7792 J. Phys. Chem. B 2001, 105, 7792-7798 Henry s Constnt Anlysis for Wter nd Nonpolr Solvents from Experimentl Dt, Mcroscopic Models, nd Moleculr Simultion Georgios C. Boulougouris,, Eponds C. Voutss, Ionnis G. Economou,*, Doros N. Theodorou,, nd Dimitrios P. Tssios Moleculr Modeling of Mterils Lbortory, Institute of Physicl Chemistry, Ntionl Reserch Centre for Physicl Sciences Demokritos, GR-15310 Aghi PrskeVi Attikis, Greece, Thermodynmics nd Trnsport Phenomen Lbortory, Deprtment of Chemicl EngineeringsSection II, Ntionl Technicl UniVersity of Athens, 9 Heroon Polytechniou Str., Zogrphos GR-15780, Athens, Greece, nd Deprtment of Chemicl Engineering, UniVersity of Ptrs, GR-26500 Ptrs, Greece ReceiVed: Februry 5, 2001; In Finl Form: My 20, 2001 Experimentl dt, equtions of stte (EoS), nd Monte Crlo simultions re used to nlyze the Henry s lw constnt of solutes in wter nd in orgnic solvents t different tempertures. EoS re incpble of correlting the perimentl dt for light hydrocrbons dissolved in wter. Novel simultion methodologies re used for methne in wter nd in ethne. Results re nlyzed with respect to the free energy of cvity formtion for hosting the solute molecule in the solvent nd the free energy of interctions between the solute molecule nd the solvent. It is shown tht the hydrophobic phenomenon is driven, to lrge tent, by the wek intermoleculr interctions between wter molecules nd nonpolr solute molecules. Introduction The Henry s lw constnt is used widely to describe the low solubility of light solutes in vriety of solvents t reltively low pressure over wide temperture rnge. 1-3 Experimentl dt nd ccurte correltions hve been reported for mny solutes in different solvents such s wter, 1 light nd hevy hydrocrbons, 2,3 nd others. 2 Typiclly Henry s constnt increses with temperture for reltively low-temperture vlues, goes through mximum, nd then decreses for higher tempertures. A reltively high Henry s constnt vlue corresponds to low solubility nd vice vers. Trditionlly, the low solubility of solute in solvent hs been ssocited with differences in energetic interctions between solvent molecules nd solute molecules or differences in moleculr size or both. A typicl mple of the first cse is the dissolution of n-lknes in wter. N-lkne molecules interct through wek London forces, wheres wter molecules hibit strong hydrogenbonding interctions resulting in three-dimensionl structures. A representtive mple for the second cse is tht of light gses (hydrogen, nitrogen, etc.) dissolved in hevy n-lkne. In this work, generlized thermodynmic frmework is presented for the nlysis of the Henry s constnt nd its vrition with temperture. Henry s constnt is pressed s product of two terms: The first term involves pure solvent thermodynmic properties only. The second term is function of the cess chemicl potentil of the solute t the stte conditions ed. The ltter is subdivided into term ssocited with the formtion of cvity in the solvent to host the solute molecule nd term tht ccounts for the free energy of inserting the solute into the cvity by turning on the energetic * To whom correspondence should be ddressed. E-mil: economou@ mistrs.chem.demokritos.gr. Ntionl Reserch Centre for Physicl Sciences Demokritos. Ntionl Technicl University of Athens. University of Ptrs. interctions between solute nd solvent molecules. In this wy, detiled nlysis of the vrious fctors ffecting Henry s constnt is performed. Experimentl dt for two representtive systems, methne in wter 3 nd methne in n-hdecne, 3 re nlyzed ccording to the proposed scheme. Mixture phse equilibrium dt re often correlted using equtions of stte (EoS). For nonpolr mixtures, cubic EoS provide ccurte estimtes of Henry s constnt. 4 In this work, EoS clcultions re presented for the two mixtures stted bove using three different models: the Sove-Redlich-Kwong cubic EoS (SRK), 5 the cubic-plus-ssocition EoS (CPA), 6 nd the sttisticl-ssociting-fluid-theory EoS (SAFT). 