Chemical Physics Letters 483 (2009) 67–71

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Chemical Physics Letters

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Role of salts on the strength of pairwise hydrophobic interaction

Giuseppe Graziano *

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

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 September 2009In final form 13 October 2009Available online 26 October 2009

0009-2614/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cplett.2009.10.040

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

The strengthening of pairwise hydrophobic interaction upon salt addition to water is due to the increasednumber density of aqueous salt solutions, according to scaled particle theory. Two CH4 molecules occupythe same volume both when they are far apart and when they are in contact, however, the contact con-figuration is favoured because the solvent-excluded volume decreases upon association and there is asubstantial gain of translational entropy of solvent molecules. The same decrease in solvent-excludedvolume upon association causes a different entropy gain depending on the number density of the solu-tion: the larger the number density, the larger will be the number of solvent particles that gain transla-tional freedom.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The hydrophobic effect is considered to be the fundamentaldriving force of a lot of biologically relevant processes, such asthe folding of globular proteins, the formation of protein–proteincomplexes, the assembly of micelles and double-layer membranes[1–4]. Unfortunately, a general consensus on the molecular-levelexplanation of the hydrophobic effect has not yet been reached,even though the results of direct structural investigations andcomputer simulation studies unequivocally indicate that thereare no ‘icebergs’ or ‘clathrates’ around nonpolar molecules in waterat room temperature and atmospheric pressure [5–7]. It does ap-pear that the molecular origin of the hydrophobic effect has tobe searched not in the structural changes induced by solute inser-tion into water (i.e., reorganization of water–water H-bonds), butin some of the peculiar geometric-structural features of water mol-ecules and, in turn, of liquid water [3,4,8]. However, the pictorial‘iceberg’ explanation is still considered right by many becausethe emerging correct rationalization is more difficult to grasp andso to accept.

A further problem is related to the basic fact that the associationof purely nonpolar molecules (i.e., the so-called hydrophobic inter-action) in water cannot be studied by direct experimental methodsbecause such molecules have a very low solubility in water at roomtemperature and atmospheric pressure. This implies that most ofthe data have been obtained by studying the hydration of nonpolarmolecules, their transfer from organic liquids to water, and theassociation of amphiphilic molecules in aqueous solution [3,4].Computer simulations have eventually offered the opportunity tostudy the association of simple nonpolar solutes, such as two

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methane molecules, and to test the validity of theoretical explana-tions [9].

It has been shown, by a long time, that the addition of salts towater markedly affects such processes, as emphasized by the Hof-meister series [10–12], even though a complete understanding ofthe role played by salts in modulating the hydrophobic effect is stilllacking. In the case of salts further pictorial models were sug-gested: the ions were divided in ‘kosmotropes’ and ‘chaotropes’depending on their ability to enhance the structure of water orto destroy it [11,13]. The former ions should strengthen the hydro-phobic effect (i.e., should stabilize the folded conformation of glob-ular proteins), whereas the latter ions should weaken it (i.e., shouldfavour the denaturation of the folded conformation). Ion classifica-tion, however, was not based on direct structural data, and is nowentirely questioned by the structural information emerging fromboth experimental measurements and computer simulations[13]. In such a situation the results of carefully performed simula-tion studies [14–16] can be the point of departure to devise andtest a theoretical rationalization of the salt effect on the strengthof hydrophobic interaction.

2. Simulation background

Jönsson et al. [17], JSL, investigated the strength of the hydro-phobic interaction between two methane molecules in pure waterand aqueous salt solutions by means of NPT Monte Carlo simula-tions at 298 K and 1 atm, using the TIP4P water model [18]. JSLconsidered a monovalent salt M+X� and a divalent salt M2+X2�

with the same non-electrostatic properties as the monovalent salt,to examine the influence of the valence of the salt species. The po-tential of mean force between the two CH4 molecules showed in allcases a global minimum at the contact distance of �3.8 Å, and thedepth of this minimum was �2.2 kJ mol�1 in water, �3.2 kJ mol�1

68 G. Graziano / Chemical Physics Letters 483 (2009) 67–71

in 3M of M+X�, and �5.5 kJ mol�1 in 3M of M2+X2�. These numbersunequivocally indicate that: (a) the addition of salts to watercauses a strengthening of pairwise hydrophobic interaction; (b)this strengthening markedly increases with the valence of the salt.These findings, absolutely not new, called for an explanation at amolecular and structural level.

