Hydrophobic lriterection in Light and Heavy Water
95
LE I: Comparison of $X Determined from P, and from Eq 10 $X determined from M
IMembrane
$X determineda from eq 10, M
-
_ _ I
PSSA
PS-1
collodion PI’#-3 Oxidized M. 1 collodion M.2 M.3 Dextran DS sulfate eo1lod i o n Protamine PF: collodion
0.091 0.014
0.005 0.014 0.021 0.0029
0.038
0.082 0.092 0.015 -0.009 0.0044-0.0058 0.0158-0.0168 0.023-0.027 0.0025 0,0027
-
0.018 for LiCl 0.024 for KF 0.034 for NaCI, KCI
a Deviation of (6X values in this column depends on the electrolyte soecies used.1xz,5
stead of eq 7. All the data studied here fall approximately on a curve which deviates systematically from the straight line of slope unity in a dilute salt solution. As discussed in previous papers,lJ,8 the degree of unbounded counterions is slightly dependent upon the external salt concentration when the concentration decreases in comparison with 4 X . Thus the deviation of the data points from the straight line stems from the concentration dependence of $X.Figure 3 shows the dependence of +X on the external salt concentration is approximately equal for all combinations of membranes and electrolytes studied here. It is noted that for the system with protamine collodion membrane in LiCl the data deviate appreciably from the straight line in the concentrated region. This is also seen in Table I where the value of $X determined from P,, us. log C plots is much smaller than that obtained from the other method in a concentrated region of LiC1, and that the value of +X for LiCl is much larger than those for KC1 or NaC1. Further study is necessary.
case where a positively charged membrane is concerned, as in the protamine collodion membranes, Ps must be de(8) M. Yuasa, fined as [ T ~ ~ ~ -- , (- I - .)]/[I - a - (1 - 2 a ) ~ ~in-~ ~ - ](1968).
Y. Kobatake, and H. Fujita, J. Phys. Chem., 72, 2871
Hydrophtobk Interaction in Light and Heavy Water A. B,ert-Naim,* J. Wilf, and M. Yaacobi Department of lnorganrc and Analyfical Chemistry, The Hebrew University of Jerusalem, Jerusalem, lsraei {Received May 3, 1972)
The solubility, free energy, entropy, and enthalpy of solution of methane, ethane, n-butane, benzene, and biphenyl were determined in light and heavy water. The hydrophobic interaction between the pairs met hane-methane, ethane-ethane, and benzene-benzene were computed in these two liquids. It was h i n d that the strength of the hydrophobic interaction for the first two pairs is larger in HzO than in DzO. The order seems to be reversed for the pair benzene-benzene.
I. Introduction Heavy water. is known to have similar properties to light water. Yet thwe are some, usually small, differences between the two liquids. This makes them an ideal pair of liquids for testing various theories and conjectures made on the molecular origin of the outstanding properties of water. This paper i s concerned with one aspect of these properties, namely, the strength of the solute-solute interaction between two simple solutes, which is referred to as hydrophobic interaction (H1[).1-5A more detailed comparison of the properties of light and heavy water has been presented by N6mei:hy and Scheraga.6 In this paper we shall refer to EBB as the indirect part of the work required to bring two solute particles from fixed positions at infinite separation to a small distance from each other. More specifically we define the HI between two simple solutes a and b at distance R by either of the following relations OAabEi*((R)cz. AAab(R)
- Uab(R)
(1.1)
(1.2) where Uab(R) is the “direct” work required to bring the two solute particles, a and b, from infinity to a distance R in vacuum. AAab(R) is the Helmholtz free energy change for the same process, but carried out within the liquid at constant temperature and volume. R) is similarly the change in the Gibbs free energy for the same process carried out within the liquid but at constant temperature and pressure. The advantage of subtracting Uab(R) in (1.1) and (1.2) has been discussed in detail in previous art i c l e ~,5. ~ (1) W. Kauzmann, Advan. Protein Chem., 14,l (1959). (2) G . Nemethy and H. A. Scheraga, J. Chem. Phys., 36, 3382, 3401 (1962); J. Phys. Chem., 66, 1773 (1962). (3) K. J. Kozak, W. S. Knight, and W. Kauzmann, J. Chem. Phys., 46, 675 (1968). (4) A. Ben-Naim, J . Chem. Phys., 54,1387,3696 (1971). (5) A. Ben-Naim, J. Chem. Phys., in press. (6)G . Nemethy and W . A. Scheraga, J. Chem. Phys., 41,680 (1964).
