J. Phys. Chem. 1980,84, 587-589
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Benzene-Benzene Interaction in Aqueous Solution Peter J. Rossky"' and Harold L. Friedman" Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794 (Received June 22, 1979; Revised Manuscript Received November 30, 1979) Publication costs assisted by the National Science Foundation
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It is shown that the recently measured deviation from Henry's law for solutions of benzene in water is in reasonable agreement with what. would be predicted from certain models for the solute-solute potential of mean force, with parameters chosen in accordance with earlier results for other hydrophobic solutes. The corresponding inference for the temperature dependence of the model benzenebenzene potential predicts a negative temperature derivative for the osmotic second virial coefficient.
Recently Tucker and Christian' reported new results for the equilibrium distribution of benzene between vapor and aqueous solution phases, showing measurable deviations from Henry'n law. As they point out, the deviation is in the direction iaxpected if painvise hydrophobic bonds2 form between benxene molecules in solution. Their result for the corresponding standard free energy of association on the mole fraction scale is about -2300 cal/mol, which is considerably less negative then an estimate given earlier by Ben-Naim and c o - ~ o r k e r s The . ~ purpose of this note is to show that the new measurements are roughly predictable based on earlier theoretical considerations of related s o l u t i o n ~ , and, ~ - ~ further, that these measurements provide a relatively sensitive test of the theoretical model. To relate the new data to earlier studies of the interaction of hydrophobic solutes in water, it is convenient to introduce the corresponding osmotic (McMillan-Mayer) second virial coefficientgJO
where Uij(r) is the effective (solvent-averaged and orientation-averaged4)potential of mean force between solute particles i and j a t a separation r, 0 = l/kBT, and kg is Boltzmann's constant. The experimental Henry's law "constant" KHBfor benzene is satisfactorily represented by1 KHB = ko + k l X B (2) where XBis the mole fraction of benzene. The pairwise benzene-benzene (BB) interaction is then reflected by the coefficientlo K, =' - k , / k o = -2pwBBB - 1 + 2 u B / u w (3) where pw is the number density of the pure solvent, UB the partial molal volume of benzene at infinite dilution (82.3 cm3), and u, is the molar volume of pure water. Tucker and Christian report' K1 = 87 corresponding to BBB = -1177 A3. It may be of interest to note that for benzene in the gas phase at 35 O C the second virial coefficient id1 about -2100 A3, corresponding to a more negative free energy of association than in solution. The comparison is, however, not very direct. The attractive part of the solvent-averaged pair potential results from solvent structural effects as well as solvent-mediated dispersion forces while, in the gas phase, the attractive part of the pair potential results Department of Chemistry, University of Texas, Austin, TX 78712. 0022-3654/80/2084-0587$0 1.OO/Q
mostly from unattenuated dispersion forces. It also may be noted that if UBBwere replaced by the interaction potential of two hard spheres, each of diameter uBB = 6 A3, corresponding roughly to a benzene molecule,6 we would have BBB = 452 A3. The difference, about -1629 A3, is a measure of the attractive force between benzene molecules in solution. A model pair potential that enables us to calculate Kl from other solution data is given by4 Uij
= CORij(r) + GURij(r)
(4)
with core repulsion CORijand a Gurney solvation co-sphere overlap term GUR,. The Gurney term4-8has the form GURij(r) = AijVmu(r)/Vw
(5)
where Vmu(r)is the volume of solvent displaced by the mutual overlap of the solvation cospheres surrounding solute particles i and j, Vwis the molar volume of water, and A, has the significance of the free energy change per mole of displaced water. In applications to ionic solutions eq 4 is extended by additional terms depending on the ionic charges. Even when i and j are both ions and Bijdoes not exist, the A, coefficients can be determined by fitting the model to appropriate experimental data.4-8 This model follows from a very simple physical picture which neglects a number of known effects, some of which are mentioned below. The neglected effects will contribute to the Gurney Aij parameters determined by fitting the models to experimental data. If the neglected effects are not too important for calculating the thermodynamic excess functions then we may expect, for example, that in the case that both i and j are hydrophobic the Aij parameters will be roughly independent of solute particle size. This behavior follows if various hydrophobic particles affect the water in their cospheres similarly, in which case the principal size effects in l 7 i j enter through the size dependence of CORij(r)and Vmu(r).4-8 Further discussion of the model assumptions is provided in the earlier ~ o r kand ~ - references ~ given there. Here we exploit the fact, not assumed but observed, that the A, parameters needed to fit the thermodynamic data are found in a certain range in cases in which particles i and j are hydrophobic or, at least, carry relatively hydrophobic groups. Thus values of the Gurney free energy parameters A, for hydrophobic species have been obtained in earlier studieskRof tetraalkylammonium ions and their mixtures with alkanes and benzene; it has been noted that these parameters do not vary greatly among species5as expected if the model is satisfactory. Further, the correspond.ing 0 1980 American
Chemical Society
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The Journal of Physical Chemistry, Vol. 84, No. 6, 1980
Rossky and Friedman
A,,/RT -0.35
-0.25
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\'
I
be(CAL/Y)LE) Flgure 1. Dependence of K , (eq 3) on the benzene Gurney parameter Ass at 35 O C , using the exponential core repulsion (gee = 6 A).