7 SAFT is sttisticl-mechnics-bsed semiempiricl EoS developed specificlly for systems contining chin or hydrogen-bonding molecules or both. CPA is n tension of the simple SRK EoS to hydrogen-bonding systems by ccounting plicitly for ssocition with n pproch similr to SAFT. It is shown here tht SRK nd CPA correlte ccurtely the Henry s constnt of methne in n-hdecne but none of the three models is ble to correlte ccurtely the Henry s constnt of methne in wter over the entire temperture rnge. Moleculr simultion hs dvnced to powerful tool for mixture thermodynmic property clcultions nd for fluid structure nlysis. In this wy, moleculr chrcteristics cn be consistently correlted with mcroscopic properties using very smll number of prmeters. Recently, the Henry s constnt of methne in wter nd of ethne in wter over the temperture rnge of 300-570 K nd pressure very close to pure wter vpor pressure ws clculted ccurtely from NPT (isothermlisobric) Monte Crlo simultion using the Widom test prticle insertion pproch. 8 For highly dense fluids, such s liquid wter, the Widom method is imprcticl nd very long simultions re required. More recently, two very efficient methods for chemicl potentil clcultion were developed on the bsis of stged nd direct deletion of the test prticle from the system, 10.1021/jp010426f CCC: $20.00 2001 Americn Chemicl Society Published on Web 07/21/2001
Henry s Constnt Anlysis for Wter nd Nonpolr Solvents J. Phys. Chem. B, Vol. 105, No. 32, 2001 7793 herefter referred to s the stged prticle deletion () nd direct prticle deletion (DPD) methods, respectively. 9,10 These methods re used here in the course of NPT simultion of methne in wter. Wter is modeled using two-body potentil. 11 The cess chemicl potentil of methne infinitely diluted in wter is clculted, s well s the free energy of cvity formtion in wter nd the free energy of converting methne molecule to cvity (hrd sphere) in wter. Furthermore, NPT Monte Crlo simultions re reported for light gs dissolved in nonpolr solvent. The specific mixture ed is methne dissolved in ethne. Simultions re performed t the sme reduced temperture t the sturtion point of the solvent for ech of the two solvents (wter nd ethne), so direct comprison is possible. Simultion dt re used to develop some insight into the hydrophobic phenomenon (hydrtion process). In the pst, hydrophobicity ws ssocited with the very low hydrocrbon solubility in wter. Furthermore, the iceberg model ws used widely to plin the structure of wter molecules surrounding smll solute molecule. 12,13 It hs been postulted tht there is n unfvorble entropy contribution to the dissolution of hydrocrbon molecules in wter relted to the formtion of cvity in the solvent in order to host the solute molecules. This unfvorble entropy ws ttributed to the smll size of wter molecules tht results in lrge number of reltively smll cvities tht prohibit solute molecule insertion 14-16 or to the strong hydrogen bonds formed by wter molecules. 17-19 Simultion results presented here for the dissolution of methne in wter nd in ethne offer new insights in this re. We define two types of cvities in wter, the polr nd the nonpolr cvity, on the bsis of the type of molecule removed for the formtion of the cvity. Simultion dt re nlyzed, nd n evlution of the free energy of cvity formtion is mde for different mixtures. Finlly, it is shown tht the wek energetic interctions re primrily responsible for the low methne solubility in wter. Henry s Lw Constnt Anlysis The Henry s lw constnt of solute in solvent (H solutefsolvent ) is given by the pression H solutefsolvent ) lim x solutef0 ( f solute x solute) (1) where x solute is the mole frction of the solute nd f solute is the fugcity of the solute. With the use of stndrd thermodynmic reltions, eq 1 cn be written s H solutefsolvent ) lim x solutef0 ( F solution β ) F solvent p(βµ, β solute ) p( ) F solvent p(βµ,cv β solute + βµ,en solute ) ) H idel H cess (2) where F solvent is the number density of the pure solvent t given temperture (T) nd pressure (P), β ) 1/(k B T), µ solute is the solute cess chemicl potentil, nd µ, solute is the sme t infinite dilution. The cess chemicl potentil, µ, of species in mixture is defined s the chemicl potentil of the species t given temperture, density, nd composition us the idel )) Figure 1. Henry s constnt (), H idel (b), nd H cess (c) of methne in n-hdecne. In ll cses, perimentl