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Chemical Physics Letters 449 (2007) 120–125

Solvation thermodynamics of methane and ethane indimethyl sulfoxide and acetone versus water

Giuseppe Graziano *

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

Received 26 August 2007; in final form 16 October 2007Available online 22 October 2007

Abstract

Methane and ethane are more soluble in dimethyl sulfoxide and acetone than in water at room temperature. This datum is rational-ized by showing that the work of cavity creation is smaller in dimethyl sulfoxide and acetone than in water. The difference betweendimethyl sulfoxide and water is not large, but is in line with the proposed direct proportionality between the work of cavity creationand (a) the surface tension, (b) the inverse of the isothermal compressibility, and (c) the cohesive energy density of the liquid.� 2007 Elsevier B.V. All rights reserved.

1. Introduction

In the last years several force fields have been developedin order to perform computer simulations on organic sol-vents, especially dimethyl sulfoxide, DMSO, and acetone[1–4]. The latter are aprotic dipolar solvents widely usedin organic chemistry. The solubility of nonpolar com-pounds in both DMSO and acetone is markedly larger thanin water. For instance, at 25 �C, the Ben-Naim standardsolvation Gibbs energy change for CH4, DG� (inkJ mol�1) = 8.3 in water [5], 5.0 in DMSO [6], and 1.2 inacetone [7]; similarly for C2H6, DG�(in kJ mol�1) = 7.6 inwater [5], 1.2 in DMSO [6], and �2.9 in acetone [7]. Theincreased solubility of CH4 and C2H6 in DMSO and ace-tone with respect to water might be related to the denatur-ing action toward globular proteins manifested by bothacetone and DMSO, with the former being much moreeffective [8,9].

It would be interesting to provide a molecular levelexplanation of the increased solubility of methane and eth-ane in DMSO and acetone with respect to water. This hasbeen the goal of a series of articles by van Gunsteren, vander Vegt and co-workers [4,10–12]. These authors, by

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means of molecular dynamics simulations, pointed outthat: (a) the work of cavity creation in acetone is markedlysmaller than in water, while the CH4-solvent attractions aresomewhat stronger in acetone than in water; (b) the workof cavity creation in DMSO is larger than in water andthe CH4-solvent attractions are significantly stronger inDMSO than in water. On reading these findings I was a lit-tle bit surprised because, to the best of my knowledge,water shows the largest magnitude of the work of cavitycreation among common liquids, with the exception ofhydrazine [13]. Thus I decided to verify the reliability androbustness of the conclusions reached by van Gunsteren,van der Vegt and co-workers by analyzing the solvationthermodynamics of methane and ethane in the three liquidson the basis of a well-established theoretical approach andcalculation procedure.

Several physico-chemical properties of the three liquidsare collected in Table 1 [14–16]. Even though all the threeliquids have molecules characterized by large dipolemoments, the boiling points are markedly different: 56 �Cfor acetone, 100 �C for water, and 189 �C for DMSO. Inaddition, DMSO has the highest density, the largest vapor-ization enthalpy and an isothermal compressibility onlyslightly larger than that of water. I have also reported thevalues of: (a) the cohesive energy density, the ratio of theinternal energy change upon vaporization to the molar

Table 1Some physical properties of water, DMSO and acetone at 25 �C

Water DMSO Acetone

d (g cm�3) 0.9972 1.095 0.7846v (cm3 mol�1) 18.07 71.37 74.01q (molecules A�3) 0.033326 0.008438 0.008137DvapH (kJ mol�1) 44.0 52.9 31.0Cp,l (J K�1 mol�1) 75.3 118.3 126.4Tc (�C) 374 447 235Tb (�C) 100 189 56a Æ 103 (K�1) 0.257 0.928 1.42bT Æ 105 (atm�1) 4.58 5.39 12.55ced (J cm�3) 2298 707 386Pint (J cm�3) 170 513 342c (dyne cm�1) 72 43 23bT Æ c (A) 0.325 0.229 0.285Viscosity (cP) 0.89 1.99 0.32er 78.5 46.0 20.7Polarizability (A3) 1.48 7.97 6.41l (debye) 1.84 4.49 2.88

Data are from Refs.[14–16]. The symbols are: d for the density; v for themolar volume; q for the number density; DvapH for the vaporizationenthalpy change; Cp,l for the constant-pressure heat capacity of the liquid;Tc for the critical temperature; Tb for the boiling temperature; a for thethermal expansion coefficient; bT for the isothermal compressibility; cedfor the cohesive energy density; Pint for the internal pressure; c for thesurface tension; er for the dielectric constant; l for the molecular dipolemoment in the gas phase.