A, values for the higher alcohols are ~ i m i l a r .Using ~ the A, values obtained previously at 25 "C to estimate A B B , we find at 35 "C, using eq 1, eq 4, and eq 5 , 7 5 > K1 > 27 or -1001 < B B B / A 3 < -276 for -190 < A B B (cd mol) < -140 and using a core repulsion potential term4
6
i8
PCORij(r) = ( B / ~ r ~ ~ T ) ( u ~ ~ / r ) ~ (6) where B = 5390.5 A deg and oBB = 6 A. Alternatively, we find 60 > K1> 39 Or -773 < &/A3 < -463 for -150 < A B B (cal/mol) < -120 and using a core term5,' /?COR&) = (T*/T)exp[(r - aij)/Rl
(7)
with Tm= 2173 deg and R = 0.344 A. An alternative choice5 for c~BBbased on the molar volume at infinite dilution in water, uBB = 5.68 A, yields values for K1 that differ from those above by only about 10%. The experimentalvalue of K1 is obtained for A B B E -175 cal/mol by using the exponential core and for A B B -200 cal/mol by using the power law core. While these values are not much larger than those estimated above, the value of K1 is sensitive to the choice of ABB, as shown in Figure 1. I t is seen that the measured value fixes relatively narrow limits on the choice of A B B within which the model might be considered to be consistent with the experimental data. Hence, the comparison between the model calculation and experiment must be considered rather satisfactory. Since the determinations of K1 are in reasonable agreement, it is natural to use the earlier result^^*^>^ to predict the temperature dependence of K1. We have, from eq 1 and 3
- ( ~ B H B / ~ =P )MdW1 + 1 - 2uB/uW)/pw)/aP)
=
As in earlier work,@we assume that the radius parameter (QB = 6 A) and the solvation co-sphere thickness (here 2.76 A) do not depend on P; we then only need the Gurney energy parameters defined as Eij = a (PAij) /a P
(9)
Assuming that EBBis similar to E , for the other hydro-
phobic solute^,^^^^^ we estimate E B B = 155 cal/mol, which is close to that obtained for Ei.at 25 "C when i = (Bu)~N+ ~ ~ . using ~ eq 7 and eq 8, we and j = i6P8or j = I Z - C ~ HThen find (35 O C ) - / ? ( d B B / a @ ) = -2704 A3; aKl/aT = 0.57 deg-l for A B B = -150 cal/mol. For the value of A B B for which the model and experimentalvalues of Kl agree $(.B = -180 cal/mol), we find the very similar values -/?(aBBB/ap) = -3330 A3; aKl/aT = 0.69 deg-l. It is important to note that the above temperature dependence corresponds to a positive enthalpy of association, and hence to increasing hydrophobic pair formation with increasing temperature, at least near 25 OC. This result follows as well from other techniques for identifying the contributions to the hydrophobic interaction for various nonionic solutes and their mixtures.1°J2 Recent theoretical con~iderationsl~ have not led to this result; the discrepancy has been attributed13 to the interference of solubilizing polar groups in the interpretation of the experimental data. Forthcoming results for the benzene system1 should provide a relatively unequivocal source of comparison. The pair potential in eq 4 is, of course, too simple. The influence of (solvent mediated) dispersion forces is included only implicitly in the Gurney term,while an explicit account of these forces may well be needed5 for an accurate treatment. However, a comparison of gas phase virial coefficients among benzene and a number of alkanes of similar size14implies that no additional special force should be attributed to the benzene-benzene interaction, and hence that the Gurney parameters should be approximately transferable from earlier work without the addition of new terms in the potential of eq 4. More important, the model potential in eq 4 does not allow for typical liquid structure effects, which tend to make Dij(r)a damped