JSL determined for each cation and anion the respective radialdistribution functions, rdfs, with water oxygen and hydrogenatoms, showing that the hydration of the ions increased with theirvalence (i.e., the height and the sharpness of the first peak of theserdfs increased significantly with the ion valence). In addition, JSLfound that the salt ions are depleted from the hydration shell ofCH4 molecules and that the depletion zone is larger for the divalentions. This means that the ions, being highly hydrated, do not di-rectly interact with the CH4 molecules. Of particular interest isthe fact that the maximum of the first peak in the ion-oxygen rdfwas located at 2.5 Å for M+–O, at 2.3 Å for M2+–O, at 3.3 Å forX�–O, and at 3.1 Å for X2�–O. These numbers, coupled to the effec-tive diameter of a water molecule [19], reff(H2O) = 2.8 Å (that prac-tically corresponds to the effective diameter of the oxygen atom),can be used to obtain reliable estimates for the effective diameterof the ions (to this aim the location of the first peak in the ion-oxy-gen rdf is more suitable than that of the first peak in the ion-hydro-gen rdf). Such estimates are: reff(M+) = 2.2 Å, reff(M2+) = 1.8 Å,reff(X�) = 3.8 Å, and reff(X2�) = 3.4 Å. It has to be remembered thatJSL fixed for both M+ and M2+ cations a van der Waals diameterrvdW(M+ or M2+) = 3.328 Å, and for both X� and X2� anions a vander Waals diameter rvdW(X� or X2�) = 4.40 Å. The comparison be-tween the effective diameter and the van der Waals one indicatesthat: (a) the strength of ion charge–water dipole interactionscauses a decrease of the ion size; (b) the ion size decrease increaseswith the ion valence. This is an important point that parallels whathappens in the case of water [20]: the effective size of an ion ormolecule depends on the strength of the interactions in which itis involved.

JSL characterized also the structure of water and aqueous saltsolutions and found that the addition of a salt to water, especiallythe divalent one, causes the breakdown of the tetrahedral H-bonded network in favour of a more densely packed structure[17]. In fact, the number density of equilibrated configurations ofthe three solutions was found to be: 0.03268 molecules Å�3 in purewater, 0.03322 molecules Å�3 in 3M monovalent salt, and0.038 molecules Å�3 in 3M divalent salt (see the second columnof Table 1). In addition, JSL calculated the average number of H-bonds formed by a water molecule normalized to the number ofwater neighbours: nHB/nNN = 0.66 in pure water, 0.64 in 3M mono-valent salt, and 0.51 in 3M divalent salt (see the last column ofTable 1). These numbers indicate that the strength of pairwisehydrophobic interaction increases on decreasing the nHB/nNN ratio(i.e., an aqueous salt solution is more hydrophobic than water,even though it possesses less H-bonds). By recognizing this point,JSL concluded that ‘the increase of the hydrophobic attractionwas associated with (i) a breakdown of the tetrahedral structureformed by neighbouring water molecules and of the H-bonds

Table 1Values of the number density, q, volume packing density, g, and average effective hard sphcreate a cavity suitable to host a CH4 molecule, DGc, calculated by means of SPT; values of dGDG(HI) for the association of two CH4 molecules, calculated by JSL from Monte Carlo simnormalized to the number of water neighbours, nHB/nNN, determined by JSL [17]. All the n

q � 102

(molecules �3)g hreffi

(Å)

H2O 3.268 0.3756 2.803M M+X� 3.322 0.4011 2.823M M2+X2� 3.800 0.4380 2.78

between them and (ii) the concomitant increase of the solutiondensity.’