G. Graziano / Chemical Physics Letters 449 (2007) 120–125 121

volume of the liquid [16], ced = (DvapH�RT)/v; (b) theinternal pressure [16], Pint = a Æ T/bT, where a is the ther-mal expansion coefficient and bT is the isothermal com-pressibility; (c) the so-called Egelstaff-Widom lengthc Æ bT, where c is the surface tension of the liquid [17]. Itproves that water has the largest ced, the smallest internalpressure and the largest Egelstaff-Widom length. The dif-ference ced�Pint has been considered an estimate of thestrength of the non-dispersion interactions existing betweenmolecules in the liquid [16]. Multiplication of such differ-ence by the liquid molar volume leads to the following esti-mates at 25 �C: 3.3 kJ mol�1 for acetone, 13.8 kJ mol�1 forDMSO, and 38.4 kJ mol�1 for water. The latter numbersindicate that: (a) acetone resembles nonpolar liquids forwhich ced � Pint [16] (this is confirmed by its ability to dis-solve nonpolar species); (b) DMSO is a special liquid some-what resembling water (for the latter, one obtains 38.4/2= 19.2 kJ mol�1 for the energy of a hydrogen bond atroom temperature, in line with theoretical calculations)due to the strength of dipolar interactions.

2. Theory of solvation

A complete exposition of this theory has already beenreported [5,18,19]. The process of inserting a solute mole-cule at a fixed position in a solvent is dissected in twosub-processes that have a clear physical meaning: (a) crea-tion of a suitable cavity to host the solute molecule; (b)insertion of the solute molecule into the cavity and turningon the solute–solvent attractive potential. The solvationGibbs energy change, DG�, is

DG� ¼ DGc þ hwaic ¼ DGc þ Ea ð1Þwhere DGc is the work of cavity creation; Æwaæc is the valueof the solute–solvent attractive potential averaged over thestatistical ensemble of solvent configurations possessing thecavity and in which the solute acts as a ghost [18,19] (i.e.,solvent molecules have not yet reorganized in response toswitching on the solute–solvent attractive potential). It isworth noting that Æwaæc ” Ea accounts for the solute–sol-vent dispersion attractions and only a fraction of dipole-in-duced dipole attractions, because in this ensemble thedipoles of solvent molecules will rarely possess the rightorientation to attractively interact with the nonpolar solutein the cavity. Clearly DGc is a positive quantity, whereas Ea

is a negative quantity; their balance determines the DG� va-lue (i.e., the solubility in the given solvent).

It has been demonstrated that DGc is entropic in nature,measuring the excluded volume effect due to the reductionin spatial configurations accessible to solvent molecules[20,21]. Monte Carlo simulations showed that the DGc val-ues, at a given number density, are similar in both watermodels and Lennard-Jones fluids constituted by particleshaving the same size of water molecules [22] (i.e., theyare independent of the intermolecular potential energyexisting among the solvent molecules). Thus, one has

DGc ¼ �TDSx ð2Þwhere DSx is the excluded volume entropy contribution.The two terms, DGc and Ea, represent the direct perturba-tion of the solvent caused by solute insertion. In responseto such direct perturbation, the solvent molecules reorga-nize producing both enthalpy DHr and entropy DSnx con-tributions. The total solvation enthalpy and entropychanges are

DH � ¼ Ea þ DH r ð3ÞDS� ¼ DSx þ DSnx ð4Þ

Note that: (a) DHr includes all interactions that a solventmolecule makes, regardless of whether the interaction part-ner is a solute or other solvent molecules [18,19]; (b) thenotation DSnx emphasises that this is the non-excluded vol-ume entropy contribution, by recognizing that also DSx

originates from a reorganization of solvent molecules[5,18]. The solvent response is a compensating process atany temperature when the solute–solvent attractive interac-tions are not large in comparison to solvent–solvent inter-actions [18,23]:

TDSnx ¼ DH r ð5Þ

Eq. (5) implies that the solvent response does not affect theDG� magnitude that, as emphasised by Eq. (1), is solelydetermined by the direct perturbation of the solvent.

