Margules Equations Applied to PAHSolubilities in Alcohol-WaterMixturesC H I H H A O F A N † A N D C H A D T . J A F V E R T *

School of Civil Engineering, Purdue University,West Lafayette, Indiana 47907

Solubilities of phenanthrene, pyrene, and perylene weredetermined in aqueous solutions containing either methanol,ethanol, or propanol at alcohol volume fractions rangingfrom zero to nearly 1, at 25 °C and atmospheric pressure.These data, and data from the literature on naphthalenewere compared to various forms of two- and three-suffixMargules equations and UNIFAC predictions. For eachsolute-solvent-cosolvent (i.e., PAH-water-alcohol) systemexamined, the solubility (S) of each PAH increased withincreasing alcohol volume fraction (v3). The UNIFAC modelpoorly estimated solubilities at cosolvent volume fractionsnear 0.5. Analysis of the data with various forms of theMargules equations shows that, in addition to solute-solventand solute-cosolvent interaction terms, inclusion ofsolvent-cosolvent (i.e., water-alcohol) and solute-solvent-cosolvent interaction terms in the model improves predic-tions, particularly at cosolvent volume fractions near 0.5. Asimplified three-suffix Margules model is proposed, whichincludes terms for all these interactions and accuratelyestimates the experimental data with a consistent set ofinteraction parameters. All sets of interaction parametersinvolving solute molecules were regressed against eachsolute’s respective log octanol-water partition coefficient,Kow. Solubilities of other PAH compounds in similaralcohol-water mixtures, obtained from the liturature, wereestimated accurately with these relationships and knowl-edge of each solute’s Kow value and enthalpy of fusion.

IntroductionAlgorithms that estimate chemical activity coefficients internary or more complex solutions may be grouped into twobasic types: (i) those that model interactions betweenmolecules at the molecular level and (ii) those that modelinteractions at the molecular-functional group level. Infunctional group-contribution algorithms, the chemicalactivity coefficient is calculated as a function of all possiblepairwise functional group interactions. Although such modelsmay be quite universal and are generally based on the samefundemental concepts as molecular level models, they requirea large training set to properly estimate the interactionparameters for all paired functional groups within the data.The UNIFAC (UNIQAC functional activity coefficient) modelis an example of this type of model.

Examples of molecular level models are the Margulesequations for binary or ternary systems (1). Although not asuniversal as functional group contribution models, these

empirical thermodynamic models often provide valuableinsight into molecular-scale interactions. In this study, weutilize both types of models to estimate solubilities of severalPAH compounds, comparing these estimates to experimentaldata. For this purpose, solubilities of phenanthrene, pyrene,and perylene were measured in methanol-, ethanol-, andpropanol-water mixtures at 25 °C and at room pressure. Inaddition, solubility data on naphthalene from the literature(2) are used to test the various models. We briefly reviewrelevant equations pertaining to ideal and real solutions,organize the perspective algorithms into the various formsof Margules equations that are tested with the experimentaldata, and report on the experimental measurements andmodel suitability for these data. In addition, the final formof the Margules equation was tested with alcohol-watersolubilities of chrysene, anthracene, and toluene. These datawere not used in the model’s construction or in theconstruction of several empirical correlations that were usedto estimate most interaction parameters involving thesesolutes.

TheoryIdeal Solubility. For compounds that are solids at a definedtemperature, the chemical potential of the solute moleculesin solution, at saturation, equals the chemical potential ofthe crystalline solute. From this equality, the ideal gas law,and the definition of Gibbs free energy, the ideal law ofsolubility (eq 1) may be derived for any solute i (1, 3-5):

where Si is the mole fraction solubility of solute i; ∆Hi,fusion

and ∆Si,fusion are the enthalpy and entropy of fusion, respec-tively; T is the temperature; Tm is the melting point of thesolute; and R is the universal gas constant. Equation 1assumes that the compound’s melting temperature is ap-proximately equal to its triple point temperature and that theheat capacities of the solid and liquid solutes are ap-proximately equal. Thermodynamically, dissolution of a solidinto a solvent is equivalent to melting of the solid followedby mixing of the resulting liquid solute molecules with thesolvent molecules. Equation 1, however, only considers thethermodynamics of melting: No terms account for inter-molecular solvation forces. Hence, eq 1 predicts that solubilityis independent of the solvent and that solubility increaseswith increasing temperature below the melting point. Table1 (6-17) reports the enthalpies and entropies of fusion andsolubilities of those PAHs investigated, as well as other usefulinformation.

Solubility in Real Solutions. For any real solution,intermolecular solvation forces are accounted for within theactivity coefficient of species i (γi). The activity coefficientequals the reciprocal mole fraction solubility, Si, multipliedby a conversion factor. This conversion factor accounts forsolute transformation from the solid phase to the corre-sponding supercool liquid phase at the specified temperature(1, 18):

where P° denotes the vapor pressure and (s) and (L) denotesolid solute and supercool liquid solute, respectively. Analgorithm from which the ratio P°(s)/P°(L) may be calculated

* Corresponding author phone: 765-494-2196; fax: 765-496-1107;e-mail: jafvert@ecn.purdue.edu.