The JSL study has attracted my attention because it offers a goodopportunity to test the validity of a simple theoretical frameworkdeveloped to rationalize at a basic level the molecular origin ofhydrophobic interaction [21].

3. Theoretical framework

The process of bringing two nonpolar solutes, such as two CH4

molecules, from a fixed position at infinite separation to a fixed po-sition at contact distance in water or aqueous solution, at constanttemperature and pressure, is the simplest example of hydrophobiceffect [22], and is called a pairwise hydrophobic interaction, HI.Ben-Naim showed that the associated Gibbs energy change canbe splitted in two terms [22]:

DGðHIÞ ¼ EaðCH4 � � �CH4Þ þ dGðHIÞ ð1Þ

where Ea(CH4� � �CH4) is the methane–methane interaction energy inthe contact configuration, and does not depend on the presence ofthe solvent and its nature; dG(HI) is the indirect part of the revers-ible work to carry out the process, and accounts for the specific fea-tures of the solvent in which pairwise HI occurs. Evaluation of thedependence of DG(HI) upon the distance between the two CH4 mol-ecules gives rise to the potential of mean force. A general relation-ship connects dG(HI) to the Ben-Naim standard solvation Gibbsenergy of the CH4� � �CH4 contact configuration and CH4 molecule,respectively:

dGðHIÞ ¼ DG�ðCH4 � � �CH4Þ � 2 � DG�ðCH4Þ ð2Þ

where DG� represents the Gibbs energy change associated with thetransfer of a species from a fixed position in the ideal gas phase to afixed position in a solvent, at constant temperature and pressure[23]. Application of a physical concept and of statistical mechanicsallows the exact division of DG� in two contributions [24,25]:

DG� ¼ DGc þ DGa ð3Þ

where DGc is the reversible work to create at a fixed position in asolvent a cavity suitable to host the solute molecule, and DGa isthe reversible work to turn on the attractive interactions betweenthe solute molecule inserted in the cavity and all the surroundingsolvent molecules (note that the second step is conditional to thefirst: there is not the additivity of independent contributions). Thereorganization of water–water H-bonds upon nonpolar solute inser-tion does not contribute to DG� because it is characterized by en-thalpy–entropy compensation [26,27]. The use of Eq. (3) in thedefinition of dG(HI) leads to:

dGðHIÞ ¼ ½DGcðCH4 � � �CH4Þ � 2 � DGcðCH4Þ�þ ½DGaðCH4 � � �CH4Þ � 2 � DGaðCH4Þ� ð4Þ

By assuming that each CH4 molecule can be treated as a hard spherethat does not have attractive interactions with the surroundingwater molecules, Eq. (4) reduces to:

ere diameter, hreffi, of the three solutions simulated by JSL [17]; values of the work to(HI) for the association of two CH4 molecules, calculated by means of Eq. (6); values of

ulations [17]; values of the average number of H-bonds formed by a water moleculeumbers refer to 298 K and 1 atm; see text for further details.

DGc

(kJ mol�1)dG(HI)(kJ mol�1)

DG(HI)(kJ mol�1)

nHB/nNN

22.2 �8.9 �2.2 0.6624.6 �9.9 �3.2 0.6431.1 �12.5 �5.5 0.51

G. Graziano / Chemical Physics Letters 483 (2009) 67–71 69

dGðHIÞ ¼ DGcðCH4 � � �CH4Þ � 2 � DGcðCH4Þ ð5Þ

Eq. (5) shows that a quantitative estimate of pairwise HI betweenhard spheres can be obtained from the calculation of DGc to create,in water or aqueous solutions, a cavity suitable to host a couple ofhard spheres in the contact configuration, and a cavity suitable tohost a single hard sphere. In this respect, by means of classic scaledparticle theory [28,29], SPT, calculations, I have shown that [21]: (a)by keeping fixed the cavity volume, the DGc magnitude depends onthe cavity shape, and proves to be roughly proportional to the wateraccessible surface area of the cavity [30], WASAc, that, in turn, is ameasure of the solvent-excluded volume due to cavity creation;(b) the value of the DGc/WASAc ratio calculated for spherical cavi-ties can be used, to a good approximation, also for non-sphericalcavities. Thus dG(HI) can be calculated from the knowledge ofWASA buried upon association:

dGðHIÞ ¼ ðDGc=WASAcÞ � DWASAðassociationÞ ð6Þ

This relationship is considered to hold for both water and aqueoussalt solutions and is the cornerstone of the present study. Since theWASA buried upon association is a negative quantity, while theDGc/WASAc ratio is a positive quantity, dG(HI) provides a negativeGibbs energy change favourable to the association of two hardspheres. Clearly, the present approach is not devised to producethe dependence of dG(HI) as a function of the distance betweenthe two interacting molecules, also because DWASA(association)does not seem to be the right variable [31,32], even though its right-ness has recently been supported by a first-principles moleculardynamics study [33].

4. Results

I have calculated, by means of classic SPT [28,29], the values ofDGc to create a cavity suitable to host a CH4 molecule in water, 3Mof M+X�, and 3M of M2+X2� at 298 K and 1 atm. In order to performcalculations, I have used the number density values obtained by JSLfor the three solutions and the effective diameters estimated fromthe location of the first peak of the ion-oxygen rdfs and reported inSection 2. The estimates are: DGc[r(CH4) = 3.73 Å] = 22.2 kJ mol�1

in water, 24.6 kJ mol�1 in 3M monovalent salt, and 31.1 kJ mol�1

in 3M divalent salt (see the fifth column of Table 1). These numbersindicate that the addition of the two salts to water causes a signif-icant decrease in the probability of finding cavities suitable to hosta CH4 molecule.

Analysis of the SPT formula for pure liquids shows that the DGc

magnitude depends on two geometric features [34]: the volumepacking density g, that represents the volume fraction really occu-pied in the liquid, and the diameter of liquid molecules. Keeping gfixed, DGc increases on decreasing the hard sphere diameter; keep-ing the hard sphere diameter fixed, DGc increases on increasing g[34]. This analysis is correct also for hard sphere mixtures (i.e.,aqueous salt solutions for the present approach), even though

Table 2Values of the experimental number density, q, volume packing density, g, and average effework to create a cavity suitable to host a CH4 molecule, DGc, calculated by means of SPT; vavalues of DG(HI) for the association of two CH4 molecules, calculated by Thomas and Elcockformed by a water molecule normalized to the number of water neighbours, nHB/nNN, deterfor further details.

[NaCl](M)

q � 102

(molecules �3)g hreffi

(Å)

0 3.333 0.3830 2.800.5 3.360 0.3880 2.801 3.390 0.3934 2.802 3.446 0.4033 2.80

one has to consider an average effective hard sphere diameterhreffi =

Pv(j) � reff(j), where v(j) is the molar fraction of the species

j in solution [35]. The trend of the calculated DGc values is ex-plained by considering that: (a) g = 0.3756 for water, 0.4011 for3M of M+X�, and 0.4380 for 3M of M2+X2�; (b) hreffi = 2.80 Å forwater, 2.82 Å for 3M of M+X�, and 2.78 Å for 3M of M2+X2� (seethe third and fourth columns of Table 1).

The water accessible surface area of a cavity suitable to host aCH4 molecule is WASAc = 4p[r(CH4) + r(H2O)]2 = 4p(1.87 + 1.40)2