dt ([) 3 nd SRK (short dshed lines), CPA (solid lines), nd SAFT (long dshed lines) predictions re shown. gs chemicl potentil of the pure species t the temperture nd moleculr density it hs in the mixture. In the rest of the pper, when we refer to cess chemicl potentil, we imply tht it is t infinite dilution, unless otherwise stted. The cess chemicl potentil cn be divided further into two terms: µ,cv solute is the free energy for the formtion of cvity in the solvent with prescribed size which is chosen t lest equl to the hrd core dimeter of the solute molecule, nd µ,en solute is the free energy of trnsforg the solute molecule to such cvity (minly due to the energetic interctions of the solute molecule with the solvent molecules). Both µ,cv,en solute nd µ solute re t infinite dilution. In eq 2, the Henry s constnt is written s product of two terms: H idel )F solvent /β which hs dimensions of pressure nd is function of pure solvent properties (nd thus it is the sme for ll solutes dissolved in solvent t the sme conditions) nd H cess which is dimensionless nd mesure of the nonidel solvent-solute interctions (cvity formtion nd energetic interctions). Most of the work presented here focuses on this nonidel contribution. In Figures 1 nd 2, perimentl dt 3 for the Henry s constnt of methne in sturted liquid n-hdecne nd of methne in sturted liquid wter, respectively, re shown s function of temperture. In both cses, mximum vlue is observed t tempertures well below the solvent criticl tem-
7794 J. Phys. Chem. B, Vol. 105, No. 32, 2001 Boulougouris et l. TABLE 1: CPA Prmeters for the Components Exed in This Work component b (L/mol) R 0 (br L 2 /mol) c 1 wter 0.0146 0.801 1.751 methne 0.0284 2.278 0.444 hdecne 0.2961 94.919 1.373 For wter, ɛ AB ) 1793.6 K nd β AB ) 0.1151. TABLE 2: SAFT Prmeters for the Components Exed in This Work component m V 00 (cm 3 /mol) u 0 /k (K) wter 2.853 3.304 167.10 methne 0.996 21.475 190.72 hdecne 11.366 11.622 209.90 For wter, ɛ/k ) 1634.7 K nd κ ) 0.3374. Figure 2. Henry s constnt (), H idel (b), nd H cess (c) of methne in wter. In ll cses, perimentl dt ([) 3 nd SRK (short dshed lines), CPA (solid lines), nd SAFT (long dshed lines) predictions re shown. perture of 722 K for n-hdecne nd 647.1 K for wter. In Figures 1b nd 2b, the H idel for the two systems is shown. In ll cses, it hibits mximum vlue driven by the combined vrition of sturted liquid density of the pure solvent (which in generl decreses monotoniclly s temperture increses, with the ception of wter tht hs mximum sturted liquid density t 277.15 K) nd the temperture (eq 2). Finlly, in Figures 1c nd 2c, the H cess for the two mixtures is presented. The mximum vlues of H cess in both cses re observed t tempertures close to the tempertures where H hs mximum. In other words, the loction of the mximum in the Henry s constnt vlue is controlled minly by the nonidelities of mixing between solvent nd solute. As one pects, H cess vlues for methne in n-hdecne re more thn n order of mgnitude lower thn those for methne in wter. Eqution of Stte Predictions Clcultions for the Henry s constnt were performed using three EoS. SRK 5 is cubic EoS used widely for fluid phse equilibrium clcultions. SRK prmeters re evluted from the criticl temperture (T c ), criticl pressure (P c ), nd centric fctor (ω) of the pure components. CPA 6 is n tension of the SRK tht ccounts plicitly for hydrogen-bonding interctions using pproprite pressions. Finlly, SAFT 7 is sttisticlmechnics-bsed EoS developed for chin molecules nd ssociting fluids using first-order thermodynmic perturbtion theory. Detiled mthemticl pressions for these equtions cn be found in the originl publictions nd re not repeted here. Both CPA nd SAFT re three-prmeter EoS for pure nonssociting fluids nd five-prmeter EoS for pure ssociting fluids (i.e., wter, lcohols, etc.). These prmeters re fitted to perimentl vpor pressure nd liquid density dt over wide temperture rnge. Prmeter vlues for the components ed in this work re shown in Tbles 1 nd 2. The models re tended to mixtures using the stndrd vn der Wls one-fluid mixing rules. For mixtures of components of considerble difference in energetic