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Chemical Physics Letters 429 (2006) 114–118

Benzene solubility in water: A reassessment

Giuseppe Graziano *

Dipartimento di Scienze Biologiche ed Ambientali, Universita del Sannio, Via Port’Arsa 11 – 82100 Benevento, Italy

Received 12 July 2006; in final form 24 July 2006Available online 5 August 2006

Abstract

It is shown that the results of molecular dynamics simulations on the hydration thermodynamics of benzene at room temperature[Schravendijk and van der Vegt, J. Chem. Theory Comput. 1 (2005) 643] are in line with a former theoretical analysis [Graziano andLee, J. Phys. Chem. B 105 (2001) 10367]. In fact: (a) the benzene–water van der Waals interaction energy proves to be larger in magnitudethan the work of cavity creation and is able to account for the experimental finding that the hydration of benzene is a spontaneous pro-cess under the Ben-Naim standard conditions around room temperature; (b) the weak benzene–water H-bonds do not provide a signif-icant contribution to benzene solubility in water because the favorable enthalpic component is almost entirely compensated for by anunfavorable entropic component. This enthalpy–entropy compensation occurs because the H-bonding potential of benzene is not strong.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

It is well established that the solubility of benzene inwater is markedly larger than that of an alkane of similarsize [1]. At room temperature, the Ben-Naim [2] standardGibbs energy change DG� associated with the hydration(i.e., gas-to-water transfer) of benzene is a negative quan-tity, whereas it is a large positive quantity for n-alkanes[3]. Clearly, it is important to provide an explanation at amolecular level of the relatively large solubility of benzenein water. Makhatadze and Privalov [3] suggested that theweak H-bonds that benzene forms with water provide afavorable energetic contribution that determines the solu-bility enhancement [4,5]. Graziano and Lee [6], G&L,reached a different conclusion on the basis of a precise divi-sion of the hydration thermodynamic quantities groundedon statistical mechanics. They concluded that: (a) the rela-tively large solubility of benzene in water is caused bystronger solute–solvent van der Waals attractive interac-tions with respect to those of alkanes; (b) the weakbenzene–water H-bonds do not afford a significant contri-bution to DG� because the favorable enthalpic term is to a

0009-2614/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2006.08.006

* Fax: +39 0824 23013.E-mail address: graziano@unisannio.it.

large extent compensated for by an unfavorable entropicterm. Graziano [7] strengthened this conclusion by pointingout that the polarizability of benzene is markedly largerthan that of an alkane of similar size.

In a recent article Schravendijk and van der Vegt [8],S&vdV, performed molecular dynamics, MD, simulationsof two benzene models in simple point charge, SPC, water[9] to try to settle down the matter. In order to address therole played by the weak benzene–water H-bonds, S&vdVstudied a real benzene model that favors the formation ofsuch weak H-bonds as well as a van der Waals, vdW, ben-zene model obtained by removing all partial charges of thefirst model and keeping the dispersion interactions unaf-fected. The main point of the S&vdV work was the calcu-lation of DG� by means of a thermodynamic integrationprocedure for the two models of benzene, using a constantpressure–temperature ensemble within the GROMACS simu-lation package [10]. They obtained, at 302 K, DG� =�4.8 kJ mol�1 for the real benzene model and +1.0kJ mol�1 for the vdW benzene model. On the basis of thesetwo qualitatively different DG� values, S&vdV [8] con-cluded that the partial charges on the benzene ring leadingto the existence of the weak H-bonds with two water mol-ecules located over the two faces of the ring are necessaryto account for the relatively large solubility of benzene in

G. Graziano / Chemical Physics Letters 429 (2006) 114–118 115

water. Note that the experimental value at 25 �C isDG� = �3.6 kJ mol�1 for benzene [3].

In the present Letter I would like to show that the num-bers coming from a careful analysis of the S&vdV work arenot in contrast with the conclusion reached by G&L [6].Specifically, the MD simulations of S&vdV confirm that:(a) the benzene–water van der Waals interaction energyoverwhelms in magnitude the work of cavity creation; (b)the enthalpic contribution of the weak benzene–water H-bonds is to a large extent compensated for by a corre-sponding entropic contribution. Since some confusionseems to be caused by an incomplete understanding ofthe approach devised by G&L, I start by spelling out indetail the fundamental points of their statistical mechanicaltheory.