3. Calculation procedure

The work of cavity creation is calculated with the fol-lowing relationship provided by scaled particle theory,

122 G. Graziano / Chemical Physics Letters 449 (2007) 120–125

SPT, by neglecting the term of pressure–volume workwhich is negligible if the pressure is fixed at 1 atm, as it isusually done [24]:

DGc ¼ RT � f� lnð1� nÞ þ ½3n=ð1� nÞ� � ðr2=r1Þ

þ ðu=2Þðuþ 2Þðr2=r1Þ2g ð6Þ

where u = 3n/(1 � n); n is the volume packing density ofpure solvent, which is defined as the ratio of the physicalvolume of a mole of solvent molecules over the molar vol-ume of the solvent, v1 (i.e., n ¼ pr3

1N Av/6v1); r1 and r2 arethe effective diameters of the solvent and solute molecules,respectively. For water, I selected r = 2.80 A (n = 0.383),close to the location of the first peak in the oxygen–oxygenradial distribution function of water [25], because the cor-responding DGc values are reliable in comparison to thoseobtained by means of computer simulations in variouswater models [19]. For acetone, I selected r = 4.76 A(n = 0.459), as determined from solubility measurementsby Wilhelm and Battino [26]. For DMSO, three differenteffective diameters have been considered: (1) r = 4.91 A(n = 0.523), as determined by Wilhelm and Battino [26];(2) r = 5.04 A (n = 0.566), that is the diameter of thesphere having the same van der Waals volume of DMSO,calculated according to the volume group contributionsof Edward [27]; (3) r = 4.78 A (n = 0.483), as determinedfrom the SPT formula for the isothermal compressibility[28].

The cavity enthalpy change is calculated from theappropriate temperature derivative of DGc [24]:

DH c ¼ ½na � RT 2=ð1� nÞ3�½ð1� nÞ2 þ 3ð1� nÞðr2=r1Þ

þ 3ð1þ 2nÞðr2=r1Þ2� ð7Þ

where a is the thermal expansion coefficient of the solvent.The cavity entropy change is:

DSc ¼ ðDH c � DGcÞ=T ð8ÞThe quantity Ea ” Æwaæc is calculated with a formula derivedby Pierotti in the assumption that (a) the solute–solventdispersion interactions are represented by the Lennard-Jones 6–12 potential, and (b) the solvent structure aroundthe solute is uniform with the density equal to that of puresolvent [24]. Pierotti’s formula is:

Ea ¼ �ð64=3Þ � n � e12 � ðr12=r1Þ3 ð9Þ

where r12 = (r1+r2)/2, e12 = (e1e2)1/2, and e1 and e2 are theLennard-Jones potential parameters for the solvent andsolute, respectively, measuring the magnitude of the maxi-mum attractive potential energy. For methane and ethanein water, DMSO and acetone, Ea ” Æwaæc should consistof dispersion attractions and a fraction of the dipole-in-duced dipole attractions [18,19]. I assume that the lattercan be absorbed into the parameterization of the dispersioncontribution because both terms depend on the inversesixth power of distance [5]. The dipole-induced dipole con-tribution is taken into account by increasing the e/k valueof water, DMSO and acetone. Specifically, for water

e/k = 120 K instead of 85 K [25], for DMSO e/k = 385 Kinstead of 333 K, and for acetone e/k = 375 K instead of362 K. There is a degree of arbitrariness in this choice that,however, may be justified by the closeness of the selectede/k values to those determined for pure dispersion contri-butions [26], and by the rough approximation of consider-ing spherical the molecules of the three liquids. Moreover,the rank order of the e/k estimates for the three liquidsagrees with that of the experimental values of molecularpolarizability, listed in Table 1, as it should be [24]. Formethane, I have fixed r = 3.70 A and e/k = 155 K in allcases, in line with literature values [10,26,29]; for ethane,I have fixed r = 4.38 A and e/k = 210 K in water, and250 K in DMSO and acetone, in line with a previous anal-ysis [5].

