is important in fixing details of the interaction. In spite of that, it is also clear that the GR ratios (eq 20) in a crude and first approximation may be described in terms of a two-atom approximation (gCH/gCD)' = (1/12 + l/l)/(l/lz + l/z) and we obtain from eq 17 and 18 ~ Y * ' ~ H ( H ~ O=) 0.088/0.0037 = 24 cm-', which yields Z*, - z' = 7.4lRT. Other plausible models for the G matrix elements such as those discussed above give values within 15% of the 24cm-I blue shift predicted for transfer from liquid hydrocarbon to water. In DzO a similar analysis yields 6v*'D 0 = 20 cm-' so Z * w - Z'= 7.58RT and Z*(HzO)- Z*(Dzd) = 4.17RT with an uncertainty in this difference estimated as 30%. The estimated error follows principally from the uncertainty in the average geometry. In SPT theory the partial molar free energy for expanding a cavity to accommodate a hard-sphere solute of
-[-% + r)z]+ 1-Y
TS/RT = 4/3(rP/RT)
1-Y
7rPa12 (23b) RT
y = TUi3p1/3
(23d)
The isotope effect on y (33 "C)is given as Ay/y = Ap/p + 3Aul/al = 0.0032 Aal/al. For HzO,y = 0.371. With alz/al taken as 1.353, numerical analysis leads to the result AGJRT = -0.17 f 0.04 = 0.090+ 60(Aa/a)
+
or Aa/a = -0.004 f 0.001 The result indicates that the effective molecular diameter of DzO is 0.4% larger than that for HzO and is in striking agreement with the value of 0.4% calculated by Choudhury and Van Hookls in their analysis of the amplitude isotope effect for oxygen motion which led directly to a semiquantitative understanding of the origin of the molar volume isotope effects of the deuterated waters and ices. In pictorial terms the amplitude describing the motion of the oxygen (convenientlyused as a reference in describing the molecular position of a given solvent molecule) about the center of mass of the molecule is larger for D,O than for HzO primarily because the distance (CM-0) is larger in DzO than it is in HzO. As far as the present purposes are concerned, the SPT calculation can be taken as confirming as reasonable the free energy difference Z*H20- Z*D~O -(0.17 f 0.04)RT deduced from the Henry's law experiments at 33 "C. Temperature and Concentration Dependences. Data on the effect of temperature on p* is available for two solutions, Bh/H20 and Bh/DzO. The I E s on the transfer free energy of Bh from DzO to HzO are -0.077, -0.096, and -0.088RT at 8,25, and 33 "C, respectively. The maximum presumably occurs at about the same temperature at which the solubilitiesgo through their minima. In the formalism above, the IE at 33 "C was shown to be a complicated function of the frequency shift on phase transfer which can be calculated from an isotope-independent (but perhaps temperature-dependent) F matrixlOJ1and an isotope-dependent G matrix describing the geometry (which may also be temperature dependent), and an isotope-dependent SPT parameter, Z*. If it is assumed that the IE on Z* is given as Z*(HzO)- Z*(DzO) = A(Ap/p) + B(Au/a),where A and B are temperature independent and have been defined by implication in the section on SPT calculations, and, if further Aula is taken as independent of temperature, then dAZ*/dT = A(Ap/p) = -0.002RT deg-' ( p is the solvent density). If one uses an equation of the form of
1720
The Journal of Physical Chemistry, Vol. 86, No. 9, 1982
eq 15 to represent the effect, IE = AZ*/RT + (6hc/ 2kT)[v*~h(H@)- v*Bh(D20)],then numerical analysis indicates that (d/dT)[(Ghc/2kt)(Av*)] N 3.8 X + (2.9 X 10"')t. The temperature dependence of the frequency shift is thus significant and, in terms of the model calculation outlined above, could result from a temperaturedependent geometry (i.e., function, see eq 20), from a shift in the magnitude of 6v*Bh = v*Bh - v*Bh, or from a combination of both effects. The present experiments do not provide enough information, nor are they of sufficient precision, to select between these alternatives. In considering the concentration dependence of the effects, it is useful to keep in mind that the excess free energy of the real solutions
=
- = - - '-'Id PEX
RT
RT
- In (1
+$X)
(24)
never rises above a few percent in the benzene/water system even though Kl/Ko is as large as lo2because the solubilities are all below 5 X lo4. Phenomonologically it would be a simple extension of the development of earlier paragraphs to reparametrize the observed excess free energies in terms of concentration-dependent frequency shifts and/or ?