= 134 Å2. Therefore, the DGc/WASAc ratio (in J mol�1 Å�2 units)amounts to 165.7 in water, 183.6 in 3M monovalent salt, and232.1 in 3M divalent salt. The WASA buried upon the associationof two CH4 molecules is obtained by considering that, in the con-tact configuration of two spheres, about 20% of total WASA isoccluded [33]. Thus DWASA(association) = �54 Å2, and, by meansof Eq. (6), one obtains dG(HI) = �8.9 kJ mol�1 in water,�9.9 kJ mol�1 in 3M of M+X�, and �12.5 kJ mol�1 in 3M ofM2+X2� (see the sixth column of Table 1). This simple approachindicates that: (a) the contact configuration of two CH4 molecules,considered to be hard spheres, is markedly favoured in water withrespect to the configuration in which the two molecules are farapart; (b) the contact configuration is even more favoured in aque-ous salt solutions: DdG(HI) = �1.0 kJ mol�1 for the transferH2O ? 3M of M+X�, and �3.6 kJ mol�1 for the transfer H2O ? 3Mof M2+X2�.

JSL, from NPT Monte Carlo simulations [17], calculated directlythe DG(HI) values that take into account all the contributions forthe association of two CH4 molecules in a liquid [see Eq. (1)]. Thus,the DG(HI) numbers of JSL, listed in the seventh column of Table 1,cannot simply be compared to the present dG(HI) ones, that, byconsidering the CH4 molecules as hard spheres, take into accountsolely the cavity contribution. The fact that the present dG(HI)numbers are larger in magnitude than the DG(HI) ones should bea simple consequence of having neglected the second term in Eq.(4) that, by measuring the loss of solute–water attractive interac-tions upon association, provides a significant positive contribution(see below for a rough estimate). In any case, it is worth noting thatthe DDG(HI) values of JSL, �1.0 kJ mol�1 for the transfer H2O ? 3Mof M+X�, and �3.3 kJ mol�1 for the transfer H2O ? 3M of M2+X2�,are close in magnitude to the DdG(HI) numbers reported above.This should imply that the present approach accounts, in a more-than-qualitatively correct manner, for the role played by salts instrengthening pairwise HI.

As a further test of the devised approach, I have calculated, at298 K and 1 atm, the dG(HI) values, by means of Eq. (6), for theassociation of two CH4 molecules in aqueous NaCl solutions overthe 0–2 M concentration range. To perform SPT calculations, I haveused the experimental density values of water and aqueous NaClsolutions, and the effective hard sphere diameters reff(Na+) =2.02 Å and reff(Cl�) = 3.62 Å [35]. The calculated dG(HI) values,listed in the sixth column of Table 2, indicate that there is a signif-icant strengthening of pairwise HI on increasing the NaCl concen-

ctive hard sphere diameter, hreffi, of some aqueous NaCl solutions [35]; values of thelues of dG(HI) for the association of two CH4 molecules, calculated by means of Eq. (6);from molecular dynamics simulations [36]; values of the average number of H-bondsmined by Thomas and Elcock [36]. All the numbers refer to 298 K and 1 atm; see text

DGc

(kJ mol�1)dG(HI)(kJ mol�1)

DG(HI)(kJ mol�1)

nHB/nNN

22.9 �9.2 �3.0 0.6723.4 �9.4 �3.3 0.6524.0 �9.7 �3.5 0.6425.2 �10.2 �3.8 0.62

70 G. Graziano / Chemical Physics Letters 483 (2009) 67–71

tration. This is in line with the DG(HI) numbers obtained by meansof molecular dynamics simulations by Thomas and Elcock [36],using the TIP3P water model [18], and listed in the seventh columnof Table 2. Since I found DGa � �15 kJ mol�1 for CH4 in both waterand aqueous NaCl solutions [35], the second term in Eq. (4) canroughly be estimated in the assumption that DGa is directly pro-portional to WASAc [i.e., something like Eq. (6)]. In this mannerthe second term in Eq. (4) is positive and equal to 6 kJ mol�1. Byadding this number to the dG(HI) estimates calculated for hardspheres and listed in the sixth column of Table 2, one obtains val-ues that are close in magnitude to the DG(HI) ones calculated byThomas and Elcock.

