Hydration of Aromatic Hydrocarbons - CiteSeerX

in magnitude to the change in the total binding energy of the solvent molecules and not to ..... ΔGc is larger, in absolute value, than Ea, and thus ...
0 downloads 0 Views 57KB Size
Download PDF
J. Phys. Chem. B 2001, 105, 10367-10372

10367

Hydration of Aromatic Hydrocarbons Giuseppe Graziano†,§ and Byungkook Lee*,‡ Faculty of Science, UniVersity of Sannio, Via Port'Arsa, 11-82100 BeneVento, Italy, and Laboratory of Molecular Biology, Center for Cancer Research, National Cancer Institute, National Institutes of Health, Bldg. 37, Room 4B15, 37 ConVent DriVe, MSC 4255, Bethesda, Maryland 20892-4255 ReceiVed: April 12, 2001; In Final Form: August 13, 2001

The transfer of benzene and toluene from gas phase into water is a spontaneous process at room temperature under Ben-Naim’s standard conditions. This strongly contrasts with the behavior of aliphatic hydrocarbons, for which the Gibbs energy of transfer is large and positive. To understand this difference in behavior between the aromatics and aliphatics, we analyze the hydration of benzene and toluene in a large temperature range using available experimental and computer simulation data. We introduce a small but important modification in the definition of the solvent reorganization enthalpy in order to handle slightly polar solutes such as the aromatic hydrocarbons, which form weak hydrogen-bonds with water. The analysis shows that, for aromatic hydrocarbons, the van der Waals interaction energy overwhelms the Gibbs energy cost for cavity creation at room temperature, thus rendering the hydration process spontaneous. The formation of the weak hydrogenbonds between the aromatic ring and water appears to be largely compensating and not to contribute significantly to the hydration Gibbs energy.

Introduction The phenomenon of hydrophobicity refers to the fact that nonpolar solutes do not dissolve in water as well as in other common organic solvents. Although this is a common everyday phenomenon, and the molecular mechanism that produces it is probably one of the most important features that determine the thermodynamic stability of globular proteins, micelles and double layer membranes,1-3 there is still much debate in the literature on its exact physical origin.4-8 Hydrophobicity is usually measured by means of the Gibbs energy change upon transferring a nonpolar solute molecule from a nonaqueous solvent to water. Hydration refers to the process of transferring a nonpolar molecule from vacuum into water. It forms one leg of the hydrophobic transfer process, the other leg being the transfer of the same solute molecule from vacuum to the nonaqueous solvent. There has been confusion in the literature on the precise definition of the term “hydration”. In this paper, we use the modified Ben-Naim standard9,10 and define hydration as the process of transferring a solute molecule from a fixed position in an ideal gas phase into a fixed position in liquid water at a constant temperature and pressure. The thermodynamic quantities for this process do not contain the effects associated with the difference in the molar volume of the solute in the two phases11 and, more importantly, do not use the ideal state standard, i.e., they do not refer to the values that they would have if the liquid phase behaved as an ideal gas or an ideal solution.10,12 In the following, only the modified Ben-Naim standard quantities, denoted by a superscript filled circle, are used. A remarkable difference exists between the hydration thermodynamics of aliphatic and aromatic hydrocarbons; at room * Corresponding author. Tel: 301-496-6580. Fax: 301-402-1344. E-mail: [email protected]. † University of Sannio. ‡ National Cancer Institute, National Institutes of Health. § E-mail: [email protected].

temperature, the hydration Gibbs energy change is large and positive for the aliphatic hydrocarbons, but negative for aromatic ones.13-15 We explore the reasons for this difference in this paper. The method we use involves dissecting the experimental thermodynamic quantities into the cavity formation, solutesolvent interaction, and solvent reorganization terms and then using available best data, experimental or otherwise, to estimate the sign and magnitude of each of these contributing terms. Since first introduced,16,17 this basic procedure has been used to gain considerable insight into the hydration and hydrophobicity of a number of nonpolar solutes.18-23 In the case of the hydration of aliphatic hydrocarbons, we had found17 that the enthalpy and entropy of solvent reorganization largely compensate so that the free energy change due to solvent reorganization is small. One of the main findings of this study is that they compensate also in the case of the aromatic solutes, which form weak hydrogen-bonds with water.24-27 However, this compensation is obtained only when the definition of the enthalpy of solvent reorganization is slightly modified. In the past, we defined the enthalpy of solvent reorganization as the change in the interaction energy between solvent molecules only.17 However, when the solute molecule is polar and can form hydrogen-bonds with water, the water molecules in the hydration shell will form hydrogen-bonds with the solute molecule and correspondingly less hydrogen-bonds with other water molecules. Therefore the solvent-solvent interaction energy increases substantially even when the binding energy of each solvent molecule, which includes interaction with the solute molecules as well as with other solvent molecules, remains largely unchanged. Predictably, and as we will show in this paper, the entropy of solvent reorganization is comparable in magnitude to the change in the total binding energy of the solvent molecules and not to the change in the solvent-solvent interaction energy alone. Therefore, consideration of the hydration thermodynamics is simpler when the solvent reorganization enthalpy change is defined as the change of the binding energy of the solvent molecules rather than that of the solvent-solvent