interctions (for mple, polr-nonpolr mixture), binry prmeter, k ij, is used tht is fitted to mixture dt. For the cse of methne in n- hdecne, both components re nonpolr nd so no binry prmeter ws used. In Figure 1, clcultions from the three models re shown for methne in n-hdecne. Interestingly, SRK predictions re in the reltively best greement with the perimentl dt becuse of cncelltion of errors. SAFT predictions for H re too low by fctor of 2 pproximtely t low tempertures nd higher thn the perimentl vlues s one pproches the solvent criticl point (t 722 K). Furthermore, SAFT predicts n H cess vlue close to 1 (Figure 1c) throughout the temperture rnge ed, which mens tht βµ is predicted to be close to zero. From these clcultions, one my conclude tht SAFT is not ccurte in predicting the infinite dilution properties of light hydrocrbons in hevy hydrocrbons. Wter-methne is highly nonidel mixture, nd therefore, binry prmeter is required. Recently, 20 SAFT nd CPA were tested tensively for the correltion of binry wter-n-lkne low- nd high-pressure phse equilibri nd corresponding binry prmeters were evluted. For the cse of wtermethne, k ij ) 0.495 for SAFT nd 0.05 for CPA. A similr nlysis ws performed for SRK, nd it ws found tht k ij ) 0.55. These prmeters were used for the clcultion of the Henry s constnt of methne in wter shown in Figure 2. SAFT nd CPA predictions gree with the perimentl dt only t high tempertures. As the temperture lowers below 400 K, both EoS predict tht H nd H cess increse monotoniclly, wheres the perimentl dt go through mximum nd then decrese. Such deficiency of EoS for wter-n-lkne mixtures hs been lso reported previously 21 nd is ttributed to the men-field nture of the mcroscopic models, which fil to ccount for the perturbtion of the locl structure of wter induced by the insertion of the solute molecule.
Henry s Constnt Anlysis for Wter nd Nonpolr Solvents J. Phys. Chem. B, Vol. 105, No. 32, 2001 7795 Monte Crlo Simultion Simultion Detils. Henry s lw constnt in wter nd in nonpolr solvents is investigted further through Monte Crlo simultions. For ll of the components ed here, twobody potentil models re used. Wter is modeled using the modified tended simplified point chrge (MSPC/E) model tht consists of Lennrd-Jones sphere (with ɛ/k ) 74.5 K nd σ ) 3.116 Å) locted on the oxygen tom, two positive prtil chrges (ech equl to 0.4108e) locted on the hydrogen toms (t 0.9839 Å from the oxygen tom), nd one negtive prtil chrge of 0.8216e locted on the oxygen tom. 11 The H-O-H ngle is 109.47. MSPC/E ws prmetrized to provide ccurte vpor pressure nd liquid density vlues of pure wter t subcriticl conditions. The intermoleculr potentil is clculted from the pression u(r) ) 4ɛ[( σ r)12 - ( σ 3 3 q γ q δ r)6] + γ)1δ)1 (3) r γδ where r is the distnce between the two oxygen centers nd the indices γ nd δ run over ll chrges on the molecules. Methne nd ethne molecules re modeled with the TrPPE united-tom Lennrd-Jones model. 22 For methne, ɛ/k ) 148 K nd σ ) 3.73 Å, wheres for ethne, ɛ/k ) 98 K, σ ) 3.75 Å, nd the C-C bond length is equl to 1.54 Å. The stndrd Lorentz- Berthelot combining rules were used for the interctions between unlike molecules: ɛ ij ) ɛ ii ɛ jj (4) σ ij ) σ ii + σ jj 2 To describe properly the long-rnge electrosttic forces, the Ewld summtion pproch ws used. For the cse of hydrocrbon interctions, the stndrd long-rnge correction method ws used 23 by setting the cutoff distnce equl to 3σ. For the cse of methne dissolved in wter, 200 wter molecules nd single methne molecule were simulted in the NPT ensemble. Simultions t five different tempertures nd t pressures equl to the pure wter vpor pressure were performed. Clcultions were mde in two stges: Initilly, very long runs, up to 500 10 6 moves in the cse of low tempertures, were performed in order to get good sttistics in the smpling of configurtions. A number of configurtions were stored in the course of the run in order to clculte the chemicl potentil of the methne using the method 9 nd the DPD method 10 in postprocessing clcultion. The method relies on the simultion of n N-molecule system (system I) nd n (N - 1)-molecule system (system III) resulting from