5592
J. Phys. Chem. B 2001, 105, 5592-5594
Reevaluation in Interpretation of Hydrophobicity by Scaled Particle Theory Masato Kodaka* Biomolecules Department, National Institute of Bioscience and Human-Technology, 1-1 Higashi, Tsukuba, Ibaraki 305-8566, Japan ReceiVed: December 31, 2000; In Final Form: March 16, 2001
Hydrophobicity is considered to be a phenomenon that nonpolar solutes have a poor solubility in water, which plays fundamental roles in many chemical and biological systems. On the basis of the scaled particle theory, it has been shown in the present study that hydrophobicity is not caused by the small size of water molecules but by their high packing density generated from strong molecular interaction. Solvophobicity, observed in some organic solvents such as ethylene glycol, can be also explained consistently by the present method.
Introduction Though hydrophobicity plays fundamental roles in many chemical and biological systems, its physical basis has not been fully understood and is still a matter of debate.1,2 On the basis of the scaled particle theory (SPT),3 some researchers have concluded that the small size of the water molecule is one of the main causes of hydrophobicity.4-8 In the present study, the author has reevaluated the physical meaning of hydrophobicity on the basis of the SPT and has concluded that hydrophobicity does not arise from the small size of water but from its large cohesive energy density (ced). Theoretical Details
where R is the gas constant, T is the absolute temperature, F is the number density of solvent molecules, a1 and a2 are, respectively, the hard sphere (HS) diameters of solvent and solute molecules, and y is the packing density (y ) πa13F/6). Here the terms including pressure are neglected.10 Equation 3 indicates that ∆Gc is a function of a1 and y under constant T and a2. Therefore, to obtain the dependency of ∆Gc on a1, y should be represented as a function of a1. Though y is equal to πa13F/6, this does not give a direct form, since F itself is a function of a1. Direct relation between y and a1 can be acquired by using the equation of state for HS fluids (equation 4)3
πa13Ph/6kT ) y(1 + y + y2)/(1 - y)3
(4)
The process of a nonpolar molecule transferring from a gas into a liquid can be broken into two steps: cavity formation and the attractive interaction between solute and solvent molecules.9 The change in Gibbs energy (∆G) of this process can be thus written as
where Ph is the pressure to be expected for HS fluids occupying the same volume as real liquids under pressure P. In weakly polar liquids, the pressure Ph is related to P as
∆G ) ∆Gc + ∆Gi
Ph ) P + (δU/δV)T
(1)
where ∆Gc is the change in Gibbs energy of forming the cavity and ∆Gi is the change in Gibbs energy performed by the attractive potential between solute and solvent molecules. Equation 1 can be further modified to
∆G ) ∆Gc + Ei
(2)
where Ei signifies the interaction energy between solute and solvent molecules.5 As Ei is not practically dependent on solvent type, the difference in ∆G between water and organic solvents is dominated by the difference in ∆Gc.5 One of the statistical procedures to calculate ∆Gc is the SPT, from which eq 3 was derived,3
∆Gc ) RT[-ln(1 - y) + (9/2){y/(1 - y)}2 {(a1 + a2)/2a1}[6y/(1 - y) + 18{y/(1 - y)}2] + {(a1 + a2)/2a1}2[12y/(1 - y) + 18{y/(1 - y)}2]] (3) * Phone: +81-298-61-6124. Fax: +81-298-61-6123. E-mail: kodaka@ nibh.go.jp.
) P + Pint Z Pint
(5)
where U is the internal energy, V is the volume, and Pint (≡ (δU/δV)T) is the internal pressure of the corresponding real liquid.3 Since Pint is much larger than P under atmospheric pressure, Ph is approximately equal to Pint as shown in eq 5. Experimentally, Pint (termed Pint(exp)) of weakly polar liquids is approximately equal to cohesive energy density (ced) Lv/V as shown by eq 6,
Pint(exp) ≡ (δU/δV)T Z Lv/V
(6)
where Lv is heat of evaporation and V is molar volume of liquids.11 Considering eq 5, eq 4 can be transformed to eq 7,
y ) {1/3(1 + m)}[{(A + B)/2}1/3 {(-A + B)/2}1/3 - 1 + 3m] (7) where m, A, and B denote the following quantities:
10.1021/jp010124d CCC: $20.00 © 2001 American Chemical Society Published on Web 05/26/2001
Letters
J. Phys. Chem. B, Vol. 105, No. 24, 2001 5593
Figure 1. Relation between experimental internal pressure (Pint(exp)) and molecular weight of cycloalkanes; 1(cyclopentane), 2(methylcyclopentane), 3(cyclohexane), 4(methylcyclohexane), 5(ethylcyclohexane), 6(cis-decahydronaphthalene), 7(trans-decahydronaphthalene), 8(bicyclohexyl).