Adopting this procedure, there is no contribution fromthe solvent reorganization upon turning on the solute–solvent attractive potential to the calculated values ofDH� and DS�. This means that DH r = DHc and DS� = DSc,as in the approach originally devised by Pierotti [24].

4. Results and discussion

The calculation of DGc by means of Eq. (6) for methaneand ethane in water, DMSO and acetone at 25 �C producesthe values reported in the fifth column of Table 2. It is evi-dent that the selection of the effective diameter for DMSOmolecules plays a crucial role, in view of the sensitivity ofSPT results to small changes in the solvent diameter [30].For the sake of clarity, I will focus the attention onmethane because the situation is qualitatively similar forethane. By fixing r(DMSO) = 4.91 A (n = 0.523), as deter-mined by Wilhelm and Battino [26], DGc = 20.2 kJ mol�1,a value smaller than that obtained for water, 22.9 kJ mol�1;a similar situation emerges by fixing r(DMSO) = 4.78 A(n = 0.483), as determined from the SPT formula for theisothermal compressibility [28]. In contrast, by fixingr(DMSO) = 5.04 A (n = 0.566), as determined from thevan der Waals volume [27], DGc = 24.6 kJ mol�1, a valuelarger than that obtained for water. The smallest DGc

value, 15.1 kJ mol�1, is obtained in the case of acetone.The work of cavity creation is not an experimentally

measurable quantity and so the reliability of the abovenumbers can be established on two grounds: (i) qualitativeagreement with empirical and/or theoretical correlations;(ii) quantitative agreement between the experimental DG�

values and the DGc + Ea sum. It has been proposed andsupported that DGc should be: (a) proportional to the sur-face tension of the liquid [31], DGc � c; (b) inversely pro-portional to the isothermal compressibility of the liquid[32], DGc � 1/bT; (c) proportional to the cohesive energydensity of the liquid [33], DGc � ced. Accepting the validityof these correlations (even though I have pointed out someimportant failures [13,25,34]), and on the basis of theexperimental data for c, bT and ced reported in Table 1,the expectation is that DGc should be larger in water thanin DMSO and acetone. It is reassuring that all the three

Table 2Estimates of DGc, DHc, DSc and Ea for the solvation at 25 �C of methane and ethane in water, DMSO and acetone

r (A) n DGc (kJ mol�1) DHc (kJ mol�1) DSc (J K�1 mol�1) e/k (K) Ea (kJ mol�1)

CH4 Water 2.80 0.383 22.9 3.7 �64.1 120 �14.5DMSO(1) 4.91 0.523 20.2 15.8 �14.8 385 �15.3DMSO(2) 5.04 0.566 24.6 21.8 �9.4 385 �16.0DMSO(3) 4.78 0.483 17.0 12.0 �16.8 385 �14.6Acetone 4.76 0.459 15.1 15.3 0.7 375 �13.8

C2H6 Water 2.80 0.383 30.2 5.0 �84.5 120 �22.7DMSO(1) 4.91 0.523 26.3 21.1 �17.4 385 �24.4DMSO(2) 5.04 0.566 32.2 29.2 �10.1 385 �25.4DMSO(3) 4.78 0.483 22.0 15.9 �20.4 385 �23.4Acetone 4.76 0.459 19.5 20.2 2.3 375 �22.1

In performing calculations I fixed for (a) methane r = 3.70 A and e/k = 155 K in all cases; (b) ethane r = 4.38 A and e/k = 210 K in water, and 250 K inDMSO and acetone. See text for further details.