the deletion of test molecule (here, the solute molecule). The reference system of N molecules should be ble to smple the complete configurtionl spce tht the perturbed system of N - 1 molecules smples. As result, n intermedite system of N - 1 molecules nd hrd core molecule (system II) is introduced nd the free energy difference between systems I nd III is clculted on the bsis of the free energy differences for the pirs I-II nd III-II. An optimum size of the hrd core molecule with respect to the size of the deleted molecule tht is system dependent nd is detered erly in the simultion ists. 9 Finlly, nd its constituent cvity term nd energetic term in the NPT ensemble re clculted from the pression (5) βµ ) βµ energy + βµ volume 1 )-ln( 1 - V N,P,T N-1 H(r i,n ) p(βu (N) (rb 1,...,rb N )) i)1 N,P,T) V N-1 ln( H(r i,n ) N-1,P,T ) (6) i)1 where H(r i,n ) is Heviside step function nd U (N) (rb 1,...,rb N ) stnds for the intermoleculr energy felt by the Nth molecule becuse of its interctions with the remining N - 1 molecules of the system. It is cler from eq 6 tht the nlysis presented here invokes clcultions in n ensemble of N molecules nd n ensemble of N - 1 molecules with ll other prmeters being the sme. Although in the thermodynmic limit, there is no difference between the two ensembles, specil ttention should be pid in the cse of moleculr simultion of finite size systems. A rigorous pproch would require two seprte simultions. However, when clcultions re performed in the NPT ensemble, the difference between the (N - 1)- nd N-ensembles corresponds only to wek size effect, wheres for clcultions in the NVT ensemble, there is smll density difference. Prcticlly, there is no difference between the two-seprte-simultion clcultion (referred to s ) nd the single-simultion clcultion (referred to s DPD), s will be mde cler in the results section. A number of NVT runs were mde in order to identify the Henry s constnt vrition with temperture t constnt volume. All of the production runs strted from well-equilibrted configurtions. Detiled results from the simultions re shown in Tble 3 for the NPT simultion of methne in wter, Tble 4 for the NPT simultion of methne in ethne, nd Tble 5 for the NVT simultion of methne in wter. For the cse of methne dissolved in ethne, system of 300 ethne molecules nd single methne molecule ws simulted. In this cse, much smller number of simultion steps ws used, on the order of 20 10 6 moves. Simultions for both mixtures were performed t the sme reduced temperture with respect to the solvent criticl temperture, so results for the two mixtures re directly comprble. In the DPD methodology, key vrible is the distnce between the center of the cvity nd the solvent tom(s). This vrible depends on the solvent hrd core size. Specil ttention should be pid when compring different systems (either with respect to solvent or with respect to solute). In the cse of queous mixtures, the hrd core of wter molecule is lrger for the cse of nonpolr solute thn for the cse of polr solute. In this work, the rdius of the sphericl cvity, R c, is defined s the distnce from the cvity center to the center of the solvent molecule, d 12, us one-hlf of the imum distnce tht two solvent molecules cn pproch ech other, r 11 (see Figure 3). Here the imum vlue for d 12, denoted s d 12, is set equl to the distnce of closest pproch between solute nd solvent molecules, detered from the corresponding pir distribution function. In the cse of nonpolr solute dissolved in wter, the wter molecules look lrger to the nonpolr solute molecules becuse of the much weker interctions between the two components. This difference is reflected in the scheme shown in the lower right prt of Figure 3 where r / 11 is lrger thn r 11. For the