m ) πa13Pint/6kT A ) -81m2 + 54m + 7 B ) 9(81m4 + 180m3 + 118m2 + 20m + 1)1/2 Equation 7 is the final form giving the direct dependence of y on a1 under the assumption that Pint is independent of a1. Results and Discussion The validity of the assumption in eq 7 that Pint is independent of a1 is actually confirmed as shown in Figure 1, where Pint(exp) obtained by using eq 6 is essentially independent of the molecular weight of cycloalkanes. This suggests that Pint is independent of a1 in a homologous series of nonpolar liquids. The curves in Figure 2 show the theoretical relation between y and a1 at T ) 298 K obtained from eq 7, in which y increases with a1 under constant Pint. This means that larger size liquids are apt to be more closely packed (viz., y is larger) than smaller size liquids. Experimental data for various liquids are also plotted in Figure 2, where a1 values were cited from the comprehensive data of Reiss3 and y values were calculated from a1 and experimental F values using the relation y ) πa13F/6. The liquids are grouped into three major categories: (a) highly hydrogen bonding liquids with strong molecular interaction, (b) liquids with weak molecular interaction, and (c) ordinary liquids with medium-strength molecular interaction. In Figure 2, type a liquids (water, hydrazine, ethylene glycol) have large Pint values, type b liquids (diethyl ether, cyclohexane, n-hexane, n-heptane, n-octane) have small Pint values, and type c liquids have intermediate Pint values. Though the Pint values calculated by eq 7 for weakly polar liquids (type b and type c) are close to the actual internal pressures, the calculated Pint value for water (7.4 × 108 Pa) is exceptionally larger than the actual internal pressure (1.7 × 108 Pa).11 In water, therefore, the Pint value obtained from eq 7 should be regarded as an apparent internal pressure, which contains contribution from the large cohesive energy density (ced) of water that can give the high packing density (y ) 0.35). Dack reported that the difference between the large ced of water (2.3 × 109 Pa) and the actual internal pressure (1.7 × 108 Pa) measures the intermolecular bonding energy owing to hydrogen bonding.11
Figure 2. Relation between packing density (y) and diameter of solvent molecule (a1) at T ) 298 K under constant internal pressure (Pint); type (a) solvents with strong molecular attractive interaction (2), type (b) solvents with weak molecular attractive interaction (9), and type (c) solvents with medium molecular attractive interaction (O); 1 (water), 2 (hydrazine), 3 (ethylene glycol), 4 (methanol), 5 (ethanol), 6 (acetic acid), 7 (carbon disulfide), 8 (acetone), 9 (ethyl bromide), 10 (ipropanol), 11 (n-propanol), 12 (chloroform), 13 (ethyl iodide), 14 (ethylene dichloride), 15 (benzene), 16 (ethyl acetate), 17 (carbon tetrachloride), 18 (n-butanol), 19 (acetic anhydride), 20 (toluene), 21 (chlorobenzene), 22 (aniline), 23 (bromobenzene), 24 (1,1,2,2-tetrachloroethane), 25 (cyclohexanol), 26 (nitrobenzene), 27 (m-xylene), 28 (acetophenone), 29 (diethyl ether), 30 (cyclohexane), 31 (n-hexane), 32 (n-heptane), 33 (n-octane). The curves are calculated with eq 7 at T ) 298 K for various Pint values.
Figure 3. Relation between change in Gibbs energy of producing a cavity (∆Gc) and diameter of solvent molecule (a1) at T ) 298 K and a2 ) 4.0 Å under constant internal pressure (Pint). The symbols and the solvent numbers are the same as those in Figure 2. The solid curves are calculated with eqs 3 and 7 and the broken curves are calculated with eq 3 at T ) 298 K and a2 ) 4.0 Å under constant y.
As shown in Figure 3, theoretical relation curves (solid curves) between ∆Gc and a1 were obtained from eqs 3 and 7 at T ) 298 K and a2 ) 4.0 Å, where experimental data are also plotted for comparison. The theoretical tendency that ∆Gc increases with a1 under constant Pint does not agree with the conventional idea that the increase in ∆Gc, viz., hydrophobicity, originates from the small size of water molecules.4-8 The present
5594 J. Phys. Chem. B, Vol. 105, No. 24, 2001 study has revealed that the large ced induced by the strong attractive interaction between water molecules dominates hydrophobicity rather than its small molecular size. In other words, the packing density of water (y ) 0.35) as shown in Figure 2, which is caused by the large ced due to many hydrogen bonds and is much higher than those of the ordinary liquids (y < 0.25) having the similar size, induces the large positive Gibbs energy of cavity formation. It is apparent from Figure 3 that water, hydrazine, and ethylene glycol have larger ∆Gc than other liquids because of their large apparent internal pressure (Pint) reflecting large ced. All of these solvents actually give poor solubility toward nonpolar solutes. The conclusion drawn by Lee et al. that hydrophobicity arises from the small size of water molecules was based on the assumption that y is constant, as illustrated by the broken curves in Figure 3.3-8 Obviously this assumption is not adequate, because y itself is dependent on a1. It is only under constant Pint that we can obtain the direct dependency of ∆Gc on a1. Water would never give large positive
Letters ∆Gc if it had smaller ced (viz., lower packing density). It should be noted furthermore that so-called “solvophobicity” can be also elucidated consistently by the high packing density of organic solvents. References and Notes (1) Graziano, G. Biophys. Chem. 1999, 82, 69. (2) Graziano, G. J. Phys. Chem. B 2000, 104, 9249. (3) Reiss, H. AdV. Chem. Phys. 1965, 9, 1. (4) Lee, B. Biopolymers 1985, 24, 813. (5) Lee, B. Biopolymers 1991, 31, 993. (6) Soda, K. J. Phys. Soc. Jpn. 1989, 58, 4643. (7) Soda, K. J. Phys. Soc. Jpn. 1993, 62, 1782. (8) Blokzijl, W.; Engberts, J. B. F. N. Angew. Chem., Int. Ed. Engl. 1993, 32, 1545. (9) Pierotti, R. A. J. Phys. Chem. 1963, 67, 1840. (10) Postma, J. P. M.; Berendsen, H. J. C.; Haak, J. R. Faraday Symp. Chem. Soc. 1982, 17, 55. (11) Dack, M. R. J. Chem. Soc. ReV. 1975, 4, 211.