10.1021/jp011382d CCC: $20.00 © 2001 American Chemical Society Published on Web 09/27/2001

10368 J. Phys. Chem. B, Vol. 105, No. 42, 2001

Graziano and Lee

interaction energy alone. In the theory section we redefine the solvent reorganization enthalpy change this way in precise statistical mechanical terms. As in many other theoretical studies of hydration,5,8,11,28,29 we break a solvation process into two steps: first a cavity is formed (introduction of the “hard” part of the potential), which is then turned into the real solute molecule by turning on the solute-solvent interaction (“soft” potential). Solvent reorganizes in both of these steps. The compensation phenomenon mentioned above applies to the solvent reorganization that happens in the second step. In the interest of completeness and in order to provide a sense of context, we begin our theory section with a short description of the compensation observed in the cavity forming step. We then present the new definition of solvent reorganization terms for the second step and compare it with the old definition. We finish the theory section by deriving a couple of relations between the noncompensating residual free energy of solvent reorganization and the average fluctuation of the solute-solvent interaction energy. These relations provide additional support for the validity of the new definition and are useful in analyzing the origin of the compensating behavior. To estimate the sign and magnitude of the various terms, we use the experimental data of Makhatadze and Privalov15 and the simulation data of Linse.25 The analysis leads to the conclusion that the hydration of aromatics is a spontaneous process at room temperature because the direct van der Waals interaction energy between benzene or toluene and water molecules overwhelms the Gibbs energy cost to create the cavity, in marked contrast to what happens for aliphatics. The weak hydrogen-bond formation between the aromatic ring and water molecules is a part of the solvent reorganization process, which produces only a compensating effect and does not contribute significantly to the Gibbs energy change. Theory and Methods (a) Solvent Reorganization in the Cavity Formation Step. The enthalpy change upon cavity formation, ∆Hc, is all due to the solvent reorganization. The entropy change upon cavity formation, ∆Sc, can be dissected into two terms,31

∆Sc ) ∆Sx + ∆Scr

(1)

where the first term on the right-hand side represents the entropy change due to simple exclusion of the solvent molecules from the cavity area and the second term that is due to additional rearrangement of the solvent molecules that generally happens because of the presence of the cavity. Although ∆Sx is formally a solvent reorganization term,17 it represents a direct effect of, rather than a response to, creating a cavity. Many will recognize only ∆Scr as the solvent reorganization term. It was shown earlier31 that ∆Scr is exactly compensating for all liquids,

∆Scr ) ∆Hc/T

(2)

where T is the absolute temperature, and that, therefore, the Gibbs energy of cavity formation, ∆Gc, is given by

∆Gc ) -T∆Sx

(3)

(b) Enthalpy of Solvent Reorganization. We recognize two contributions to the enthalpy change, ∆Ha, upon turning on the

solute-solvent attractive interaction:

∆Ha ) c + ∆Har

(4)

The first term on the right-hand side is the solute-solvent interaction energy averaged over the ensemble of the initial system. This is the system that has a cavity but in which the solvent molecules have not yet reorganized in response to the perturbation ψa. The c term, therefore, represents the direct interaction energy between solute and solvent molecules, excluding the effect from any solvent reorganization. The effect from the solvent reorganization is given by the second term, ∆Har. It can be easily shown that the ∆Har term is given by

∆Har )

∫[H(X) + ψa(X)][Fs(X) - Fc(X)] dX )

s - c (5)

where X stands for a particular configuration of solvent molecules in the system, integration is over the ensemble of the system, H(X) ) E(X) + PV(X), E(X) is the interaction energy among the solvent molecules only, P and V(X) are the pressure and volume of the system, respectively, Fs(X) and Fc(X) are the probabilities of the system being in the configuration X in the final solution and in the solution with only the cavity, respectively, and the angled brackets with the subscripts s and c indicate ensemble averages over the solvent configurations in the final solution and in the solution with only the cavity, respectively. The ψa term is written as ψa(X) here in order to emphasize the fact that it depends on the solvent configuration. Note that c is the enthalpy of the system after the perturbing potential ψa(X) has been turned on, but before the solvent reorganizes in response. The ensemble of the solvent in this state is identical to that of the initial system, which is the equilibrium system with the cavity but without ψa. Thus the perturbing potential ψa(X) acts as a test30 or ghost interaction that does not alter the distribution of solvent configurations. Equation 4 effectively defines the solvent reorganization term ∆Har. This definition is different from that we used in the past,17 namely, Here, the first term on the right-hand side is given as