2. Theory of solvation

Solvation corresponds to the transfer of a solute mole-cule from a fixed position in the ideal gas phase to a fixedposition in a solvent at constant temperature and pressure[2]. The process can be treated as the insertion of an exter-nal perturbing potential W(X), where X is a vector repre-senting a single configuration of the N molecules of thesolvent [11,12]. By using the Widom’s potential distribu-tion theorem [13], DG� is given by:

DG� ¼ �RT � lnhexp½�WðXÞ=RT �ip ð1Þ

where the subscript p means that the ensemble average isperformed over the pure solvent configurations, assumingan NPT ensemble (i.e., keeping fixed the number of mole-cules, the pressure and the temperature), whose probabilitydensity function is:

qpðXÞ ¼ exp½�HðXÞ=RT �=Z

exp½�HðXÞ=RT �dX ð2Þ

where H(X) = U(X) + P Æ V(X) is the enthalpy function ofa configuration, U(X) and V(X) are the correspondinginternal energy and volume, and the denominator is theisobaric partition function of the system.

A liquid is a condensed state of the matter and the inser-tion of a solute molecule requires the exclusion of the sol-vent molecules from a region of space that is suitable tohost the solute. Therefore, the perturbing potential canbe expressed as [11]:

exp½�WðXÞ=RT � ¼ fðXÞ � exp½�wðXÞ=RT � ð3Þ

where f(X) can assume only the values of one or zerodepending on whether or not there is a suitable cavity tohost the solute molecule in the given solvent configuration;and w(X) represents the attractive potential between thesolute molecule and the surrounding solvent molecules.Insertion of Eq. (3) into Eq. (1) leads to:

DG� ¼ �RT � lnhfðXÞip � RT � lnhexp½�wðXÞ=RT �ic ð4Þ

where the subscript c means that the ensemble average isperformed over the solvent configurations possessing a

suitable cavity to host the solute whose probability densityfunction is:

qcðXÞ ¼ fðXÞ � exp½�HðXÞ=RT �=Z

fðXÞ

� exp½�HðXÞ=RT �dX ð5Þ

Such configurations are only a small fraction of the totalsolvent configurations (i.e., those for which the functionf(X) is equal to one). Eq. (4) indicates that DG� is thesum of two terms: the work to create the cavity and thework to turn on the attractive solute–solvent potential.However, this does not mean additivity of contributions:the attractive solute–solvent potential is switched on giventhat the cavity has already been created, and the average iscalculated by using the conditional probability densityfunction qc(X).

The work to create the cavity is:

DGc ¼ �RT � lnhfðXÞip � �RT � ln pinsertion ð6Þ

It is the reversible work to single out the configurationscontaining the cavity from the ensemble of pure solventconfigurations [14,15] and, according to a theorem of sta-tistical mechanics [16], it is exactly related to the probabil-ity that the desired region of space in the liquid is devoid ofliquid molecules (i.e., the so-called insertion probability).

The work to turn on the attractive solute–solvent poten-tial is:

DGa ¼ �RT � lnhexp½�wðXÞ=RT �ic ð7ÞBy setting j = w(X) � Æw(X)æc, expanding in power seriesthe exponential function, and keeping in mind that Æjæc ” 0,one obtains [6,7]:

DGa ffi hwðXÞic � ½hj2ic=2RT � ð8Þif higher order terms are ignored. It is simple to show thatthe fluctuation term �Æj2æc/2RT is of entropic origin[11,12]. When the attractive potential w(X) is weak, thefluctuations in the value of Æw(X)æc are small, and the sec-ond term on the right-hand side of Eq. (8) can be neglected.Under this condition, the solvation Gibbs energy change isgiven by:

DG� ¼ DGc þ hwðXÞic ð9ÞFor the hydration of alkanes and noble gases, which areunable to form H-bonds with water, the fluctuation term�Æj2æc/2RT is small because w(X) represents the solute–water van der Waals interactions which are weak comparedto the water–water H-bonds [17]. In such cases it should besafe to assume that Æw(X)æc is equal to the direct solute–water interaction energy DUsw (i.e., the subscript sw standsfor solute–water interaction).