G. Graziano / Chemical Physics Letters 449 (2007) 120–125 123

correlations lead to the same rank order of the three liquidsfor the DGc magnitude. This would imply that the correctchoice for the effective diameter of DMSO molecules is4.91 A, the value determined by Wilhelm and Battino [26].

The Ea estimates calculated for methane in three liquidsby means of Pierotti’s formula are listed in the last columnof Table 2; the values of the DGc + Ea sum are listed in thethird column of Table 3 to be contrasted with the experi-mental DG� numbers in the fourth column. It is evident thatthere is agreement for water and acetone, and also DMSOif r(DMSO) = 4.91 A. By fixing r(DMSO) = 5.04 A, toreach agreement, Ea has to be significantly larger in magni-tude, and this implies that its Lennard-Jones parameter e/kshould be equal to 580 K, a value markedly larger than thatselected, 385 K. On the other hand, by fixing r(DMSO) =4.78 A, to reach agreement, Ea has to be significantlysmaller in magnitude and this implies e/k = 260 K, a valuemarkedly smaller than that selected, 385 K. Therefore, alsoon the basis of the agreement between calculated andexperimental values of the Ben-Naim solvation Gibbs ener-gies, it emerges that the effective diameter of DMSO mole-cules is 4.91 A. Obviously, this implies that, for a givencavity size, DGc is larger in water than in DMSO at roomtemperature.

The circumstance that the difference ced�Pint is signifi-cant in the case of DMSO, resembling water (see the last

Table 3Comparison between calculated thermodynamic quantities and experimental oacetone

DGc + Ea

(kJ mol�1)DG�

(kJ mol�1)Ea

(kJ

CH4 Water 8.4 8.3 �1DMSO(1) 4.9 5.0DMSO(2) 8.6 5.0DMSO(3) 2.4 5.0 �Acetone 1.3 1.2

C2H6 Water 7.5 7.6 �1DMSO(1) 1.9 1.2 �DMSO(2) 6.8 1.2DMSO(3) �1.4 1.2 �Acetone �2.6 �2.9 �

Experimental data are from References [5–7]. See text for further details.

part of Section 1), may provide a further means to discrim-inate among the three diameters considered for DMSOmolecules. For water, it has unequivocally been shownthat the hydrogen bonds are so strong to bunch up inter-acting molecules well beyond their van der Waals diameter[22]. The effective diameter of water molecules is 2.80 A,whereas the van der Waals diameter is about 3.20 A [22].Since DMSO somewhat resembles water for the strengthof non-dispersion interactions, one can argue that abunching up mechanism is operative also in this liquid,rendering the effective diameter of DMSO molecules smal-ler than the van der Waals one, 5.04 A. Thus, the valuedetermined by Wilhelm and Battino [26], 4.91 A, wouldbe the correct estimate for the effective diameter of DMSOmolecules.

Moreover, it is important to recall that Ea ” Æwaæc doesnot represent the solute–solvent interaction energy in anoptimized relative geometry/orientation. It is the interac-tion energy of the solute molecule with the surrounding sol-vent molecules averaged over the ensemble of solventconfigurations possessing the desired cavity, and in whichthe solvent molecules have not yet reorganized in responseto switching on the solute attractive potential [18,19]. Thismeans that for the magnitude of Ea, the dispersion contri-bution is fully operative because it weakly depends on therelative molecular orientations, whereas only a fraction of

nes for the solvation at 25 �C of methane and ethane in water, DMSO and

+ DHc

mol�1)DH�

(kJ mol�1)DSc

(J K�1 mol�1)DS�

(J K�1 mol�1)

0.8 �10.9 �64.1 �64.40.5 – �14.9 –5.8 – �9.4 –2.6 – �16.8 –1.5 �1.4 0.7 �8.7

7.7 �17.2 �84.5 �83.23.3 – �17.4 –3.8 – �10.1 –7.5 – �20.4 –1.9 �7.2 2.3 �14.4

124 G. Graziano / Chemical Physics Letters 449 (2007) 120–125

the dipole-induced dipole contribution should be active.On this basis, it should not be a surprise that the Ea valuesfor methane in the three liquids are similar (see the last col-umn of Table 2), even though the DMSO and acetone mol-ecules have markedly larger dipole moments than the watermolecules in the gas phase (see Table 1).