& ratios. It is to be recognized that the required concentration dependences are sizable. For example, consider the isotopic difference at 33 "C between Bh/H20 and Bd/H20. For this system ApEX/RT= (-60 f 13)X, which at X = 4.104,not too far from saturation, decreases the zero concentration isotope effect by nearly 30%. Similarly the Bh/D20, Bd/D20 effect at 33 "C and X = 4 X lo-' is about 30 f 8% larger than at zero concentration. The temperature dependence of the excess effects is also marked. Data are only available for the Bh/HZO, Bh/D2O pair, but for these molecules the correction at 33 "C, which increases the magnitude of the effect by about 50%, falls off to approximately 10% a t 8 "C. It is important to recognize that the existence of a nonzero solvent isotope effect, p'sh(HzO) - pLEXsh(D2O), has the same implications as were discussed for the zero concentration phenomena. They thus strengthen the conclusion that the thermodynamic properties of these solutions can only be rationalized in terms of some type of intimate coupling between solute and solvent. The marked concentration dependence of this coupling as demonstrated by the excess free energy in turn implies that a single benzene interacts with a large number of solvent molecules (markedly perturbs the water "structure" in its neighborhood if you prefer). This follows from the idea that the standard-state properties, J*, monitor the solute-solvent interaction exclusively, and the excess properties reflect the perturbation of the resulting complex by neighboring solute particles. Alternative Interpretations. Another interpretation of the excess effects employed by TC2 uses mass-action concepts to derive a value for the equilibrium constant for benzene dimerization, 2B s Bz. In the limit it is readily demonstrated that Kdimer = Xdher/Xmonome? = K1/2K0. The data in Table I obviously lead to large temperature and isotope dependences for Kdimer.Conversion of the Henry's law parameters into dimerization constants offers no advantage as far as the detailed molecular interpretation of the effects are concerned. Rossky and Friedman' (RF) suggest an interpretation of the excess free energies based on the McMillan-Mayer theory of solutions.% The McMillan-Mayer second virial ~~~
(34) McMillan,
W.;Mayer, J. J. Chem. Phys.
Dutta-Choudhury et al.
TABLE V : McMillan-Mayer Second Virial Coefficients for Some Solutions o f Isotropic Benzenes and Waters 10-2. 10-2. BB!/ a Bww: a3 system temp, "C Bh, in H,O
8 15 25 33 3 5 (ref 2)
Bh, in D,O
8
Bd,inH,O Bd, in D,O H,OinBh, D,OinBh,
25 33 33 33 33 33
- 6.4 -4.8 - 5.2 -15.6 -11.8 - 3.9 -2.5 t 2.2 6.3 -6.9
-
- 51
- 56
(I Uncertainties readily calculable from data in Tables I and 11.
coefficient and the Henry's law constants are related by the expression -K1/Ko = -2713bb - 1 i~VB/V~
where 71, is the number density of the pure solvent, VB is the partial molar volume of benzene at infinite dilution, and w-is the molar volume of water. Although the present VB data are limited to 25 "C, we do not expect a very sensitive temperature dependence. Also the last term in eq 25 is small compared to Kl/Ko and we have proceeded to calculate McMillan-Mayer virial coefficients for the data reported in Table I, taking VB/ V,,, independent of temperature. These coefficients are given in Table V. Obviously they reflect the marked temperature and solvent isotope dependences discussed earlier in terms of the Kl/Ko ratios. RF develop the virial coefficients in terms of a model pair potential containing a core repulsion and Gurney solvation co-sphere terms. Apparently the complicated temperature and isotope dependences of the excess effects will ultimately be described in terms of an isotope-dependent Gurney term in the RF formalism. Free Energy Differences in Benzene-Rich Solutions b*Hp(Bha) - p*w(BhG). Henry's law constants, KO and K ~ for dilute solutions of H20 or D20 dissolved in B 4 have been reported in Table I. The isotopic free energy dif& ) ] is in reaference, h [ K " H O H ( B ~ ) / K " ~ D (=B-0.012 sonable agreement with the values reported by Moule2Iand as calculated from solubility meaBackx and G01dman~~ surements assuming (improperly)solution ideality. Those authors actually report values of -A In Y, which, more directly, are to be compared with the present A In K O + (K~/K")HY,(HZO) - ( K ~ / K " ) Y ~values. ( D ~ ~The ) E"$ are solubilities. The isotope effect on K ~ / K Ois zero within experimental error. It is interesting that K J K " for these solutions of water in benzene is on the same order of magnitude as the K1/Ko ratios characterizing benzene in water solutions, so the ratio of solute dimers to monomers (should that formalism be elected) is about the same in the two systems at equivalent concentrations. Even so, the solubility of water in benzene is larger than that of benzene in water by about a factor of 10, and therefore near saturation these solutions deviate much further from Henry's law, f, = K O Y ~ . The most important feature of the data is the observation that the isotopic free energy difference between H20and DzO changes from a very large and positive values of +0.132 (33 "C) unit in the pure liquid to a small negative value at high dilution in benzene.