These authors found also a decrease in the nHB/nNN ratio onincreasing the NaCl concentration (see the numbers listed in thelast column of Table 2), and suggested a correlation between thereduction in the number of water–water H-bonds and thestrengthening of the association between two CH4 molecules[36]. On the basis of the present approach, the suggested correla-tion should be interpreted in the following manner. The strongion charge–water dipole interactions perturb the tetrahedralH-bonded network, causing the breaking of several water–waterH-bonds and so an increase in number density that renders morecostly the process of cavity creation.

5. Discussion

The devised approach is grounded on the basic notion that cav-ity creation in a liquid, at constant temperature and pressure, givesrise to a solvent-excluded volume (even though there is an in-crease in the average volume of the liquid phase by a quantityequal to the cavity volume): the centre of each solvent particle can-not enter the shell region between the cavity van der Waals surfaceand the solvent accessible surface of the cavity [37]. This shell re-gion is inaccessible to solvent particles simply because the cavityregion has to be empty by definition; thus one speaks of solvent-excluded volume for the process of cavity creation (see Fig. 1). Inaddition, by means of SPT calculations [21], I have shown that thisinaccessible shell region can reliably be approximated by WASAc.

Two CH4 molecules occupy the same volume both when theyare far apart and when they are in contact, however, the latter con-figuration is favoured from the thermodynamic point of view be-cause the solvent-excluded volume decreases upon association(i.e., a WASAc decrease according to the present approach), andthere is a substantial gain of translational entropy of solvent mol-ecules (see Fig. 1). This is the entropic driving force for the associ-ation of two hard spheres in a liquid phase. In water, this entropicdriving force is operative and is enlarged by the small size of watermolecules [34]; i.e., it is larger the number of solvent moleculesthat gain translational freedom upon association.

The fact that pairwise HI is strengthened by the addition of saltsto water can readily be rationalized along the same geometricarguments. The two aqueous salt solutions, 3M of M+X� and 3Mof M2+X2�, have hreffi practically identical to the effective size of

Fig. 1. The inner circle represents a hard sphere (i.e., a cavity), while the shellbetween the outer and inner circles represents the volume inaccessible to thecentre of solvent particles for the simple presence of the hard sphere. Thus hardsphere insertion in a liquid produces a solvent-excluded volume. In the contactconfiguration of the two hard spheres there is a marked decrease in the inaccessiblevolume for simple geometric reasons, leading to a substantial gain of translationalentropy of solvent molecules.

water molecules, 2.80 Å, and so the fundamental variable is thevolume packing density g. The latter increases markedly on pass-ing from water to the two aqueous salt solutions, reflecting the in-creased number density of aqueous salt solution (see the secondand third columns of Table 1); the same reasoning holds for theaqueous NaCl solutions, as can be appreciated by looking at thenumbers reported in Table 2. Therefore, the same reduction in sol-vent-excluded volume (i.e., WASAc) upon association of the twoCH4 molecules gives rise to a different entropy gain dependingon the solution number density: the larger the number density,the larger will be the number of solvent particles that gain transla-tional freedom. The addition of salts to water causes a structuralperturbation in the tetrahedral H-bonded network that, by break-ing some H-bonds, produces a more densely packed structureand so an increase in the number density that, in turn, strengthenspairwise HI. The reduction in the number of H-bonds due to saltshas not a direct effect on the strength of pairwise HI; the effect isindirect by causing an increase in the solution number density.This is the basic link between the structural modifications pro-duced by salt addition and the thermodynamics of pairwise HI,and is in line with arguments advanced by Docherty et al. [38],and Paschek and co-workers [39,40].

It is important to note that the structural findings emerged fromcomputer simulations [17,36] are in line with the results of neu-tron scattering measurements on several aqueous salt solutions.It has been found that: (a) there is a marked increase of water den-sity around the ions compared to pure water at the same temper-ature; (b) this density increase resembles that occurring when thepressure applied on water is markedly increased at constant tem-perature; (c) this density increase is due to a partial disruption ofthe tetrahedral H-bonded network of water [41,42].