When the solute molecule can form H-bonds with water,the condition of small fluctuations for the solute–waterinteraction energy could not hold [17]. However, in theensemble of the water configurations possessing a suitablecavity to host the solute, the quantity Æw(X)æc should notaccount for the solute–water H-bond energy. A negligible

116 G. Graziano / Chemical Physics Letters 429 (2006) 114–118

fraction of water molecules in the qc(X) distribution wouldhave the correct orientation to form a H-bond with the sol-ute molecule inserted in the cavity because water moleculeshave not yet reorganized in response to the attractivepotential of the solute, since the latter acts as a ghost[6,7]. Thus Æw(X)æc should account only for the van derWaals component of the solute–water interaction energy,Æw(X)æc = DUsw(vdW), the fluctuation term is expected tobe small, and the contribution of solute–water H-bondsshould be included in the solvent reorganization term.

By adopting this line, G&L found that [6,7]: (a) the ben-zene–water van der Waals interaction energy is larger inmagnitude than the work of cavity creation and is able toaccount for the experimental negative DG� value; (b) thefluctuation term proves to be small for the hydration ofbenzene in view of the weakness and negligible anisotropyof the van der Waals component of the benzene–waterpotential; (c) the reorientation of water molecules to formH-bonds with the aromatic ring is characterized byenthalpy–entropy compensation because the H-bondingpotential of benzene is not strong.

Further light can be shed on the matter by using aninverse relationship proved by Widom [13], so that thework to turn on the solute–solvent attractive potential is:

DGa ¼ �RT � lnhexp½�wðXÞ=RT �ic¼ RT � lnhexp½wðXÞ=RT �ia ð10Þ

where the subscript a means that the ensemble average isperformed over the solution configurations possessing thecavity occupied by the solute molecule that interacts withthe surrounding solvent molecules. The probability densityfunction of such configurations is:

qaðXÞ ¼ fðXÞ � expf�½wðXÞ þ HðXÞ�=RTg=Z

fðXÞ

� expf�½wðXÞ þ HðXÞ�=RT gdX ð11Þ

In other words, in such ensemble the solute molecule doesnot act as a ghost but fully interacts with and influences thesurrounding solvent molecules. In general, it should benoted that the Widom’s inverse relationship is valid onlywhen the external potential W(X) is not infinite, and thatW(X) to insert a cavity at a fixed position is infinite formany solvent configurations if the position is occupied bysolvent molecules. This is why cavity creation is a specialprocess and is separated out first, before applying the Wi-dom’s inverse relationship.

By setting c = w(X)�Æw(X)æa, expanding in power seriesthe exponential function in the last expression of Eq. (10),and keeping in mind that Æcæa” 0, one obtains:

DGa ffi hwðXÞia þ ½hc2ia=2RT � ð12Þ

if higher order terms are ignored. Even though Eq. (12)looks like Eq. (8), its physical meaning is entirely different:(a) Æw(X)æa accounts for the full solute–water interactionenergy, including possible H-bonds, Æw(X)æa = DUsw(full);(b) the fluctuation term Æc2æa/2RT, always of entropic ori-

gin, should be a large quantity for a solute able to formH-bonds with water since the H-bonding potential is strongand is characterized by a marked anisotropy due to thegeometric requirements for the occurrence of H-bonds.Actually, for strongly H-bonding solutes such as n-alco-hols, the truncation of the power series expansion cannotbe a correct procedure [18].

If the fluctuation term Æc2æa/2RT is not a negligiblequantity, the solvation Gibbs energy change is given by:

DG� ¼ DGc þ hwðXÞia þ ½hc2ia=2RT � ð13ÞFurthermore, since both Eqs. (8) and (12) are correct fromthe statistical mechanical point of view, by equating thetwo expressions, one obtains [6]:

�hj2ic=2RT ¼ ½hc2ia=2RT � þ ðhwðXÞia � hwðXÞicÞ ð14Þthat should allow a reliable evaluation of the fluctuationterm �Æj2æc/2RT. Now, the numerical values calculatedby S&vdV can be inserted in the above equations in orderto test the validity of the G&L analysis of benzene solubil-ity in water.

3. Analysis of the S&vdV work

Beyond to calculate DG�, S&vdV [8] computed directlyfrom the MD simulation runs the benzene–water interac-tion energy DUsw. It resulted that, at 302 K, DUsw = �59.4kJ mol�1 for the real benzene model and �45.6 kJ mol�1

for the vdW benzene model, respectively. S&vdV assumedalso that the water reorganization process is characterizedby a perfect enthalpy–entropy compensation [19] (i.e., thesubscript ww stands for water–water interaction):

DH ww ¼ T � DSww ð15ÞTherefore DG� was solely given by the enthalpic and entro-pic contributions due to solute–water interactions (note, ingeneral, that in a condensed phase of the matter, such as aliquid, the difference between energy and enthalpy is verysmall):