† Present address: Water Resources Bureau, Ministry of EconomicAffairs, Taipei, Taiwan.

ln Si ) -∆Hi,fusion

R (1T

- 1Tm

) )∆Si,fusion

R (1 -Tm

T ) (1)

γi ) 1Si

P°(s)

P°(L)(2)

Environ. Sci. Technol. 1997, 31, 3516-3522

3516 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 31, NO. 12, 1997 S0013-936X(97)00261-7 CCC: $14.00 1997 American Chemical Society

from other thermodynamic data is provided by Prausnitz etal. (1):

Combining eqs 2 and 3 and rearranging yields

Activity coefficients are directly related to excess statefunctions. An excess state function is defined as the differencebetween the state function of the real solution and thecorresponding state function of the ideal solution andindicates relative deviation from ideal behavior. For phase-equilibrium calculations, the most useful excess functionrelated to the activity coefficient is the partial excess Gibbsfree energy, gi

E:

where nT and ni are the mole numbers of the mixture andcomponent, respectively, and gE is the molar excess Gibbsfree energy of the total system. This equation is directlyapplied later in the development of the Margules equations.

UNIFAC. In the UNIFAC model, the activity coefficientof any component in the system is calculated as the productof two terms: a combinatorial term γcombinatorial, which resultsfrom entropy of mixing, and a residual term γresidual, whichresults from the enthalpy of mixing (1):

In this model, the Flory-Huggins athermal equation (1, 19-21) is used to calculate the excess entropy of mixing betweenall functional groups of all component compounds, and thesum of these values equals γcombinatorial. The excess enthalpyof mixing between all functional groups of all componentcompounds is calculated with the Wilson equation, and thesum of these values equals γresidual. Hence, enthalpy andentropy are calculated independently, and both with equa-tions which have in their development assumed that thecontribution of the other (enthalpy or entropy) to the totalexcess free energy is zero. UNIFAC also assumes that allinteractions are pairwise (i.e., three-way and higher-orderinteractions are negligible).

The Margules Equations. Although mole fraction (x) unitswere employed in the original derivation of Margules equa-tions, we derive the appropriate equations here with volumefraction (v) units due to the large disparity in the molecularsizes of those molecules considered. The choice of volumefraction units is similar to the application of effective volumefractions in Wohl’s equation, yet the advantage over effectivevolume fractions is that the number of unknown or fittingparameters is not increased. For mixtures of two componentsranging in volume fraction values from zero to one, the

following two boundary conditions must be satisfied if eachpure solution is considered as an ideal solution: (1) gE ) 0at v1 ) 0, and (2) gE ) 0 at v2 ) 0. The simplest empiricalexpression that obeys these boundary conditions is

where v1 and v2 are the volume fraction concentrations ofcomponents 1 and 2, respectively; the constant A in eq 7 isthe interaction parameter between component moleculetypes 1 and 2. Substituting eq 7 into eq 5 (on a volume basis)and taking the derivative with respect component i ) 1 or 2,results in the volume fraction-based two-suffix Margulesequations (1):

The above empirical method may be extended to morecomplicated systems. Expanding to a three-componentsystem (solute, solvent, and cosolvent) and assuming onlytwo-component interactions occur results in

Again, substituting eq 10 into eq 5 (on a volume basis) andtaking the derivative with respect to species 1 results in

where, in our case, the subscripts 1, 2, and 3 refer to thesolute (PAH), solvent (water), and cosolvent (alcohol) and A,B, and C are interaction parameters between solute-solvent,solute-cosolvent, and solvent-cosolvent molecules, respec-tively. These parameters may be obtained from experimentaldata on binary systems (1).

We refer subsequently to eqs 10 and 11 as a modifiedtwo-suffix model for a ternary system (modified in the sensethat vi values rather than xi values are used). Combining eqs4 (for component 1) and 11 results in (dropping the subscript1 on S):

Parameter A may be determined directly from the solubilityin pure solvent (v3 ) 0), parameter B may be determinedfrom the solubility in pure cosolvent (v2 ) 0), and parameterC may be determined from vapor pressure data on solvent-cosolvent solutions.