∆Ha ) s + ∆Har(old)

(6)

the solute-solvent interaction energy after the solvent reorganizes upon the introduction of ψa. The second term is then given by

∆Har(old) )

∫ H(X) [Fs(X) - Fc(X)] dX,

(7)

which includes the change in the interaction energy between solvent molecules only. This ∆Har term is large and positive because it excludes the hydrogen-bonding interaction of water molecules with the aromatic ring. In the new definition, ∆Har includes all interactions that a solvent molecule makes, regardless of whether the interaction partner is a solute or other solvent molecules (see eq 5). In the case of aliphatic hydrocarbons and noble gases, ψa represents the solute-solvent van der Waals interaction, which is weak compared to the hydrogen-bonds. Since the perturbation is small, there is little solvent reorganization upon turning on this potential8,17 and c can be expected to be nearly equal to s. Therefore, the latter can be used for c and the new and the old definitions of the enthalpy of solvent reorganization give the same results. However, when the solute molecule can form hydrogen-bonds with water molecules,

Hydration of Aromatic Hydrocarbons

J. Phys. Chem. B, Vol. 105, No. 42, 2001 10369

c will be different from s; s includes the solute-solvent hydrogen-bond energy whereas c will contain little of it, since water hydrogens in this ensemble will rarely be directed toward the solute cavity. Thus, the enthalpy due to the hydrogen-bonds between solute and water molecules is included in ∆Har in the new definition whereas it is included in the s term in the old definition. The total enthalpy of hydration is given by

∆H• ) ∆Hc + ∆Ha ) c + ∆Hr

(8)

where ∆Hr ) ∆Hc + ∆Har is the enthalpy change associated with the solvent reorganization that happens in response to both cavity creation and turning on the solute-solvent interaction. (c) Gibbs Energy of Solvent Reorganization. Using eq 4, the Gibbs energy change upon turning on the solute-solvent interaction, ∆Ga, can be written as

∆Ga ) ∆Ha - T∆Sa ) c + ∆Gar

(9)

where ∆Sa is the entropy change upon turning on the solutesolvent interaction and

∆Gar ) ∆Har - T∆Sa

(10)

is the Gibbs energy of solvent reorganization upon introduction of the perturbation c. The total Gibbs energy of hydration is given by

∆G• ) ∆Gc + ∆Ga ) ∆Gc + c + ∆Gar

(11)

The first two terms in the last expression, ∆Gc and c, can be considered to represent the entropic (see eq 3) and enthalpic direct perturbations, respectively. The ∆Gar term represents the total Gibbs energy change due to the solvent reorganization in response to the direct perturbations. ∆Gar can be expressed in terms of the fluctuation of the perturbation, when the latter is small. Thus, the Gibbs energy change upon turning on the solute-solvent interaction is given by17,32

∆Ga ) -RT lnc ) RT lns (12) where R is the gas constant. Defining x ) ψa(X) - c, expanding the middle expression of the above equation in power series of x and ignoring terms beyond the second order, and using eq 9, one obtains

∆Gar ≈ -c /2RT 2

(13)

for small x values. This expression shows that ∆Gar is always negative, i.e., that the solvent reorganization process is a spontaneous process. This is different from using the old definition of the enthalpy of solvent reorganization, where ∆Gar was always positive.17 Defining y ) ψa(X) - s and applying a similar procedure, but using the third expression of eq 12, one obtains

∆Gar ≈ [s /2RT] + [s - c]

(14)

for small y values. It can be seen from eqs 13 and 14 that s

TABLE 1: Thermodynamic Data for the Hydration Process of Benzene (a) and Toluene (b) from Ref 15 T °C