It is worth noting that, using the force field recentlydevised by van Gunsteren and co-workers for DMSO [4],the work of cavity creation proves to be larger in suchDMSO model than in the SPC model of water, at roomtemperature. In particular, Ozal and van der Vegt obtainedfor methane, at 25 �C, DGc = 22.5 kJ mol�1 in such DMSOmodel versus 20.1 kJ mol�1 in SPC water [12]. This resultcontrasts with that emerging from my analysis and shouldbe rationalized. Actually, the DMSO model of van Gunst-eren and co-workers shows, at 25 �C, bT = 4.85 Æ 10�5 bar�1 that is lower than the bT value of the SPC modelof water, 5.44 Æ 10�5 bar�1 [4,12]; note that the experimen-tal values of bT, listed in Table 1, show the reverse order.On the basis of the correlation DGc � 1/bT [32], the lowerisothermal compressibility of the DMSO model withrespect to the SPC model of water, at room temperature,could be one of the reasons rendering DGc larger in suchDMSO model than in SPC water.

In addition, the extent of structural reorganization ofsolvent molecules upon turning on the solute attractivepotential depends on the strength of solute–solvent interac-tions with respect to that of solvent–solvent interactions. Inview of the strength of water–water hydrogen bonds withrespect to methane–water van der Waals attractions, watermolecules reorganize to a little extent. This is why there isagreement between my Ea value, �14.5 kJ mol�1, and thosecalculated by full accounting of dipole-induced dipole inter-actions: Guillot and Guissani obtained�15.0 kJ mol�1 [29],and Ozal and van der Vegt obtained �13.5 kJ mol�1 [12].On the other hand, the van der Waals interactions of meth-ane with acetone and DMSO molecules have a strengthcomparable to that of acetone–acetone and DMSO–DMSOinteractions, and so a significant reorganization occursupon turning on the solute attractive potential. This iswhy my Ea values, �15.3 kJ mol�1 for methane in DMSO,and �13.8 kJ mol�1 for methane in acetone, are signifi-cantly smaller in magnitude than those obtained by vanGunsteren, van der Vegt and co-workers, �20.8 kJ mol�1

for methane in DMSO, and �16.5 kJ mol�1 for methanein acetone [10–12].

This reasoning is important also to rationalize the agree-ment/disagreement between the experimental values of DH�

and DS� and the calculated ones. In the case of water theagreement is good because the structural reorganization(i.e., the response) of water molecules, in practice, is solelyassociated with the process of cavity creation, and the rela-tions DH� = Ea + DHc and DS� = DSc work well. In thecase of acetone the latter relations do not work wellbecause a significant structural reorganization of acetonemolecules is also associated with turning on the soluteattractive potential. This means that a negative term should

be added to both the enthalpy and entropy changes. Infact, for methane in acetone the difference DH��(Ea +DHc) = �2.9 kJ mol�1 is in line with the difference betweenmy Ea estimate and the one calculated by van der Vegt andvan Gunsteren [10], �16.5 + 13.8 = �2.7 kJ mol�1. Thelatter quantity should be a measure of the energy gain asso-ciated with the structural reorganization of surroundingacetone molecules to strengthen their interactions withthe methane molecule (a quantity largely included in theDHr term of the present theory). A similar situation shouldoccur in DMSO, even though there are no experimentaldata for DH� and DS� of methane and ethane to verify thisstatement.

In conclusion, the present analysis suggests that theincreased solubility of methane and ethane in DMSO andacetone with respect to water is mainly because the workof cavity creation is smaller in the two organic solvents.Even though for DMSO this result is strongly dependenton the effective diameter assigned to DMSO molecules,the performed choice does appear to be reasonable onthe basis of proposed correlations between DGc and (a)the surface tension, (b) the inverse of the isothermal com-pressibility, and (c) the cohesive energy density of theliquid. In any case, further experimental and computa-tional studies on the solvation properties of DMSO andacetone are welcome.

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