In conclusion, the devised approach, grounded on the basic no-tion that the solvent-excluded volume decreases upon the associ-ation of two solute molecules, shows, in line with the results ofcomputer simulations, that: (a) the contact configuration of twoCH4 molecules is markedly favoured in water with respect to theconfiguration in which the two molecules are far apart; (b) thecontact configuration is even more favoured in aqueous salt solu-tions due to the number density increase. The driving force of pair-wise HI is entropic, as originally argued by Kauzmann [1], but doesnot originate from the release of ‘structured’ water molecules fromthe overlapping hydration shells of the two CH4 molecules. It orig-inates from the gain in translational freedom of water moleculesfor the increase in available space as a consequence of the decreasein solvent-excluded volume upon association.

Appendix. On the validity of the Setchenow equation

The effect of salts on the water solubility of a given solute is de-scribed by the empirical Setchenow equation [12,43]:

log½cAðwaterÞ=cAðsaltÞ� ¼ kS � cS ðA1Þ

where cA(water) and cA(salt) are the molarity of the A solute inwater and aqueous salt solutions, respectively; cS is the molarityof the salt in the aqueous solution, and kS is called the Setchenowconstant.

The chemical potential of the A solute in water and aqueous saltsolution, respectively, is given by [23]:

lAðwaterÞ ¼ l�AðwaterÞ þ RT � ln cAðwaterÞ ðA2ÞlAðsaltÞ ¼ l�AðsaltÞ þ RT � ln cAðsaltÞ ðA3Þ

where l�AðwaterÞ and l�AðsaltÞ are the Ben-Naim standard chemicalpotentials of A in water and aqueous salt solutions, respectively. Ina thermodynamic system in which the translational degrees offreedom can be treated classically [23], the Ben-Naim standard

G. Graziano / Chemical Physics Letters 483 (2009) 67–71 71

chemical potential is the reversible work necessary to transfer a sol-ute molecule from a fixed position in the ideal gas phase to a fixedposition in the liquid phase, at constant temperature and pressure.Thus l�AðwaterÞ represents the coupling work between an A mole-cule and the surrounding water molecules, while l�AðsaltÞ repre-sents the coupling work between an A molecule and thesurrounding water molecules and salt ions.

In the thought case of a special membrane, permeable to the Amolecules but impermeable to water molecules and salt ions, sep-arating the two liquid phases, it is possible for the A species to at-tain chemical equilibrium between the two solutions, keepingtemperature and pressure constant [44]. Thus, the chemical poten-tials of A in the two liquid phases have to be equal, lA(water) =lA(salt), so that:

ln½cAðwaterÞ=cAðsaltÞ� ¼ ½l�AðsaltÞ � l�AðwaterÞ�=RT ðA4Þ

According to a well-founded theory of hydration [20], Eq. (A4)becomes:

ln½cAðwaterÞ=cAðsaltÞ� ¼ ðDDGc þ DDGaÞ=RT ðA5Þ

where DDGc = DGc(salt) � DGc(water) is the difference in thereversible work to create a cavity suitable to host the A moleculein the aqueous salt solution and in water, respectively; andDDGa = DGa(salt) � DGa(water) is the difference in the reversiblework to turn on attractive interactions between the A moleculeand the surrounding solvent particles in aqueous salt solution andin water, respectively. By comparing Eqs. (A1) and (A5), oneobtains:

ðDDGc þ DDGaÞ ¼ 2:3 � RT � ks � cs ðA6Þ

This relationship implies that the function DDGc + DDGa should lin-early depend on the molarity of the salt in the aqueous solution, at afixed temperature. According to my approach [35,44], grounded onclassic SPT and Pierotti’s formula for DGa, the functionDDGc + DDGa depends on (a) the volume packing density g of theaqueous salt solution, and (b) the effective size of the particles con-stituting the solution. For a selected salt, the effective size of theparticles is fixed, and the only variable is g, that is more-or-less lin-early related to the molarity of the salt in the aqueous solution (seethe first three columns of Table 2 and [35,38]). This should be thephysical origin of the validity of the Setchenow equation.

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