∆Cp• J K-1 mol-1

∆H• kJ mol-1

∆S• J K-1 mol-1

∆G• kJ mol-1

a

5 25 50 75 100 125

319 292 268 248 231 201

-35.7 -29.6 -22.6 -16.2 -10.2 -4.8

-108.4 -87.2 -64.6 -45.3 -28.7 -14.7

-5.6 -3.6 -1.7 -0.4 0.5 1.1

b

5 25 50 75 100 125

386 361 333 307 283 260

-41.4 -33.9 -25.2 -17.2 -9.8 -3.1

-127.4 -101.4 -73.4 -49.5 -29.0 -11.4

-5.9 -3.7 -1.5 0.0 1.0 1.5

- c cannot be positive and that it becomes zero if and only if c ) s ) 0. Results (a) Estimation. The procedure is to obtain estimates of the direct perturbation terms (c, s , and ∆Gc) by basically theoretical means and then obtain the solvent reorganization terms (∆Hr and ∆Gar) by the difference from the experimental values. Makhatadze and Privalov33 used a precise scanning microcalorimetric technique to determine the temperature dependence of the hydration heat capacity change for benzene and toluene in a large temperature range. Subsequently, the same authors15 combined their ∆Cp• values with the ∆H•, ∆S•, and ∆G• data at 25 °C34,35 to evaluate these latter functions in the temperature range of 5-125 °C. These data are given in Table 1. Linse25 performed molecular dynamics simulations on two related systems. One was a benzene molecule in 256 TIP4P water molecules36 at constant temperature and pressure, using a benzene-water intermolecular potential energy function derived from ab initio quantum chemical calculations. The other was a similar one except that the partial charges of the benzene molecule were set to zero. This modified benzene molecule cannot form hydrogen-bonds with water molecules. He reported the benzene-water interaction energies, Ea(F) and Ea(M), for the systems with benzene and modified benzene molecule, respectively. We assume that these energies are fair approximations to s and c, respectively, since these latter two quantities represent solute-solvent interaction energies, with and without the benzene-water hydrogen-bonds, respectively. The Ea(F) and Ea(M) values reported are -61.6 ( 0.6 and -42.6 ( 0.3 kJ mol-1, respectively, at 25 °C. The value of Ea(M) for toluene was obtained by the following relation:

Ea(M, toluene) ) Ea(M, benzene) + [Ea(ethane)/2] Ea(hydrogen atom) ) - 42.6 - 10.0 + 2.0 ) -50.6 kJ mol-1

(15)

where Ea(hydrogen atom) ) Ea(methane) - [Ea(ethane)/2], and Ea(methane) and Ea(ethane) are the solute-solvent interaction energies computed by Jorgensen and co-workers37 for the aqueous solutions of methane and ethane, respectively. The Ea(M) values, and the ∆Hr values obtained by the difference using eq 8 and c ≈ Ea(M), are summarized in Table 2. As in the case of the hydration of aliphatic hydrocarbons, ∆Hr values are positive at 25 °C. Had we used Ea(F) and the old definition (eq 6), the ∆Hr values would be even more positive.

10370 J. Phys. Chem. B, Vol. 105, No. 42, 2001

Graziano and Lee

TABLE 2: Components of the Enthalpy Change at 25 °C (All values are in kJ mol-1 units) benzene toluene

c

∆Hr

∆H•

-42.6 -50.6

13.0 16.7

-29.6 -33.9

TABLE 3: Gibbs Energy of Cavity Creation for the Hydration of Benzene (a) and Toluene (b) at Different Temperatures (All Gibbs energy values in kJ mol-1 units) T (°C)

∆Gc(1)a

∆Gc(2)b

∆Gc(3)c

a

5 25 50 75 100 125

37.0 39.0 40.9 42.2 43.1 43.7

37.4 39.7 41.9 43.6 44.6 45.4

35.6 38.0 40.5 42.3 43.8 44.9

b

5 25 50 75 100 125

44.7 46.9 49.1 50.6 51.6 52.1

44.7 47.2 49.7 51.6 52.6 53.6

39.5 42.9 45.7 47.8 49.4 50.6

a ∆G (1) is calculated as ∆G (1) ) ∆G• - . b ∆G (2) is c c a c c calculated by means of eq 17 at 25 °C and similar expressions valid at the other temperatures. c ∆Gc(3) is calculated by means of the scaled particle theory.