DG� ¼ DU sw � T � DSsw ð16Þ

S&vdV, by using Eq. (16), obtained the solute–water entro-py contribution at 302 K: T Æ DSsw = DUsw � DG� =�54.6 kJ mol�1 for the real benzene model and �46.6kJ mol�1 for the vdW benzene model, respectively. In orderto provide an interpretation for these large and negativevalues of the benzene–water entropy change, S&vdV useda theoretical relationship derived by Sanchez and col-leagues [20]:

T � DSsw ¼ RT � ln pinsertion � ½hc2ia=2RT � ð17Þ

where the first term on the right-hand-side is minus thework of cavity creation, DGc, and the second term on theright-hand-side is a fluctuation term identical to that ofEq. (12) – it accounts for the fluctuation of the attractivesolute–solvent potential energy w averaged over the ensem-ble of solution configurations in which there are no solute–

G. Graziano / Chemical Physics Letters 429 (2006) 114–118 117

solvent repulsions and the attractive solute–solvent poten-tial is turned on.

S&vdV performed MD trajectories 90 ns long for thetwo models of benzene, and found that the fluctuation termamounts to 12.8 kJ mol�1 for the real benzene model and3.9 kJ mol�1 for the vdW benzene model [8]. This meansthat the work to create in SPC water at 302 K a cavity suit-able to host benzene is: (a) DGc = �RT Æ ln pinsertion =54.6 � 12.8 = 41.8 kJ mol�1 for the real benzene model;(b) DGc = �RT Æ ln pinsertion = 46.6 � 3.9 = 42.7 kJ mol�1

for the vdW benzene model. It is important to note thatthese two values should be identical because DGc is a prop-erty of the pure liquid and cannot depend on the potentialenergy of the solute molecule [14,15]; in the following thecorresponding mean value, DGc = 42.3 kJ mol�1 will beused. The two numbers derived from the simulations ofS&vdV are close to each other and are reliable. Indeed(a) from the DGc values calculated by van Gunsteren andcolleagues [21] in SPC water at room temperature,DGc � 43 kJ mol�1 for a cavity suitable to host a benzenemolecule; (b) according to scaled particle theory [22],DGc = 41.3 kJ mol�1, using the experimental density ofwater at 25 �C, and the hard sphere diameters r(ben-zene) = 5.26 A and r(water) = 2.80 A [6,23].

Clearly, it is now possible to apply Eqs. (9) and (13) tothe two benzene models of S&vdV. By using Eq. (9) for thevdW benzene model and considering that Æw(X)æc =DUsw(vdW), it results:

DG� ffi DGc þ DU swðvdWÞ ¼ 42:3� 45:6 ¼ �3:3 kJ mol�1

ð18ÞBy using Eq. (13) for the real benzene model and consider-ing that Æw(X)æa = DUsw(full), it results:

DG� ¼ DGc þ DU swðfullÞ þ ½hc2ia=2RT �¼ 42:3� 59:4þ 12:8 ¼ �4:3 kJ mol�1 ð19Þ

These calculations indicate unequivocally that both Eqs. (9)and (13) are right and provide a similar numerical resultand the same physical explanation for the relatively largesolubility of benzene in water. Eq. (18) indicates that theexperimental negative DG� value of benzene in water is ac-counted for by simply considering the balance between thework of cavity creation and the contribution of van derWaals interactions. But this is the same explanation emerg-ing from Eq. (19), because the favorable enthalpic contribu-tion of the weak benzene–water H-bonds is compensatedfor by a corresponding unfavorable entropic contribution:

DU swðfullÞ þ ½hc2ia=2RT � ¼ �59:4þ 12:8 ¼ �46:6 kJ mol�1

� DU swðvdWÞ ð20Þ

The entropic fluctuation term practically cancels thedifference in the benzene–water interaction energy calcu-lated for the two different benzene models by S&vdV. Thisis exactly the manifestation of the enthalpy–entropy com-pensation characterizing the weak benzene–water H-bonds,

as indicated by G&L [6]. The magnitude of the fluctuationterm Æc2æa/2RT expresses the extent to which the attractivesolute–solvent interactions bias positions and orientationsof the vicinal solvent molecules, causing a reduction ofthe accessible configuration space that, in turn, leads toan entropy decrease.