The Journal of Physicai Chemistry, Val. 77, No. 7. 1973
96
A. Ben-Naim,J. Wilf, and M. Yaacobi
The process carried out at constant volume is usually slightly simpler for a statistical mechanical treatment than the one at constant pressure. The latter is, however, more directly related to the experimental quantities which are measured a t constant pressure. Using the approximate relation, derived previously,4.s between HI and measurable quantities we write 6Aabi"(i71)
=Z
apa+bo
- ApaO - A&'
(1.3)
+-
where a b stands for a single molecule produced by "fusing" together7 the two particles a and b to a distance CQ. dpmO is the s t a n d a ~ dfree energy of solution of the solute a. Since the temperature dependence of Apa0 is usually determined at constant pressure, it is appropriate to define the entropy and the enthalpy of HI by the relations dSabH' (A') =
- (a6GH'( R )/dT) p
(1.5)
These quantities are then estimated at a particular distance 6 1 by
In this paper we report the values of the HI and the corresponding entropy and enthalpy of HI for the pairs methane-methane, ethane-ethane, and benzene-benzene in light and heavy water. T o the best of our knowledge, the only previous work with direct bearing to the present one has been published by Mresheck, Schneider, and Scheraga 8 In the next section, we summarize some of the experimental aspects of our work. Section I11 reports the results of the solubilities, free energies, entropies, and enthalpies of solution of some simple solutes in light and heavy water. Section IV presents the corresponding values for the hydrophobic interaction and its temperature dependence. Possible interpretations of these results are briefly discussed in sect ion V
11, Experimeintsd Section A. Solubilii'ies of Methane, Ethane, and n-Butane. The solubilities of methane, ethane, and n-butane were determined according to the method developed by Ben-Naim and Baer.9 All iche gases were purchased from Matheson (with the following purity: methane 99.9770, ethane 99.970, and a-butane 99.95%'1. D20, for this purpose, was from Darmstadt (9!3.?5%) and was used as received. Some modifications in the apparatus were introduced according to suggestions made by Wen and Hung.10 These modifications have improved the accuracy of the measurements especially for the hydrocarbons. (It is worth noting that the solubilities of argon, determined in the new apparatus, were the same as in the older one.) We believe that the major improvement is in reducing the total amount of gas adsorbed on the surface of the vacuum grease, a phenomenon that seems to be more important as the chain length of lche paraffin molecule increases. (This was checked directly by letting the gas in contact with the liquid for 24 hr before starting the dissolution process. In this way we could mmimize the error due to adsorption.) Our values of the Ctstwald absorption coefficient were systematically higher than those reported by Wen and Hunglo (methane -0.3'70, ethane --8.5%, and n-butane -1.2%). The Journal of Physical Chemistry, Vol. 77, No. 1, 1973
Figure 1. Three cells connected by bridges. The initial solution was placed in cell A whereas cell C contained the pure solvent. In cell B the concentration of the solute vapor was measured. Equilibrium is reached when the solute concentrations in A and C are equal.
The discrepancies, however, are quite small compared to the isotopic difference for the solubilities of these gases in H20 and in DzO. B. Solubilities of Benzene and Biphenyl. The solubilities of benzene and biphenyl were determined spectroscopically. Benzene was purchased from Fluka (Puriss, 99.94%, used as received) as was biphenyl (Purum, recrystallized from ethanol, washed and drie effect on the spectrum of its solutions could be detected). Distilled water was purified by further distillation from alkaline KMn04 and acid KzCrz07 solutions, the conductivity of the water finally used was (4.8 X 0hm-I cm-1. D20 was also purchased from Fluka (99.75%) and used as received. The absorbancies of the solutions and of their vapors were measured directly at Xmax with a Perkin-Elmer spectrophotometer (Model 450), operated with a slit width of 0.05 mm at 250 nm at low scanning speed, and fitted with a thermostat-controlled cell holder. The temperature was maintained within 0.05" in the range of 1-50'. Saturated solutions of benzene were obtained in two ways: (1) by direct mixing of benzene and water (or DzO) for about 48 hr and then measuring the absorbancy of the aqueous solution in equilibrium with the benzene (liquid) phase; (2) by dissolution of benzene through the vapor phase, in this case, there is no contact between the two liquids. Equilibrium is established between the aqueous solution and the gaseous phase. The two methods gave identical solutions when the vapor pressure of benzene above the solution was equal to the vapor pressure of pure benzene. However, the latter method is superior for the purposes bf the present study. Here, we are interested in the Iimiting value of the Ostwald absorption coefficient (see next section for more details) at a very low concentration of solute. li successive series of solutions with decreasing concentrations of benzene, obtained by the second procedure, showed no deviation from Beer's law. (There has been some controversy in the 1 i t e r a t ~ r e l l - regarding l~ deviations from Beer's law of aqueous benzene solutions. At low concentrations of benzene, in which the Ost,wald absorption coefficients have (7) For more details of the process ascribed by "fusion," and the nature of the approximation involved in (1.3) and (1.4),the reader is referred to ref 4 and 5. (8) G.C. Kresheck, H. Schneider, and H. A. Scheraga, J. Phys. Chem., 69, 3132 (1965). (9) A. Ben-Naim and S. Baer, Trans, Faraday Soc., 59,2735 (1963). (IO) Wen-Yang Wen and J. H. Hung, J. Phys. Chem., 74, .I70(1970). ( 1 1 ) D. S. Arnold, C. A. Plank, E. E. Erickson, and F. P. Pike, Ind. Eng. Chem., Chem. Eng. Data Ser., 3,253 (1958). (12) L. J. Andrews and R. M. Keefer, J. Amer. Chem. SOC.,71, 3644 (1949). (13)J. D. Worley, Can. J . Chem., 45, 2465 (1967). (14) F. Franks, M. Gent, and H. H. Johnson, J. Chem. Soc., 2716 (1963). (15) 0.G.Marketos, Anal. Chem., 41,195 (1969).