The Ea(M) values can also be estimated using Pierotti’s expression for the dispersive interaction energy.38 Using the σ and  values reported by Wilhelm and Battino,39 the values obtained using this formula are -42.3 and -50.6 kJ mol-1 for benzene and toluene, respectively, which are nearly identical to those obtained using the simulation data. The ∆Gc values were calculated using two different methods. One method was based on the linear correlation observed between ∆Gc and the hydration number, NH2O, for aliphatic hydrocarbons. Using the values of ∆Gc for five aliphatic hydrocarbons already reported,17 and the values of NH2O estimated by Dec and Gill for these compounds,40 one obtains at 25 °C

∆Gc ) -(14.80 ( 1.86) + (2.04 ( 0.08) ‚ NH2O [r ) 0.9980] (16) The NH2O values used for benzene and toluene were 26.7 and 30.4,41 respectively. The ∆Gc values at other temperatures were obtained using expressions analogous to eq 16 but valid at other temperatures (not shown). The calculated values are given in Table 3. Another method is the scaled particle theory (SPT).28,42 We used the following hard-sphere diameters for these calculations: σ ) 5.26 Å for benzene; σ ) 5.64 Å for toluene; σ ) 2.75 Å for water.28,39 These values were considered temperature independent. The ∆Gc still depends weakly on temperature because it contains the RT term and also because it depends on the density of water. We used the experimentally determined density of water compiled by Kell.43 It is well-known that the estimates of ∆Gc from SPT strongly depend on the values of solute and solvent hard-sphere diameters.44-46 The values listed above are the usual customary ones. In particular, the value used for benzene, σ ) 5.26 Å, is close to the position of the first peak of the ring center-ring center radial distribution function of liquid benzene at room temperature, as determined from X-ray scattering,47 neutron scattering,48 and computer simulations.49 The value selected for water, σ ) 2.75 Å, is also close to the location of the first peak of the oxygen-oxygen radial

distribution function of liquid water,50 and has been used by many authors.28,44 The ∆Gc values calculated by this method are also given in Table 3. (b) Experimental Hydration Gibbs Energy Is Fully Accounted for by the Direct Perturbation Terms Alone. The direct perturbation is given by the sum of ∆Gc and c. If c is approximated by Ea(M), then the magnitude of direct perturbation is nearly the same as the experimentally observed total Gibbs energy of hydration. The remainder, which approximates ∆Gar according to eq 11, is -0.7 or +1.0 kJ mol-1 for benzene at 25 °C, depending, respectively, on whether eq 16 or SPT is used to calculate ∆Gc. The estimated ∆Gar value for toluene is similarly small, -0.3 or +4.0 kJ mol-1 at 25 °C using eq 16 or SPT, respectively. Thus, both methods of calculating ∆Gc yield small values for ∆Gar. This means that the solvent reorganization upon turning on the attractive potential is a compensating process. Of particular interest is the fact that the compensation is obtained using Ea(M), not Ea(F), for c. This implies that the weak hydrogen-bond formation between the aromatic ring and water is among the compensating processes. (c) The van der Waals Term Dominates over the Cavity Term in the Hydration of Aromatics. Assuming perfect compensation (∆Gar ) 0), ∆Gc was calculated at all temperatures using eq 11 with the experimental values of ∆G• and the temperatureindependent Ea(M) value for c. The results are shown in Table 3. It is clear that the three independent methods of estimating ∆Gc give essentially the same result at all temperatures both for benzene and for toluene. Importantly, these estimates of ∆Gc values show that ∆G• is negative around room temperature for aromatic hydrocarbons because the Ea(M) values are larger in absolute value than ∆Gc. The excluded volume effect due to cavity creation is more than counterbalanced by the van der Waals interactions between the aromatic ring and water. This is opposite to what happens for aliphatics, where ∆Gc is larger, in absolute value, than Ea, and thus ∆G• is always positive.17 ∆G• for aromatics does become positive at around 100 °C for both benzene and toluene because ∆Gc slowly increases with temperature while Ea(M) is assumed to be independent of temperature. Discussion (a) Role of the Hydrogen-Bonds between the Aromatic Ring and Water. One simple but important result of this study is the fact that the hydration Gibbs energy is nearly exactly given by the sum of c and ∆Gc, when the c term is approximated by the Ea(M) reported by Linse from simulation calculation. The ∆Gc values used were calculated in two different ways and without regard to the compensation consideration at all. It is true that one of the two methods, the SPT, depends on the size of the molecules used. However, we used sizes that are typically used by other authors, without any attempt to fit any thermodynamic data of the aromatic hydration. The Ea(M) value is also probably reliable, since Pierotti’s analytical formula gives nearly identical results. Ea(M) is the solute-solvent interaction energy when the partial charges on the aromatic ring are turned off. It, therefore, represents only the van der Waals interaction. The full interaction, Ea(F), between an aromatic ring and water involves weak hydrogen-bonds in addition to the van der Waals interaction. Both experimental and theoretical studies have led to the conclusion that benzene acts as a hydrogen-bond acceptor for two water molecules located above and below the aromatic ring: both hydrogens of each water molecule interact with the