~
1945, 13, 276.
(25)
(35) Backx, P.;Goldman, S.J. Phys. Chem., 1981,85, 2975.
,
Isotope Effects In Aqueous Systems
TABLE VI: Transfer Volumes at 25 "Ca soln -
v*, F2
Bh/H,O Bh/D,O Bd/H,O Bd/D,O H,O/Bh D,O/Bh In cm3/mol.
-6.9 t 0.1 -6.6 f 0.2 -6.4 2 0.2 -4.9 f 0.2 6.3 + 0.4 6.9 f 0.5
The Journal of Wysical Chemistry, Vol. 86, No. 9, 1982 1721
V,(satd) - F,
soln
-6.2 f 0.4 -15.6 i 2.0 -5.3 0.6 -9.5 f 1.2 -2.2 f 2.0 -1.0 f 2.0
H,O/Bd D,O/Bd Th/H,O Th/D,O Td/H,O Td/D,O
No entry, solubility not available.
_
-
v*,- v', C C
-9.4 f 0.2 -8.5 t 0.2 -8.6 i 0.2 -8.2 f 0.2
-
V,(satd)
b b -6i 1 -11 f 1 -6 i 1 -9 f 1
Insufficient number of data points for extrapolation t o
Molecular interpretations have been advanced by Moule,2l Van Hook,%and most recently Goldman%in terms of the details of spectroscopic frequency shifts on phase change, etc. The phenomenon is therefore well understood in the context of the present discussion. It is doubtful that minor changea in the parameters of fit in order to force agreement with the present value of -0.012 (as compared to that of -0.03 f 0.02 derived from the solubility will improve the physical understanding of the effects. The most important aspect of the present results is that they demonstrate appreciable and isotope-independent nonideality of the aqueous component in the solution. We regret that data on thermodynamic activities of H20 in Bd are not available as this would enable the presence (or absence) of dynamical solute-solvent coupling to be assessed (vide infra). Volumetric Effects. Apparent molar volume data for solutions of Bh and Bd as well as toluene-h8 (Th) and toluene-d8 (Td) in H20 and D20 have been presented in Table 11. Apparent molar volumes of H20 or D20 in Bh or Bd are also given, but the later set of solutions only have one data point each. The data clearly establish an important concentration dependence for the excess volumes, which is isotope dependent. This holds true for the three different kinds of solution investigated. As mentioned in the early part of the paper, this observation alone verifies the earlier conclusion, based on free energy measurements, that these solutions are not ideal in the Henry's law sense. The point is important since a number of claims to the (36) Van Hook,W. A. Zsotopenpraxk 1968,4, 168. (37) Ben-Naim, A,; Wilf, J. J.Phys. Chem. 1980,84, 583. (38) G r w , P.M.;Saylor, J. H.J. Am. Chem. SOC.1931,53, 1744. (39) Veda, A. Thesis, University of Turku, Finland, 1973.
- 7,
v*,.
contrary have been made in the relatively recent literat~re.~J~ The emphasis in the present paper has been on the transfer properties, pure liquid solute to solution. Transfer volumes to the dilute solution reference state, and to the saturated solution, have been dculated where possible and are found in Table VI. The transfer volumes are large and isotope dependent. The effect of concentration is striking (compare transfer to the saturated solutions with that to the infinitely dilute reference state). Insofar as the excess partial molar volumes reflect the concentration dependence of the parameters describing the geometry of the solutions, they reenforce the earlier conclusion (based on the free energy measurements) that hydrophobic ordering phenomena are very sensitive to isotopic substitution on either solvent or solute. That in turn was earlier rationalized, at least in a qualitative sense, in terms of dynamical coupling between solute and solvent species.
Conclusion Neither hydrocarbon-rich nor water-rich solutions of benzene or deuteriobenzene, toluene, or deuteriotoluene, in H20 or D20 are ideal in the Henry's law sense. Both the standard-state and excess partial molar free energies and volumes display marked isotope dependences which have been interpreted in the framework of the theory of isotope effects in condensed-phase sytems. The analysis indicates that significant dynamical vibrational coupling exists between solute and solvent normal modes. Acknowledgment. Financial support from the National Science Foundation, under Grant CHE81-12965, and from the National Institutes of Health, Institute of General Medical Sciences, is appreciated.