Hydrophobic Interaction in Light and Heavy Water been evaluated, we have found no curvatures in the plot of absorbancies us concentration.) To ensure that the system has reached equilibrium with its vapor, we have connected a series of three quartz cells by means of two bridges as depicted in Figure 1. The first cell contained aqueous solutions of benzene (below saturation, and with different concentrations at different runs). The second cell contained only the vapor of the solution. The third cell was initially filled with pure water (or D20).The three cells formed a closed system so that the total amount of benzene was fixed. The absorbancies of the three cells were determined while stirring the solutions. The final reading was taken when the concentrations of benzene in the two liquid phases were identical. This indicates that the system has reached complete equilibrium, partii-ularly between the solution and its vapor. (In a preliminary stage of this work we have tried to determine direct!y the concentration of benzene in the vapor in equilibrium with the solution. The ratio of these two concentrations approaches a constant value after about 24 hr. However il is difficult to be certain when the system bas reached complete equilibrium. In the three-cell method, we hitve quite a sharp indicator for the equilibrium point. since the third cell is being saturated through the gaseous phase. It is clear that once the two liquid cells have equal solute concentrations, the solute is also in equilibrium with its vapor.) Saturated biphenyl solutions were obtained by stirring the solid brphenyl with water (or DzO) for about 60 hr. The absorbancies of these solutions were determined at equilibrium with excess of the solid phase. Here, since the saturation concentrations are quite small, one may neglect solute-solute i d eraction at these concentrations. In order to determine the Ostwald absorption coefficient of these solutions, we have also measured the optical density of the vapor of pure biphenyl in a 10-cm quartz cell. From these measurements, we obtained the concentrations of biphenyl in t k gaseous phase, which were used to compute the Ostwald absorption coefficients. The overall accuracy of the Ostwald absorption coefficient for benzene and biphenvl was lower than that for the gaseous solutes and the rslisibility is believed to be within &5%. C Determanation of the Molar Sxtinction Coefficients The molar exlinction coefficients, e, of benzene in H20 and in DzCl were determined by measuring the absorbancies of un%atmrai,edsolutions with known concentrations. The average value of E at 254 nm in both H2O and B& was found to be 180 at 20°, in satisfactory agreement with the rrported d u e of Marketos.l5 Similarly, the ~ ~E128 ! ~ and l DzO has been detervalue of c for ~ i p ~ e in mined. The average value of t (in both HzO and D2O) a t Xnrax 247 nm was 1 0 5 x lo4 and a t Xmax 200 nm was 2.58 x (The latter was more convenient at very low concentrations of biphenyl.) To determine the e of benzene vapor we have adopted the following procedure. The absorbance of a sample of benzene vaFlor at low pressure (about 4.5 mm pressure in a 18-cm quartz cell at 20") was determined at Xmax 252 nm. Then, the total quantity of benzene was solidified by introducing thc cell into liquified air. At this state the cell was quickly filled with hexane and the temperature raised to 20", so tha: the total quantity of benzene that had occupied the vapor is now dissolved in hexane. Next, the absorbance of the solution was measured, and using the value18 of t 250 at )airiax 254 we could determine the con-
97
700
-
5
600
r ._
-, -0
-E
w
500
400
to
c
Temperature dependence of the molar extinction coefbenzene vapor at Xmax 252 nm (see section I I for more details).
Figure 2. ficient of
centration of benzene in the solution and, hence, in the vapor phase as well. Repeating the same procedure with ten samples of benzene vapor (at low pressures) we obtained the average value 5 515 (at Xmax 252 nm at 20"). Once we had determined the concentration of benzene vapor a t one temperature, we could proceed to evaluate the temperature dependence of t from l to 50". These values are reported in Figure 2. As a further check of consistency, we have determined the vapor pressure of benzene in equilibrium with the pure liquid (using the values of t obtained by the above procedure). Assuming that the gas phase is ideal, the vapor pressures, computed by p = D R T / t l (with D as the absorbance, 1 the optical way, and R the gas constant), were found to agree within 10% with the values reported in the literature. The t of biphenyl vapor (in a 10-cm quartz cell) was determined by measuring the € of biphenyl in hexane, cyclohexane, ethanol, methanol, and methylcyclohexane. As no regular changes in t a t Xmax 247 nm in the various Solvents were observed, we have assumed that the same value of E is also valid for the vapor phase, namely, t 17,000 at Xmax 236 nm. We have repeated the same measurements on samples of biphenyl. at very low pressures (much lower than the equilibrium pressure in the presence of the pure solid) between 5 and 50". No significant temperature effect was detected. By using this value of t , we could also determine the vapor pressure of the pure biphenyl. A nice linear plot of In P us. T-3. was obtained (Figure 3), which coincides with the curve obtained from data from another source.17 111. Thermodynamic Quantities of Solution The basic experimental ingredient that is employed in (16) "U. V. Atlas of Organic Compounds," Vol. 11, Rutterworths, 7966. (17) R. S. Bradley and T. G. Cleasby, J. Chem. SOC., 1690 (1953). The Journai of Physicai Chemistry, Voi. 77, NU. 7, 7973
98
A. Ben-Naim, J. Wilf, and M. Yaacobi -1
e
-2
r
e
i
-3
-4
tn P -5
Figure 4. Dependence of the Ostwald absorption coefficient ys of benzene in water at various concentrations (at t = 20'). T h e least-squares fit of a straight line is also shown.