Hydration of Aromatic Hydrocarbons high electron density of the aromatic π electron cloud.24-27 From a geometric point of view, the two water molecules located over the two faces of the aromatic ring, with their hydrogen atoms pointing in, will promote the formation of a good network of hydrogen-bonds on a flat surface. These observations suggested that the weak hydrogen-bond formation might be the cause for the improved solubility of the aromatic hydrocarbons compared to the aliphatics.15 However, the fact that all of experimental ∆G• is given by the sum of ∆Gc and Ea(M), not Ea(F), means that the weak hydrogen-bonds between an aromatic ring and water do not contribute to the hydration Gibbs energy change; the hydrogen-bond formation must be a fully compensated process. One expects that the compensation will break down when the solute-solvent interaction is stronger.32 For example, in the case of the hydration of alcohols and carbonyls, there is a large Gibbs energy associated with the hydrogen-bond formation between the OH or the CO group of the solute molecule and water.22,51 But the hydrogen-bonds between an aromatic ring and water appear to be weak enough for the compensation to be maintained. In the case of aliphatic hydrocarbons and noble gases, we have already noted in the Theory section that s is expected to be nearly equal to c, in which case c ) s ) 0 according to eqs 13 and 14. The simulation calculations from Jorgensen and co-workers37 indeed show that s/2RT is negligibly small for aliphatic hydrocarbons. These observations justify the assumption of ∆Gar ) 0 for the aliphatics and noble gases.16-22 For the aromatics, on the other hand, c and s are given approximately by Ea(M) and Ea(F), respectively, and [s - c] is approximately equal to Ea(F) - Ea(M) ) -19 kJ mol-1. This must be essentially the enthalpy of hydrogen-bonds between the aromatic ring and water and is no longer negligible. The fact that ∆Gar is again small even in this case means that the formation of the aromatic-water hydrogenbonds is accompanied by other processes that produce unfavorable Gibbs energy changes. At least two such processes are immediately apparent. First, the formation of solute-solvent hydrogen-bonds will reduce the number of solvent-solvent hydrogen-bonds. In fact, the old definition of the solvent reorganization enthalpy change gives the change in the solventsolvent interaction energy, which will essentially be given by the change in the effective number of solvent-solvent hydrogenbonds. This is given by ∆H• - Ea(F), which equals +32 kJ mol-1 for benzene at 25 °C. Thus, a large amount of solventsolvent hydrogen-bonds are broken at room temperature. Since ∆Hr ) ∆H• - Ea(M) is 13 kJ mol-1 (Table 2), only about (32 - 13)/32 ) 59% of the hydrogen-bonds are recovered by the formation of the solute-solvent hydrogen-bonds. Essentially, two water molecules, which previously hydrogen-bonded other water molecules, reorient themselves on benzene insertion to form (weaker) hydrogen-bonds with the aromatic ring. Second, hydrogen-bonds between the aromatic ring and water molecules will restrict rotational degree of freedom of the solute molecule with respect to the solvent. In fact, Linse25 found that the tumbling motion of benzene (reorientation of the C6 axis by means of rotations around the C2 axes) is slower when the full benzene-water potential is used than when the modified potential is used or in pure benzene. These counterbalancing effects must all be included in the [s/2RT] term in eq 14, although eq 14 itself may not be valid because [s/2RT] term is no longer small. Assuming that eq 14 is valid, perfect compensation requires s/2RT to be 19 kJ mol-1 and the root-mean-square value of y to be

J. Phys. Chem. B, Vol. 105, No. 42, 2001 10371 9.7 kJ mol-1 at 25 °C. This compares with 7.5 kJ mol-1 value for the root-mean-square fluctuation of the enthalpy of pure water at room temperature, estimated from its heat capacity.52 The fact that the fluctuation in y is larger may indicate that the aromatic-water hydrogen-bonds are weaker than those between water molecules. Linse’s simulation data do not suggest such a large fluctuation for ψa(X), but it is difficult to obtain the breath of fluctuation of a quantity from a simulation calculation. The closest value suggested from the simulation is the standard deviation of the mean value of Ea(F) from 30 intervals of 3 ps simulations each from a run of 90 ps total length. The value obtained is only 3.3 kJ mol-1, but 90 ps is probably too short a time for measuring the full extent of fluctuations in a hydrogen-bonded system. Recently, Gallicchio et al.8 introduced a computational technique in which only the solvent molecules in the hydration shell are included in the calculation. This technique should allow a more accurate calculation of these quantities. (b) Large van der Waals Term for the Aromatics. Another simple observation is that the van der Waals term, Ea(M), is larger in absolute value than the cavity term, ∆Gc, at room temperature. This makes ∆G• to be negative for aromatics. This is in contrast to the case of aliphatics and noble gases, wherein ∆G• is always positive because Ea is always smaller than ∆Gc. This difference in behavior arises not because ∆Gc is smaller, but because Ea(M) is larger, for the aromatics than that for the aliphatics. The fact that the magnitude of the van der Waals interaction with water is so large for aromatics is striking. To gain perspective, it is useful to compare the Ea values of benzene and toluene with those for the aliphatics. A generalized function for Ea can be calculated on the basis of a linear correlation with the hydration number NH2O of aliphatic hydrocarbons. Using the values of Ea calculated by Jorgensen and co-workers37 for seven aliphatics in TIP4P water, and the corresponding values of NH2O,40 one obtains