7796 J. Phys. Chem. B, Vol. 105, No. 32, 2001 Boulougouris et l. TABLE 3: NPT Monte Crlo Simultion of Methne Infinitely Dilute in Wter temp (K) pressure (br) DPD DPD,en DPD,cv 300 0.02 3.80 ( 0.33 3.27 ( 0.25-4.34 ( 0.13 7.61 ( 0.29 333 0.14 3.70 ( 0.18 3.83 ( 0.44-3.69 ( 0.18 7.52 ( 0.40 371 0.80 3.58 ( 0.11 3.58 ( 0.11-3.17 ( 0.12 6.75 ( 0.16 400 2 3.30 ( 0.05 3.38 ( 0.07-2.83 ( 0.03 6.21 ( 0.07 470 15 2.48 ( 0.18 2.64 ( 0.19-2.23 ( 0.15 4.50 ( 0.08 The nonpolr cvity rdius is R c ) 1.79 Å. TABLE 4: NPT Monte Crlo Simultion of Methne Infinitely Dilute in Ethne temp (K) pressure (br) Widom,en,cv 125 0.013-4.11 ( 0.49-3.84 ( 0.84-13.41 ( 0.84 9.56 ( 0.04 150 0.91-2.93 ( 0.18-2.92 ( 0.49-10.83 ( 0.49 7.91 ( 0.07 200 3-1.83 ( 0.05-1.99 ( 0.14-7.42 ( 0.13 5.43 ( 0.06 250 17-1.25 ( 0.04-1.21 ( 0.09-4.86 ( 0.05 3.65 ( 0.07 The nonpolr cvity rdius is R c ) 1.79 Å. TABLE 5: NVT Monte Crlo Simultion of Methne Infinitely Dilute in Wter t 1.062 g/cm 3 temp (K) systems ed here, it is r wter-wter ) 1.14 Å, wheres r / wter-wter ) 1.23 Å. In the simultions performed in this work, the imum distnce between different molecule pirs is d wter-wter 0.74σ wter-wter, d wter-methne 0.84σ wter-methne, d methne-methne 0.86σ methne-methne, d methne-ethne 0.87σ methne-ethne, nd 0.83σ ethne-ethne. The difference between d ethne-ethne d wter-wter nd d methne-methne with respect to the corresponding σ vlues clerly reflects the strong effect of polr interctions between wter molecules on the wter hrd core dimeter compred to the wek interctions between methne molecules.,en,cv 300.13 4.66 ( 0.33-4.61 ( 0.21 9.27 ( 0.25 371.15 6.11 ( 0.19-3.34 ( 0.15 9.44 ( 0.13 470.13 7.22 ( 0.16-2.41 ( 0.04 9.63 ( 0.16 All chemicl potentil clcultions re from DPD. The nonpolr cvity rdius is R c ) 1.82 Å. Figure 3. Schemtic representtion of the vrious size prmeters used in this work: d ij is the imum distnce between two molecules i nd j, r ii is the contribution to the imum distnce due to molecule i, nd R c is the cvity rdius. Furthermore, the d wter-methne vlue shows tht the prtil chrges in the solvent molecule hve very wek effect on the hrd core distnce between nonpolr molecule nd wter Figure 4. Excess chemicl potentil (top) nd reduced cess chemicl potentil (bottom) of methne in wter: perimentl dt (line) 3 nd Monte Crlo simultion results using the method 9 (b) nd the DPD method 10 (O). Monte Crlo simultions re shown for the cvity formtion contribution (]) nd the energetic contribution (4) for ech digrm. The rdius of the nonpolr cvity is R c ) 1.79 Å. molecule. Note tht this is not to be pected for the cse of polr molecules dissolved in wter. From the bove rguments, it is cler tht the ccessible volume for nonpolr molecules dissolved in wter is much smller compred to the ccessible volume for polr molecules, nd thus, the insertion of nonpolr molecules requires lrger d vlues compred to the insertion of polr molecules of the sme size. In the literture, it hs been proposed to set the hrd core dimeter of the solute molecule equl to the Lennrd-Jones σ prmeter of the solvent. 15 Although such definition my be resonble for the cse of nonpolr solutes in wter, it is highly unphysicl for polr solute molecules, including wter itself. Discussion of Results. The infinite dilution chemicl potentil of methne in wter nd in ethne long the sturtion curve of the solvent ws clculted using the method 9 nd the DPD method. 10 In Tble 3 nd in Figure 4, perimentl dt nd simultion results re shown for µ nd βµ of methne in wter. The two different sets of clcultions gree with ech other within sttisticl uncertinty, nd the sttisticl errors re similr. Furthermore, simultion results re shown for the cvity,cv,en nd energy terms. Both βµ methne nd βµ methne show monotonic vrition with density, wheres their combintion results in mximum vlue for βµ methne. Moleculr simultion ws used to clculte the infinite dilution chemicl potentil of methne in light, nonpolr solvent, nmely ethne. Simultions were performed t the sme reduced temperture for ethne s tht for wter, long the sturtion curve of ethne. In this wy, direct comprison cn be mde between the queous solvent nd the orgnic solvent for the different structurl nd thermodynmic phenomen observed. In Figure 5, µ nd βµ of methne in ethne re shown t different tempertures from the Widom insertion