-6
TABLE I: Polynomial Coefficients in (3.3) and the Standard Deviation for Each Set of Measurements
-7
l l _ _ l _
System
Figure 3. Vapor pressur13 (in mm) of biphenyl as a function of temperature. Open circles are values from ref 17 and full circles are values obtained speclroscopically (see section I I ) . computing the thermodynamic quantities of solution is the Ostwald absorption coefficient, defined for a solute s in a liquid 1by 'YS
= (Ps'/psg)eq
(3.1)
where psl and p s g are the molar concentrations of the solute s in both the liquid and the gas at equilibrium, respectively. It is important to realize that although ys is a well-defined quantity without any requirements on the absolute magnitude of the concentrations of s in the two phases, one must take the proper limiting value of ys for computation of the si andard thermodynamic quantities. More specifically, the standard free energy of solution of s from the gas to the 1iyc;id i s tiefined by
Note, that since the limit is taken along an equilibrium line, it is sufficient to let one concentration, p J or psg, tend to zero. In order to assure that we obtain the limiting value as in (3.2j7 we have made a series of measurements of ys a t Pow concentrations. Figure 4 shows ys of benzene in water a t various concentrations of psl. It is clearly seen that the value of ys approaches a constant value. We have, in most of our work, determined y s at about half of the saturation ~ o n c e n ~ ~ ~ tAs i o for n . the solubilities of the gases methane, ethane, and butane, we have found that the value of ys is almost constant when the total pressure is reduced to one-half or to one-third of the atmospheric pressure. Thmefore, one may safely assume that the saturated solutions of' these gases are, to a good approximation, dilute idea' solutions. About ten values of A p s o were determined in the range of temperatures from 5 t o 25" for the gases, and between 5 and 50" for the benzene and biphenyl. The temperature dependence of A,ugO was fitted to a second-degree polynomial of the form
+ bT+
cT2 ,Tis the temperature in "C) Apso = a
(3.3)
from which the entropy and the enthalpy of solution were The Journal ot Phys,ca/ Chemisfry, Vol. 77, No. 7, 1973
Methane in H20 Methane in D 2 0 Ethane in H20 Ethane in D 2 0 n-Butane in H 2 0 &Butane in D20 Benzene in H20 Benzene in D 2 0 Biphenyl in H 2 0 Biphenyl in D20
C
- 0.09371 - 0.09 185 -0.1 161 1 -0.21988 -0.1 5648 -0.1 2702 0.02666 0.03864 - 0.04126 0.01 945
-
b
a
71.359 71.233 89.432 149.86 118.82 104.26 37.122 44.618 52.057 39.059
- 10944.7 -1 5102.5
- 14509.6 -23335.8
- 19425.0
- 17694.0 -9536.2 -10657.5 - 14740.4 - 7 2827.6
Standard deviation 0.807 1.36 2.039 2.041 2.945 4.124 8.788 15.103 22,794 27.015
computed. The values of the coefficients in (3.3) as well as the standard deviation in each case are reported in Table I. The value of Apso defined in (3.2) may be interpreted in two different ways.ls (I) Apso is the change of the Gibbs free energy19 for transferring 1 mol of solute s from the gas to the liquid at constant P and T. In addition, we require that psl = psg (provided that both ps1 and psg are small enough that the phases are dilute-ideal). (11) Apso/NO (with NO being the Avogadro number) is the change in the Gibbs free energy for transferring one solute particle from a fixed position in the gas phase to a fixed position2*in the pure solvent a t constant P a n d T. The first interpretation is closer to an experimental procedure, and hence the more commonly used; the second one, on the other hand, has some advantages from a mo9 this quanlecular point of view. It can be ~ h o w n l " ~ that tity may be expressed as
where Us, (Xs,Xw) is the solute-solvent pair potential a t the specific configuration denoted by X,, Xw. p (X,/X,, I ) is the singlet conditional distribution function for solvent molecules around a given solute at a fixed configuration Xs and coupled to the extent l . Note that since Us, has a range of a few molecular diameters and since p(X,/X,,C;) (18) A. Ben-Naim in "Wafer and Aqueous Solutions," R. A. Horne, Ed., Wiley-lnterscience, New York, N. Y., 1972, Chapter 11. (19) In ref 18, the treatment is confined to the TVN ensemble where the Helmholtz free energy changes are examined. In a similar fashion one may consider changes in the Gibbs free energy in the T f " ensemble. (20) In the case of simple spherical particles, it is sufficient to require a fixed position only. For more complex solute rnolecules, one has to require a fixed orientation of the molecule a ~ also d a proper average over all internal rotations. For more details on this averaging procedure the reader is referred to ref 4 and 5.