Ea ) (20.83 ( 1.56) - (1.94 ( 0.06)‚NH2O [r ) -0.9977] (17) Application of this equation to benzene (NH2O ) 26.7) and toluene (NH2O ) 30.4) results in Ea ) -31.0 and -38.2 kJ mol-1, respectively. The estimate for benzene is about 12 kJ mol-1 smaller in absolute value than the Ea(M) value Linse obtained from his molecular dynamics simulations. It should be noted that, Jorgensen and Linse both used the same TIP4P water model. The large difference is surely due to the very large polarizability of the aromatic ring, arising from the delocalized π-electrons that strengthen the dispersive interaction with water molecules. The large value of Ea for the aromatics is striking, but it is hardly a new observation. For example, using an expression similar to our eq 11 with ∆Gar set to zero, and using the SPT to calculate ∆Gc, Wilhelm and Battino39 obtained values for the  parameter in the Lennard-Jones potential for both aromatics and aliphatics. The /k value for benzene (531 K) is significantly larger than that for the corresponding aliphatic hydrocarbon, n-hexane (517 K), even though benzene is considerably smaller than n-hexane (σ ) 5.26 and 5.92 Å, respectively, for benzene and n-hexane). The /k value for toluene (σ ) 5.64 Å; /k ) 575 K) is only slightly larger than that for the corresponding aliphatic, n-heptane (σ ) 6.25 Å; /k ) 573 K), but toluene is much smaller than n-heptane. The fact that aromatics have a large  value, and the fact that the hydrogen-bonds do not contribute to the Gibbs energy, means that they interact favorably in both aqueous and nonpolar

10372 J. Phys. Chem. B, Vol. 105, No. 42, 2001 media. This is in contrast to aliphatics, which have a strong preference for the nonpolar environment only, and to the charged and strongly hydrogen-bonding groups, which strongly favor the aqueous phase only. This property of the aromatics may explain why aromatic residues are often found abundantly in the protein-protein interface, such as the antigen-binding site of antibodies.53,54 Residues in such interfaces are exposed to water when the protein is monomeric, but buried in the hydrophobic environment when the protein forms a protein complex. Many aromatic residues are also found to line the position of the integral membrane proteins where the lipid bilayer of the membrane meets the aqueous phase.55 Because of the fluctuations in the structure of the lipid bilayer,56 residues at this position will frequently go in and out of the lipid phase. In both of these types of sites, the aromatic residues will be preferred because they interact favorably in either the polar or the nonpolar environment. Acknowledgment. G.G. is supported by a P.R.I.N. grant from the Italian Ministry for the University and the Scientific and Technological Research (M.U.R.S.T., Rome). References and Notes (1) Kauzmann, W. AdV. Protein Chem. 1959, 14, 1-63. (2) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980. (3) Dill, K. A. Biochemistry 1990, 29, 7133-7145. (4) Blokzijl, W.; Engberts, J. B. F. N. Angew. Chem., Int. Ed. Engl. 1993, 32, 1545-1579. (5) Wallqvist, A.; Covell, D. G. Biophys. J. 1996, 71, 600-608. (6) Silverstein, K. A.; Haymet, A. D. J.; Dill, K. A. J. Am. Chem. Soc. 1998, 120, 3166-3175. (7) Mountain, R. D.; Thirumalai, D. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 8436-8440. (8) Gallicchio, E.; Kubo, M. M.; Levy, R. M. J. Phys. Chem. B 2000, 104, 6271-6285. (9) Ben-Naim, A. J. Phys. Chem. 1978, 82, 792-803. (10) Lee, B. Methods Enzymol. 1995, 259, 555-576. (11) Ben-Naim, A. SolVation Thermodynamics; Plenum Press: New York, 1987. (12) Lee, B. Biophys. Chem. 1994, 51, 263-269. (13) McAuliffe, C. J. Phys. Chem. 1966, 70, 1267-1275. (14) Ben-Naim, A.; Marcus, Y. J. Chem. Phys. 1984, 81, 2016-2027. (15) Makhatadze, G. I.; Privalov, P. L. Biophys. Chem. 1994, 50, 285291. (16) Lee, B. Biopolymers 1985, 24, 813-823.