It is also possible to obtain an estimate of the fluctuationterm �Æj2æc/2RT in Eq. (8) using Eq. (14) and the valuescalculated by S&vdV:

� hj2ic=2RT ffi ½hc2ia=2RT � þ DU swðfullÞ � DU swðvdWÞ¼ 12:8� 59:4þ 45:6 ¼ �1:0 kJ mol�1 ð21Þ

This estimate indicates that such fluctuation quantityshould be small [6]. It is also important to note that�Æj2æc/2RT is a negative quantity implying that the reorga-nization of water molecules to form the weak H-bonds withthe benzene ring is a spontaneous process, even thoughcharacterized by an almost complete enthalpy–entropycompensation.

S&vdV determined the fluctuation term also for thevdW benzene model and the value was 3.9 kJ mol�1, thatcannot be considered negligible. However, according tothe statistical-mechanical approach of G&L, the quantity�Æj2æc/2RT has to be calculated over the ensemble of sol-vent configurations possessing the suitable cavity, whichis different from the ensemble of solution configurationsused by S&vdV to calculate the fluctuation term for thevdW benzene model.

4. Discussion

The performed analysis indicates that the valuesobtained from the MD simulations of S&vdV are not incontrast with the G&L approach: the benzene–water vander Waals interaction energy is larger in magnitude thanthe work of cavity creation and is able to account for theexperimental negative DG� value. It is worth noting thatthe van der Waals benzene–water interactions are strongerthan those of an aliphatic hydrocarbon of similar molecu-lar size with water, due to the larger polarizability of thearomatic ring [7]. Consider, for instance, that, even thoughthe polarizability of c-hexane, 10.78 A3, is similar to that ofbenzene, 10.32 A3 [22], the c-hexane molecule is signifi-cantly larger than that of benzene. The effective hardsphere diameters are 5.63 A for c-hexane and 5.26 A forbenzene [12], so that the molecular volume of the latter is82% of that of the former. Such a value should correspondalso to the polarizability ratio because this physical quan-tity is proportional to molecular volume. However, exper-imental data indicate that the polarizability of benzene is96% of that of c-hexane [22]. In addition, van Gunsterenand colleagues [24] obtained, at room temperature,DUsw = �35.0 kJ mol�1 for i-butane and �36.4 kJ mol�1

for n-butane; these values are about 10 kJ mol�1 smallerin magnitude than the benzene–water van der Waals inter-action energy, DUsw(vdW) = �45.6 kJ mol�1. This finding

118 G. Graziano / Chemical Physics Letters 429 (2006) 114–118

is in line with the analysis by G&L [6,7], but was not under-scored by S&vdV.

A further point should be considered: if the contributionof the weak benzene–water H-bonds is small due toenthalpy–entropy compensation, the thermodynamic inte-gration procedure should produce the same DG� estimatefor the two models of benzene devised by S&vdV. Indeed,thermodynamic integration takes into account solely theeffective contributions to the DG� quantity. The differencefound by S&vdV, DG� = �4.8 kJ mol�1 for the real benzenemodel and+1.0 kJ mol�1 for the vdW benzene model, is notsmall, but cannot be considered to be large. In order to gainperspective, one has to remind that, for an alkane having asize comparable to that of benzene (i.e., i-butane and n-butane), DG� amounts to 9–10 kJ mol�1 at 25 �C [2,3].The latter numbers indicate that the vdW benzene modelof S&vdV is markedly less hydrophobic than an alkane ofsimilar size. This result was neglected by S&vdV, eventhough it emerged directly from their MD simulations.

In addition, S&vdV showed that a change of GROMOSparameters produces a marked effect on the DG� value cal-culated for the real benzene model: DG� = �4.8 kJ mol�1

for the first-set, and �6.7 kJ mol�1 for the second-set [8].S&vdV did not show the DG� value obtained for thevdW benzene model using the second-set of GROMOSparameters. One could speculate that, by using the sec-ond-set of GROMOS parameters, DG� would be a negativequantity also for the vdW benzene model.

In conclusion, the performed analysis points out that thenumerical values obtained from MD simulations byS&vdV indicate that: (a) the benzene–water van der Waalsinteraction energy overwhelms the work of cavity creation,accounting for the experimental negative DG� value; (b) theweak benzene–water H-bonds do not really influence thebenzene solubility in water because the favorable enthalpiccomponent is almost entirely compensated for by an unfa-vorable entropic component.

Acknowledgement

I thank Dr. B. Lee (Center for Cancer Research, NCI,NIH, Bethesda, MD) for reading earlier drafts of the man-uscript and providing useful suggestions.

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