Hydrophobic Interaction in Light and Heavy Water
99 TABLE IV: Values of the Ostwald Absorption Coefficient y s , Standard Free Energy of Solution, and Standard Entropy and Enthalpy of Solution for Processes I and II (Defined in Section Ill) for n-Butane in H 2 0 and in D 2 0 *
LE ! I : Values ob the Ostwald Absorption Coefficient ys, Standard Free l ~ n e of ~ Solution, ~ y and Standard Entropy and Enthalpy of Soltition for Processes I and II (defined in Section Ill) for Methane in H20 and in DaOa _ l _ _ _ l l l _ l
:,"C
y+
Apso
SS,O(II)
AH,O(I)
AH,o(ll)
-19 -18 -17 -16 -15
-4200 -4000 -3700 -3400 -3100
-3700 -3400 -3200 -2900 -2600
5 10 15 20 25
0.0641 0.0511 0.0416 0.0346 0.0293
1519 1674 1821 1960 2092
-34 -32 -31 -29 -27
-22
-20
-21
-19
-20 -19 -18
-18 -17
-4600 -4300 -4000 -3700 -3500
-4000 -3700 -3500 -3200 -2900
5 10 15 20 25
0.0689 0.0539 0.0430 0.0350 0.0289
1479 1644 1802 1954 2100
-36 -34 -33 -32 -30
ASS0(l)
HzO 5 10 15 20 25
00502
5
00549 0 04.65 00454 00393 0 0360
00448 00405 0 0370 0 03L12
1654 1747 1836 1921 2001
-21 -20 -59 -18 -17
H20
10
15 20 25 a
The units for
-16
a
LE I!!: Values of the Ostwald Absorption Coefficient ys, Standard Free Energy of Solution, and Standard Entropy and Enthalpy of Sol~itionfor Processes I and I! (Defined in Section Ell) fox Ethane in H 2 0 and in DnOa
y,3
'&\c
______ IO 15
20 25
-7000 -6500 -6000
-34 -32 -31 -30 -29
-8500 -8100 -7700 -7300 -6900
-5500
-7900 -7500 -7100 -6800 -6400
AS,O(I)
AS\"(Il)
AH,@(l)
-
Units as in Table I.
TABLE V: Values of the Ostwald Absorption Coefficient yss Standard Free Energy of Solution, and Standard Entropy and Enthalpy of Solution for Processes I and I I (Defined in Section I ! I) for Benzene in HzQ and in D20*
AS,O(II)
t, "C
y.
10
20 30 40 50
79 5.1 3.4 23 1.6
-1163 -945
-526 -324
-24 -23 -23 -22 -21
10 20 30 40 50
73 47 3.1 21 15
-1122 -898 --682 -474 -274
-25 -24 -23 -22 -21
aiL.0
AS?O(I)
I
Hz0 5
-29 -27 -2
-7300 -6900 -6400 -6000
arid AH,O are cal/moi and for AS," are cal/mol
deg
t"C
-7900 -7400
DzO
D70 16C5 1?C3 1797 1886 1971
-32 -30
0 0820 00691 00591 0 0574 0 0453
1383 1504 I620 1730
-27 -26 -24 -23
-25 -24 -23 -21
I834
-22
-20
00889 00723 00621 0 0539 0 0479
1338
-30 -27 -25 -23 -20
-6100 -5700 -5400 -5100 -4700
-5500 -5200 -4900 -4500 -4200
-6900 --6300 -5600 -5000 -4300
-6300 - 5700 -5100 -4400 -3800
AS,O(ll)
~
-
AH,"l)
_
_
AH,"(II)
_
H20
-734
-22 -21
-8000 -7800
-21
-7600
-20
-7400 -7300
-20
-7400 -7200 -7100 -6900 -6800
Dz0 5 10 15
20
25 LL
1471 1592 1702 7801
-28 -25 -23
-21 -19
Units as in Tahle I
D2Q
a
tends to a coniitant for large distances between the solvent and the solute, the integration over the location of the solvent moleciile practically extends over a finite and small region about the location o f the center of the solute. This "local c h a r ~ ~ c t t ~ofr "Apso makes it a suitable quantity for studying solute-solvent interactions. It is worth noting that although the two processes I and II produce the same value of Apso (per one mole or per one particle), the situation is somewhat more complex for the corresponding changes of the entropy and the enthalpy, for which we h t ~ e * ~ , ~ 9
(3.8)
where p' and pg are the densities of the two phases, respectively, and the derivatives are taken at constant pressure. The second process (11) is more advantageous for the
-23 -22 -21 -20 -20
-8100 -7900 -7600 -7400 -7190
-7600 -7300 -7100 -6900 -6600
Units as in Table I