Henry s Constnt Anlysis for Wter nd Nonpolr Solvents J. Phys. Chem. B, Vol. 105, No. 32, 2001 7797 Figure 6. (O), βµ,cv solute (]), nd βµ,en solute (4) of methne (for rdius of the nonpolr cvity R c ) 1.79 Å) infinitely dilute in wter from DPD nd βµ (b), βµ,cv solute ([), nd βµ,en solute (2) of methne in ethne from t sturtion. T/T c is the reduced temperture with respect to the solvent criticl temperture from the moleculr model. Dshed lines connect the simultion points nd re drwn to guide the eye. Figure 5. Excess chemicl potentil (top) nd reduced cess chemicl potentil (bottom) of methne in ethne: Monte Crlo simultion results using the Widom insertion technique (0) nd the scheme 9 (b). Monte Crlo simultions re shown for the cvity formtion contribution ([) nd the energetic contribution (2) in ech digrm. Dshed lines connect the simultion points nd re drwn to guide the eye. method nd the method. Clcultions bsed on the use of the two methods re in very good greement. It should be noted tht, in the temperture rnge ed, is monotonic function of temperture. However, t higher tempertures pproching the criticl point of ethne, decreses with infinite slope t the criticl point. 24 Anlysis of the Hydrophobic Phenomenon. The seprtion of βµ into cvity formtion term nd dispersion energy term (eq 2) provides useful insight into the hydrophobic phenomenon, which is of immense importnce to biology, chemistry, nd physics. A number of perimentl nd theoreticl investigtions hve been published concerning the hydrophobic effect. Despite this effort, number of importnt issues relted to this phenomenon re still unresolved. It hs been climed tht the free energy for cvity formtion is much higher for the cse of wter compred to tht for n orgnic solvent either becuse of the smll size of the wter molecules, 14-16 which results in the formtion of severl smll cvities insted of few lrge ones, 25-28 or becuse of the ordering of wter molecules round the cvity. 17-19,25-28 To obtin more meningful results, comprisons for the different solvents (queous versus orgnic) should be mde for the sme thermodynmic conditions, i.e., comprison of the reduced chemicl potentil for the sme reduced temperture t sturtion. In this work, simultions for methne in the two solvents (wter nd ethne) were performed t sturtion for the sme reduced tempertures with respect to the solvent criticl temperture. In Figure 6, simultion results for, βµ,cv solute, nd βµ,en solute re presented. It is cler tht βµ,cv solute for nonpolr cvity in wter is close to tht for cvity of equl rdius in ethne for ll tempertures. There is difference in the temperture dependence of βµ,cv solute in wter nd in ethne (not very clerly shown in Figure 6) which is due to the unique temperture dependence of wter sturtion density. 27,28 Simultion results shown in Figure 6 verify tht the bsolute vlue of βµ,en solute is much lower for the cse of wter compred to the cse of ethne. This leds to very high vlues for nd, consequently, for H of methne in wter. In other words, simultion results support the ide tht intermoleculr interctions re, to lrge tent, responsible for the low solubility of methne in wter compred to tht of methne in n orgnic solvent such s ethne, under corresponding thermodynmic conditions. It should be emphsized here tht βµ,en solute is free energy term tht ccounts for both energetic nd entropic contributions. The reltively smll intermoleculr energetic interctions cn be esily plined by compring the Lennrd-Jones ɛ/k prmeter vlue used for the vrious components, which is mesure of the dispersion interctions: ɛ/k ) 74.5 K for wter, ɛ/k ) 148 K for methne, nd ɛ/k ) 98 K for ethne. Although polrizbility effects re not ccounted for plicitly in the model, it is not pected tht they would ffect these results considerbly.,en The difference in between methne dissolved in wter nd methne dissolved in ethne cn be plined lso on the bsis of the iceberg model. 12 According to this model, the highly structured wter molecules round centrl methne,en molecule result in lower entropic contribution to the (dissolution becomes less fvorble) compred to the cse of n orgnic solvent. Although the ct vlues for βµ,en solute nd βµ,cv solute depend on the definition of the hrd core rdius of the solvent, 14-15,17,25-28 Figure 6 clerly demonstrtes tht the energetic term is mostly responsible for the difference in vlues for the cse of wter solvent nd ethne solvent. Temperture Dependence of. From the nlysis of perimentl dt for methne in wter nd in n-hdecne, it ws mde cler tht the vrition of with temperture hibits mximum vlue, t lest for the cse where the solvent is t (or close to) sturtion. To e the effect of density, simultions for methne in wter were performed t different tempertures by keeping the wter density constnt t 1.062 g/cm 3. In Tble 5 nd Figure 7, simultion results re presented for, βµ,cv solute, nd βµ,en solute from the method. Interestingly, constnt density clcultions hibit monotonic increse for, unlike the clcultions long the solvent sturtion curve. This behvior supports previous mcroscopic