oscillatory function of r, nor does it allow for the distinction between contact and solvent-separated hydrophobic bonds, such as one may expect on the basis of various experimental15 and more recent theoretical13J6evidence. However, the use of eq 4, supplemented by additional charge-dependent terms,7 affords an interpretation of thermodynamic data for ionic systems in terms of Gurney Aijcoefficients that characterize the solvation contribution to the average i-j interaction in solution. While there is some evidence17that the oscillatory contribution in Dij(r) may be much larger when both i and j are ions than when both are uncharged, the Aij coefficients are not very sensitive to the charges, as noted above. In this sense, then, the thermodynamic excess functions used in adjusting the Aij cpefficients are not sensitive to the oscillatory features in Uij(r). Finally, it should be remarked that the fixed parameters in eq 6 and 7 originally were fixed by consideration of the dimensions of alkali halide crystals. For other types of solute pairs further adjustment of these parameters on the basis of nonsolution data is justified but has not been attempted. However, the trends in Aij coefficients for hydrophobic pairs are little changed when different types of ionic radius parameters are used.6 So, even in view of these complications, the present comparison with experiment indicates that, in the case of the interaction between two hydrophobic solutes, the A, coefficients derived from data for ionic solutions, electrolyte--nonelectrolyte mixtures, and solutions of alcohols afford a reasonable characterization for the simplest apolar solutes as well. A c k n o w l e d g m e n t . The financial support of this research by the National Science Foundation i s greatly appreciated. One of us (P.J.R.) acknowledges support by a National Science Foundation National Needs Postdoctoral Fellowship. We also thank Professor Chandler and Rerne for
J. Phys. Chem. 1980, 84, 589-593
sending us reports of their work in advance of publication. References a n d Notes (1) E. E. Tucker and S. U. Christian, J. Phys. Chem., 83, 426 (1979). (2) F. Franks in “Water, A Comprehensive Treatise”, F. Franks, Ed., Vol. 4, Plenum Press, New York, Chapter 1. (3) A. Bendaim, J. Wilf, and M. Yaacobi, J. Phys. Chem., 77, 95 (1973). (4) H. L. Friedman and C. V. Krishnan, J. Solution Chem., 2, 119 (1973). (5) C. V. Krishnan and H. L. Friedman, J. Solution Chem., 3, 727 (1974). (6) P. S. Ramanathan, C. V. Krishnan, and H. L. Friedman, J. Solution Chem., 1, 237 (1972). (7) H. L. Friedman, C. V. Krishnan, and C. Jolicoeur, Ann. N. Y. Acad. Sci., 204, 79 (1973). (8) W. H. Streng and W.-Y. Wen, J . Solution Chem., 3 , 865 (1974). (9) W. McMiliani and J. Mayer, J. Chem. Phys., 13, 276 (1945).
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(10) J. J. Kozak, W. S.Knight, and W. Kauzmann, J . Chem. Phys., 48, 675 (1968). (11) J. H. Dymund and E. 8. Smith, “The Virial Coefficients of Gases”, Oxford University Press, London, 1969. (12) B. Y. Okamoto, R. H. Wood, and P. T. Thompson, J . Chem. Soc., Faraday Trans. 7 , 74, 1990 (1978), Table 9. (13) (a) L. R. Pratt and D. Chandler, J . Chem. Phys., 67, 3683 (1977); (b) J . Solution Chem., 9, 1 (1980). (14) 2. S. Belousova and Sh. D. Zaalishvili, Russ. J. Phys. Chem., 41, 1290 (1967); Sh. D. Zaalishvili, 2. S. Belousova, and V. P. Verkhova, /bid., 45, 149, 894, 902 (1971). (15) (a) W.-Y. Wen and S. Saito, J. Phys. Chern., 68, 2639 (1964). (b) For a relatively recent review, see ref 2, pp 77-78. (16) C. Pangali, M. Rao, and B. J. Berne, J. Chem. Phys., 71, 2975, 2982 (1979). (17) D. Levesque, J. J. Weis, and G. N. Patey, phys. Lett., 86A, 115 (1978).