following reasons. (1) The thermodynamic quantities are simpler to obtain from measurable quantities. (Particularly, one does not need the coefficient of thermal expansion of the solvent for evaluating the entropy associated with process 11.) (2) Because of the local character of ApsO/NO, the thermodynamic quantities associated with process 11 are more suitable for interpretation in terms of solute-solvent interaction. (3) The quantities associated with process I1 are naturally involved in the approximate expressions for the strength of the hydrophobic interaction435 and its temperature dependence (see next section for more details). In Tables 11-VI we report the values of ys Apso, ASSo, and AHsO (for the two processes I and 11 as indicated) for methane, ethane, n-butane, benzene, and biphenyl in H20 and in DzO. There are a few points that are worth noting regarding these quantities. (1)The values of ys for methane and its temperature dependence for N28 and DzO are quite similar to the corresponding values for argon in these two solvents.21 (2) The Ostwald absorption (21) A Ben-Naim, J. Chem Phys., 42, 1512 (1965)
lhe Journal of Physical Chemisiry, Vol 77, No. 1, 1973
A. Ben-Naim, J. Wilf, and M. Yaacobi
100 TABLE VI: Values lref the Ostwald Absorption Coefficient ys, Standard Free Energy of Solution, and Standard Entropy and Enthalpy of Solution for Pirocesses I and I I (Defined in Section Ill) lor Biphenyl in H20 and in D 2 0 a
TABLE VIII: Hydrophobic Interaction and the Corresponding Entropy and Enthalpy Values for the Reaction Z(Ethane) -n-Butane (eq 4.5) in H20 and D20a
H20
10 20 30 40
50
3574 180.0 96.2 542 32.0
-3309
-3026 -2752 -2485 -2227
-31 -30 -29 -28
-29 -28 -27 -26 -25
-27
- 1248
5 -12,000 -11,700 -11,500 -11,200 -10,900
-11,400 -11,200 -10,900 -10,700 -10,400
H20
- 1335
10 15 20 25
-1419 -1499 - 1576
18 18
3700
17
3400 3000 2900
15 ld
3500
DzQ B20
10 20 30
40 50
369.5 187.3 99.9 55.8 32.5
-3328
-3049 -2775 -2504 -2237
-30 -30 -29 -29 -28
5 -28 -28 -27 -27 -26
-11,800 -11,700 -11,600 -11,400 -11,300
-11,300 -11,200 -11,000 -10,900 -10,800
10 15 20 25 a
a Units as in Table
-1198 -1297 -1381 -1450 - 1502
22 18
15 12
4700 3900 31 00 2000
9
'1 200
Units as in Table I
I
LE '1111: Hytlrolihobic Interaction and the Corresponding Entropy and Enthalpy Values for the Reaction Z(Methane) --Ethane (eq 4.4) in H20 and D20a
TABLE IX: Hydrophobic Interaction and the Corresponding Entropy and Enthalpy Values for the Reaction 4(Methane) n-Butane (eq 4.6) in H20 and in D20a
-
~~
f, " C
6G HI
5 10 15
-- '1924
6SH'
6HH'
t, " C
6GH'
5
- 5095
10 15 20 25
-5316 -5525 5724 -591 1
Hzo
20 25
-1990 - 2053 -2112 - 2168
13 12 11 11 10
1900 1600 1300 1000
12 13 13 13 13
1700 1700 1900 2000 2000
1500
-
a
- 1872 - 1936
25
-2141
- 2002
- 2071
6H"'
44 42 39 37 34
7500 6700 6400 5600 4900
46 44 41 38 35
8700 7300 6900 6000 5200
D20
D20
5 10 15 20
6s"
Units as in Table I
coefficients oE methane, ethane, and n-butane are smaller in HzO than in Dzs3 (except for n-butane at 25"); the difference between the values in the two liquids tend to diminish as temperature increases. (3) The Ostwald absorption coefficient of benzene is larger in H20 than in DzO. In spite of the relativeffy low accuracy of these results, we believe that the sign of the isotope effect on ys is correct. The sign of the isotope effect on ys for biphenyl is estimated to be within the limits of our accuracy. (4) The standard entropy and enthalpy of solution are usually (but there are exceptions) more negative in DzO than in Hz0. This result is in accordance with similar ones obtained for argon,21 for which more accurate results could have been obtained. XV. Hydrophobic Interaction
The hydrophobic interaction (HI) in this section refers to the indirect part of the Gibbs free energy change for bringing two solute particles from fixed positions a t infinite separation to some close distance, the whole process being carried o~utwithin the liquid at constant pressure and temperature. For a particular close distance, say 1.53 A for two methane molecules, an approximate measure of the strength of the HI has been suggested4.5 The Journal of Physical Chemistry, Vol. 77, No. 1, 1973
5
-4941 -5170 - 5386 -5591 - 5784
10 15 20 25 a Units as
in Table I.