7798 J. Phys. Chem. B, Vol. 105, No. 32, 2001 Boulougouris et l. interctions between wter nd nonpolr solutes results in smll negtive βµ,en solute vlues tht re not sufficient to overcome the unfvorble highly positive βµ,cv solute vlues. This nlysis supports the rgument tht hydrophobic hydrtion is driven to lrge tent by the energetic interctions between wter nd nonpolr solutes. Figure 7. (b nd O), βµ,cv solute ([ nd ]), nd βµ,en solute (2 nd 4) of methne infinitely dilute in wter t sturtion conditions (open symbols) nd t constnt density (full symbols). The rdius of the nonpolr cvity is R c ) 1.82 Å. clcultions by Prusnitz nd co-workers. 29 Furthermore, β µ,en solute for given temperture is prcticlly independent of the density. As pected, the difference is considerble in the cse of the cvity formtion term. The free energy of cvity formtion is strong function of the density nd is prcticlly independent of the temperture, t lest for the temperture rnge ed. In generl, βµ,cv solute nd βµ,en solute vry monotoniclly with density. For the cse in which clcultions re performed t solvent sturtion, the temperture dependence of βµ,en solute dotes the temperture dependence of t low tempertures. As temperture increses, the cvity formtion term becomes progressively more importnt s result of the considerble density decrese nd so goes through mximum nd then decreses. Conclusions A thorough nlysis of perimentl dt for the Henry s constnt of methne in wter nd in n-hdecne with respect to n idel nd n cess contribution ws presented. It ws shown tht mcroscopic models such s cubic EoS predict dequtely the Henry s constnt of light gses in hevy nonpolr solvents, s, for mple, in the cse of methne in n- hdecne. However, in the cse of queous solvents, EoS re incpble of predicting ccurtely the perimentl dt over the entire temperture rnge. Moleculr simultion ws used to evlute the free energy of cvity formtion nd the free energy of solute prticle to cvity trnsformtion for the cse of methne in wter nd of methne in ethne. These two contributions provide the cess Henry s constnt. It ws shown tht the free energy of cvity formtion for cvity sizes on the order of the size of methne molecule in wter nd in ethne solvents t the sme reduced temperture is surprisingly close for the cse of nonpolr cvity in wter nd tht of n equl rdius cvity in ethne. For polr cvity, the cost is lower becuse wter molecules pper to interct with polr molecules with significntly smller hrd core dimeter. The lck of strong energetic Acknowledgment. G.C.B. would like to cknowledge the support of the Europen Commission through Trining nd Mobility of Reserchers (TMR) Grnt No. ERB FMGE CT950051 (the Trining nd Reserch n Advnced Computing Systems (TRACS) Progrmme t Edinburgh Prllel Computing Centre (EPCC) nd the centrl computing center of the Ntionl Technicl University of Athens, Greece. Professor J. M. Prusnitz, University of Cliforni t Berkeley, is cknowledged for helpful discussions nd suggestions. References nd Notes (1) Tsonopoulos, C.; Wilson, G. M. AIChE J. 1983, 29, 990. (2) Prusnitz, J. M.; Lichtenthler, R. N.; de Azevedo, E. G. Moleculr Thermodynmics of Fluid-Phse Equilibri, 3rd ed.; Prentice Hll: Englewood Cliffs, NJ, 1999. (3) Hrvey, A. H. AIChE J. 1996, 42, 1491. (4) Schulze, C.; Donohue, M. D. Fluid Phse Equilib. 1998, 142, 101. (5) Sove, G. Chem. Eng. Sci. 1972, 27, 1197. (6) Kontogeorgis, G. M.; Voutss, E. C.; Ykoumis, I. V.; Tssios, D. P. Ind. Eng. Chem. Res. 1996, 35, 4310. (7) Hung, S. H.; Rdosz, M. Ind. Eng. Chem. 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