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Thermal Transpiration Effects for Gases at Pressures above 0.1 torr Isao Yasumoto Department of Chemistry, Yonago Technical College, Yonago, 683, Japan (Received Aprll 12, ‘1979) Publication costs assisted by Yonago Technical College
Thermal transpiration effects have been measured by a relative method with a high-sensitivity mercury U-tube manometer for helium, neon, argon, krypton, xenon, hydrogen, deuterium, oxygen, nitrogen, carbon monoxide, nitric oxide, carbon dioxide, nitrous oxide, hydrogen sulfide, ammonia, hydrogen chloride, water vapor, methane, hydrogen bromide, acetylene, ethylene, ethane, methyl chloride, methyl alcohol, ethyl alcohol, propane, n-butane, and isobutane in a pressure range from 10 to 0.1 torr (1.3 X lo3-13 Pa). The Takaishi-Sensui equation was applied to the data obtained. Values of constants in their equation, A, B, and C, were determined for the gases mentioned above. Empirical relations were obtained between the diameters of molecules and the values of constants A and C.
Introduction The importance of applying the corrections due to thermal transpiration effects for gases has been amply demonstrated in such cases as in adsorption studies and low pressure measurements, where the temperature difference exists between the part of a system in which the pressure is to he measured and a pressure measuring device. Many investigators L-12 measured the effects for noncondensable gases, and obtained experimental curves or empirical equations. However, few measurements of the effects for condensable eases have been made because of technical difficulties. Recentlv, Takaishi and SensuilO DroDosed from their hydrogen measurement one of the mosi useful empirical equations, which is a modification of that due to Liang,4 and is expressed as
The equation has the following excellent advantage: when it is difficult to measure the effect for an appropriate gas under the condition of Tl < T2,we can estimate the effect by using the equation whose constants A, B, and C were obtained under the condition T1> T2. The aim of this study is to examine the general adaptability of the equation. When a high-sensitivity mercury U-tube manometer was used, measurements of the effelcts for a number of condensable gases as well as noncondensable ones have been made by the so-called relative method4 under a variety of temperature conditions in a pressure range from 10 to 0.1 torr. The data obtained were found to be fitted with the Takaishi-Sensui equation. In connection with the diameters of molecules, discussion has been made on the values A, B, and C which were obtained for a variety of gases used.
(1 - (Pi/P2))/(1 - (T1/7’2)”2)= 1/(AX2
Water Vapor. The Takaishi-Sensui equation states that thermal transpiration values, P,/ Pa, are described as a function of the ratio of the product P2d and the mean temperature T. The adaptability of this equation was examined experimentally for water vapor, one of the most condensable gases. The equation was found correct for explaining the results. The experimental procedure and the results are presented below. The experimental arrangements used are illustrated in Figure 1. The pressure difference was measured between the upper ends of a capillary tube and a wide tube which were joined together at their lower ends to form a U-tube. This was installed in a brass box 80 mm wide, 300 mm
-
+ BX + CX112+ 1)
for T , < T2and
(1 - (P2/P1))/(1- (7‘2/TJ1/’)= 1/(AX2 + B X
+ CX1I2+ 1) for Tl > T2with X = P,(d/T) and T = ( T , + T2)/2,where
PI and P2 and T1and T’, denote pressure and temperature in the respective parts of a two-part system which are connected by a narrow tube, and d denotes the diameter of the connecting tube along which the temperature gradient exists. A , B, and C are specific constants for an appropriate gas.
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Experimental Section
0 1980 American
Chemical Society