6GH' = A ~ E ~-O2A/.lMe0
(4.1)
The entropy and the enthalpy corresponding to this process are computed by 6SH' = -(SGGH'/dT)p
=ASE~O(II)- 2ASm0(n) (4.2)
6 l P = GGH1 -+ TGSnr
(4.3)
Note that for the purpose of computing the entropy and the enthalpy of the HI, only process I1 described in section I11 is of relevance. The HI and the corresponding entropies and enthalpies were computed for the following "reactions"4~~ B(methane) -ethane
(4.4)
2(ethane)
14.5)
-
4(methane)
n-butane n-butane
B(benzene) -biphenyl
(4.6) (4.9)
and are reported in Tables VII-X. The most conspicuous result is that the HI for the first three reactions, (4.4), (4.5), and (4.6), is stronger in € 3 2 0 compared to DzO, in agreement with a conclusion reported in ref 8. The isotope
Hydrophobic lnleractian in Light and Heavy Water
101
TABLE X: Hydrophobic Interaction and the Corresponding Entropy and Enthalpy Values for the Reaction Z(6enzene) Biphenyl (eq 4.7) in W 2 0 and D 2 0 a
-
H20
10
15 14 15
30 40 50
-983 -1136 -- 1284 -.1434 1579
10
- 1085
18
20
-.1253
30 40
-1410
16 15
50
-.1689
20
14
15
3400 3200 3300 3100 3200
DzO
-. 1556
13 14
3900 3400 3300 2900 2400
a Units as in Table I
effect for the last reaction seems to have reversed its sign. The entropy rind the enthalpy of the HI are all positive. This may well be attributed to the decrease of the "structure of the water" as the two solutes approach each other to a close distance.22 The strength of the HI for reactions 4.4, 4.5, and 4.7 are in decreasing order. This effect was found earlier,b and was attributed to the relative extent of penetration of' the two molecules compared to their molecular dimensions. We have estimated that the isotope effect on the entropy and the enthalpy of the HI are well within the lim>tsof experimental error.
V. Discussion mil Conclusion The HI, being viewed as a property of the solvent, may be best investigated by using a pair of inert solutes. In fact, we have already argued4 that the simplest solute for this purpose could he hard spheres.23 Solutions of saturated paraffins may serve as a reasonable model for inert molecules for estimating the strength of HI. The case of benzene solution seems not to fall into this category probably because of the stronger inkeractiori one expects between benzena>and water. The results obtained for the HI between a pair of benzene molecules, though of some value in the rontex t of the general solute-solute interaction problem, cannot convey faithful information on the properties of the solvent. The question of interpreting the isotopic effect on the HI is a difficult one, since we have as yet no general relation between HI anid molecular properties of the solvent. We should like to point out some possible sources for the isotopic effect. though it seems that no one points to the correct sign of his effect. a. l2ensit.y The density of DZQ, in grams/ml, is about 10% higher than that of W20. However, expressing the densities in number of particles per unit volume, we get almost exactly the same value for the two liquids. In ref 5 we have s t u d i d the effect of the density of the solvent on the solute-solute interaction at short distances. That study was confined t o a system of Lennard-Jones particles in two dimenbions. 1k was found that the strength of the solute-solute nnteraction increases with the density of the solvent. Althougb i t i s difficult to infer from this result any sound conclusion for aqueous solutions, we may expect that, other things being equal, the small difference in the number density between H2Q and D2O could not account for the observed isotopic effect on HI.
b. Cohesive Energy. In a simple solvent composed of, let us say, Lennard-Jones particles, the average cohesive energy may be represented by the depth of the pair potential function. Using again the simplified model in ref 5 , we found that the solute-solute interaction increases with the cohesive energy of the solvent. In water it is difficult to point out a particular parameter that conveys the cohesive energy. As a rough measure of this, one may take the hydrogen bond energy which is estimated6 to be larger in DzO than in HzO. Therefore, on the basis of hydrogen bond energy alone one would expect a stronger HI in DzO, in contrast to the experimental findings. c. Q u a n t u m Effects. Although it i s extremely difficult to estimate the isotopic effect on the HI for such a complex fluid like ~ a t e r , it ~ 5is believed that the difference in mass of about 10% between the two liquids may be importaht in establishing the isotopic effect on pendix we have shown that the first-order rection to the HI even for simple fluids may not be estimated without a detailed knowledge of the intermolecular potential. This difficulty becomes even more acute for complex fluids, such as water. All the above fractional information is strictly relevant to a simple solvent with pairwise additive potential. The fact that none of these points out the source of the isotopic effect on HI indicates that more complicated features of the solvent. are involved. Using classical arguments only, it is possible that the peculiar mode of packing of the water molecules in the liquid state, a feature that is currently referred to as the structure of water, plays an important role in determining the strength