Graziano and Lee (17) Lee, B. Biopolymers 1991, 31, 993-1008. (18) Lee, B.; Graziano, G. J. Am. Chem. Soc. 1996, 118, 5163-5168. (19) Graziano, G. Can. J. Chem. 1998, 76, 437-444. (20) Graziano, G. J. Chem. Soc. Faraday Trans. 1998, 94, 3345-3352. (21) Graziano, G. Phys. Chem. Chem. Phys. 1999, 1, 1877-1886. (22) Graziano, G. Phys. Chem. Chem. Phys. 1999, 1, 3567-3576. (23) Graziano, G. Biophys. Chem. 1999, 82, 69-79. (24) Levitt, M.; Perutz, M. F. J. Mol. Biol. 1988, 201, 751-754. (25) Linse, P. J. Am. Chem. Soc. 1990, 112, 1744-1750. (26) Atwood, J. L.; Hamada, F.; Robinson, K. D.; Orr, G. W.; Vincent, R. L. Nature 1991, 349, 683-684. (27) Suzuki, S.; Green, P. G.; Bumgarner, R. E.; Dasgupta, S.; Goddard, W. A., III; Blake, G. A. Science 1992, 257, 942-945. (28) Pierotti, R. A. J. Phys. Chem. 1965, 69, 281-288. (29) Pratt, L. R.; Chandler, D. J. Chem. Phys. 1977, 67, 3683-3704. (30) Widom, B. J. Phys. Chem. 1982, 86, 869-872. (31) Lee, B. J. Chem. Phys. 1985, 83, 2421-2425. (32) Lee, B. Biophys. Chem. 1994, 51, 271-278. (33) Makhatadze, G. I.; Privalov, P. L. J. Chem. Thermodyn. 1988, 20, 405-412. (34) Wauchope, R. D.; Haque, R. Can. J. Chem. 1972, 50, 133-138. (35) Gill, S. F.; Nicholls, N. F.; Wadso, I. J. Chem. Thermodyn. 1976, 8, 445-452. (36) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926-935. (37) Jorgensen, W. L.; Gao, J.; Ravimohan, C. J. Phys. Chem. 1985, 89, 3470-3473. (38) Pierotti, R. A. Chem. ReV. 1976, 76, 717-726. (39) Wilhelm, E.; Battino, R. J. Chem. Phys. 1971, 55, 4012-4017. (40) Dec, S. F.; Gill, S. J. J. Solution Chem. 1985, 14, 417-429. (41) Privalov, P. L.; Gill, S. J. AdV. Protein Chem. 1988, 39, 191-234. (42) Reiss, H. AdV. Chem. Phys. 1966, 9, 1-84. (43) Kell, G. S. J. Chem. Eng. Data 1975, 20, 97-105. (44) Wilhelm, E.; Battino, R. J. Chem. Phys. 1972, 56, 563-566. (45) Morel-Desrosiers, N.; Morel, J. P. Can. J. Chem. 1981, 59, 1-7. (46) Tang, E. S.; Bloomfield, V. A. Biophys. J. 2000, 79, 2222-2234. (47) Narten, A. H. J. Chem. Phys. 1977, 67, 2102-2108. (48) Cabaco, M. I.; Danten, Y.; Besnard, M.; Guissani, Y.; Guillot, B. J. Phys. Chem. B 1997, 101, 6977-6987. (49) Linse, P. J. Am. Chem. Soc. 1984, 106, 5425-5430. (50) Sorenson, J. M.; Hura, G.; Glaeser, R. M.; Head-Gordon, T. J. Chem. Phys. 2000, 113, 9149-9161. (51) Ben-Naim, A. J. Phys. Chem. 1991, 95, 1437-1444. (52) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water; Oxford University Press: London, 1969. (53) Padlan, E. A. Proteins 1990, 7, 112-124. (54) Mian, I. S.; Bradwell, A. R.; Olson, A. J. J. Mol. Biol. 1991, 217, 133-151. (55) White, S. H.; Wimley, W. C. Annu. ReV. Biophys. Biomol. Struct. 1999, 28, 319-365. (56) Merz, K. M.; Roux, B. Biological Membranes: A molecular perspective from computation and experiment, Birkhauser: Boston, 1996.