of the HI. Thus far no general relation is available between HI and the structure of water. Therefore, it is impossible at present to explain the isotope effect on H by classical arguments. Moreover, the fact that water as small moments of inertia, and having a strong orientational dependence interaction, may indicate that quantum correction to the rotational degrees of freedom may be very important in establishing the differences between the two liquids. Acknowledgment. We are grateful to Professor Avner Treinin for discussions and suggestions concerning the spectroscopic aspects of the experimental work. We are also indebted to Dr. F. H. Stillinger for valuable comments on the manuscript. This work was supported in part by the Central Research Fund of the Hebrew University of Jerusalem, for which the authors are very grateful. Appendix
First-Order Q u a n t u m Correction to t h e Free Energy of Cavity Formation and Hydrophobic Interaction at Zero Separation. Consider a system of N simple molecules contained in a volume V and at temperature T. Assuming (22) Here we us0 the undefined concept of the "structure of water." The statement made in the article is of course meaningless if one does not define this concept. For a reasonable definition of the structure of water, argument may be given to show that two methane molecules will enhance the structure more than a singie ethane molecules. This point has been shown for a specific model by Nemethy and Scheraga.2 A more general argument, based on an exact mixture model approach to liquid water, wiil be published elsewhere. (23) The HI between two hard-sphere solutes is the same as ?he cavitycavity i n t e r a ~ t i o nFor . ~ such a pair the Hi at zero separation may be estimated using the scaled particle theory. I t has been shown previously4 that the HI between two cavities at zero separation is indeed stronger in water than in some nonaqueous solvents. This point has been further confirmed recently by Wilhelm and 13attin0.'~ (24) E. Wilhelm and R. Battino, J. Chem. Phys., 58,563 (1972). (25) H.Friedmann, Physica, 30, 921 (1964). The Journal of Physical Chemisiry, Vol. 77, No. 7, 7973
A. Ben-Naim, J. Wilf, and M. Yaacobi
102
pairwise additivity of the total potential energy, the Helmholtz free energy may be expanded in power series of the Planck constant h. The first-order correction to the classical value of the free energy, A c l , is26 A(T,V,N)
=A,cl(T,V,N
+
free energy of solution of an HS in the two fluids must be due to quantum effect, the first order of which is27 ApesO(inOD) - ApesO(in(us)= [h2/24(2*kr)'][(ljmD)(Z1(V iu)')cav (l/mD)(
(h2/24(2xhT)2m)( Z , = I ~ ( V , V ) ~ ) ( A . l ) Here, m is the mass of a single particle, V ,U is the gradient of the total potential function with respect to R,, and the average isi taken over all the configurations of the system of the cla.~sic*alT, V,Nensemble, namely (Zt=1N(V,U)2) =z .f
1
0
JdRNP(RN) Z l , 1 N [ V 1 U ( R N ) ] 2 (A.2)
zi(via2) -
(l/mH)(Zi(ViCr)')mv
( l / m H ) ( Z i ( v i a 2 ) ] (-4.5)
where ( )cav is the same average as in (A.2), but excluding the region U~ over which the integrations carried out in (A.2) and in (A.3). Relation A.5 may be rearranged to ApHs0(in CYD)- ApHsO(inCXH) = [h2(mD - (mH)/24(2Tkr)2mDmH] x [ ( Z i ( V , i W 2) (Zi(ViU):')cavl
(-4.61
with
In ref 4 we have shown that the HI between two cavities at zero separation is given by
P(Rn) = exp[ - BU(RhJ1]/.f.-.J d R v exp[ - P U R N ) ] (A.3) and /? = ( k T ) -1. The free energy of cavity formation in the system, at constant TVlV is the same as the standard free energy of solution of a hard sphei~e(HS) having a suitable radiusls
Hence
Lp -[bo
A ( T,V , N ; F )- A( T, V,N)
(A.4)
where A ( T ,V,N u ) denotes the Helmholtz free energy of the same system but having a cavity of radius u a t some fixed position. This is the same as imposing a restriction on all the cerkew of the particles, R,, to be excluded from the spherical region u0 of radius u about a fixed position. Now compare two solvents, let us say CYH and CYD,which are identical except for having different molecular masses, m~ and mD, respectively. The classical free energy contains the maw o f the particles only in the momentum partition function, which is the same for the system with or without the cavity. Hence, the classical part of Apeso is independent of the mass. The difference in the standard
The Journal of Physical Chemistry, Vol. 77, No. 1, 7973
SAHS~'(O)= - A ~ H s O
(A.7)
GAHs~'(O,in CYD)- GAHs~I(O,in CYH) = -[Apeso(in CYD)- Apwso(in C Y H ) ] (AB) Thus we see that the first-order quantum correction to the free energy of solution and to the HI of hard sphere will depend on the difference in masses, mD - me, and on the type of the pair potential operating between the particles. The latter cannot be estimated without a detailed knowledge of the pair potential even for a simple fluid of spherical particles. (26) See, for example, L. D. Landau and E. M. Lifshitz, "Statistical Physics," Pergamon Press, London, 1959, p 100. (27) A strickt cavity produces a discontinous potential into the system, and therefore (A.2) may not be applicable. This difficulty may be overcome by considering a "soft cavity," in which the repulsive potential is built up continuously as the particles approach the region VU.