Straight Line Log S - Arithmetic v3 Model. In dilutesolutions, the volume fraction of the solute is negligible ascompared to solvent and cosolvent volume fractions:

TABLE 1. Properties of Perylene, Pyrene, Phenanthrene, and Naphthalene

PAH perylene pyrene phenanthrene naphthalene

molecular weight 252.32 202.26 178.24 128.12entropy of fusion ∆Smelt,Tm (cal/mol K) 10.2a 9.6b 10.5c 13.1d

entropy of fusion ∆Hmelt,Tm (kcal/mol) 5.62a 4.45e 4.56c

octanol-water partition coefficient, log Kow 6.50f 5.22g 4.46h 3.37i

melting point (K) 550f 423j 374k 353e

measured S (mg/L) 0.00021 0.135 0.703 30l

a Tsonopoulos and Prausnitz (6). b Casellato et al. (7). c Wauchope and Getzen (8). d Ubbelohde (9). e Parks and Huffman (10). f Yalkowsky andValvani (11). g Hutchinson et al. (12). h Leo et al. (13). i Chiou et al. (14). j Cleland and Kingsbury (15). k Weast (16). l Gunther et al. (17).

lnP°(s)

P°(L))

∆Smelt,Tm

R (1 -Tm

T ) (3)

ln Si ) -ln γi +∆Si,fus

R (1 -Tm

T ) (4)

∂(nTgE)

∂ni) gi

E ) gi,real - gi,ideal ) RT ln γi (5)

ln γi ) ln γi,combinatoial + ln γi,residual (6)

gE

RT) Av1v2 (7)

ln γ1 ) Av22 (8)

ln γ2 ) Av12 (9)

gE

RT) Av1v2 + Bv1v3 + Cv2v3 (10)

ln γ1 ) Av22 + Bv3

2 + (A + B - C)v2v3 (11)

Av22 + Bv3

2 + (A + B - C)v2v3 ) -ln S -∆Smelt,Tm

R (Tm

T- 1)(12)

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Assuming negligible interactions between solvent and co-solvent molecules occur, eq 11 simplifies to

where again the subscripts 1, 2, and 3 refer to solute, solvent,and cosolvent, respectively. If the volume fraction of thecosolvent, v3, equals zero, the interaction parameter A is easilycalculated with eq 12, where v2 ) 1 and S is the solute’ssolubility in the pure solvent. Similarly, if v2 ) 0, B may becalculated from the solute’s solubility in the pure cosolventphase. Combining eqs 4, 14, and 12 (at both limits) resultsin

where S2 and S3 are the solubilities in pure water and alcoholphases, respectively. Combining with eq 13 and writing inbase-10 log form results in

where S2 and S3 are the solubilities in the pure solvent (water)and cosolvent (alcohol), respectively. This equation definesa linear relationship between logarithmic solubility andarithmetic cosolvent volume fraction. It was first proposedby Yalkowsky (22) and has been applied to other solubilitydata (23, 24); however, it has not previously been derivedthrough this appoach. In these references, this simple modelis referred to as a “log-linear” model.

Materials and MethodsPerylene (>99.5%), phenanthrene (>98%), and reagent-gradeethanol were purchased from Aldrich Chemical Company.Pyrene (99%) was obtained from the EPA’s Pesticides andIndustrial Chemical Repository. All were used as receivedwithout further purification. Methanol was purchased fromFisher Scientific. Acetonitrile (ACN), the mobile phase inHPLC analysis, was purchased from Mallinckrodt Chemical.The water to prepare all solutions and to perform allexperiments was first deionized by H+ and OH- ion exchangeand subsequently triple-distilled in glass. Mixed solventsolutions at desired cosolvent concentrations were preparedin 50-mL Pyrex glass test tubes equipped with Teflon-linedcaps for 24 h prior to experiments. For each solubility datumpoint, a mass of solid PAH and a volume of mixed solventwere added to a 25-mL Corex centrifuge tube and aged at 25°C within a constant temperature room for 1 week on a wrist-action shaker. Samples were prepared in duplicate. Selectedtubes sampled with time confirmed that 1-week equilibrationwas sufficient. On day 7, all tubes were visually examined toconfirm the presence of excess solid. Tube were centrifugedat 9000 rpm for 2 h in a Sorvall SA-600 rotor. A 2-mL portionof the supernatant was transferred to a second tube with aglass autopipet, rinsing the pipet with 2 mL of ACN to thesecond tube. These samples (1:1 water-ACN volume ratio)were analyzed by reverse-phase high-pressure liquid chro-matography (HPLC) with 100% ACN as the mobile phase.Sample peak areas were compared to those of PAH standardsdissolved in 1:1 water-ACN (v/v). The HPLC consisted of aWaters 600 pump and a SLM Amino Bowman Series 2luminescence spectrometer, equipped with a Hellma flow-through cell (no. 176.353-QS, 8 µL chamber volume) and afocusing lens within the path of the excitation light. Thecolumn was a 25 cm × 4.6 mm LC-PAH column purchasedfrom Supelco. The configuration of the fluorescence detectorwas as follows: sensitivity: 650, 500, and 650 V for perylene,pyrene, and phenanthrene, respectively; refresh period: 0.5s; band width: 8 (for both emission and excitation). For

perylene, pyrene, and phenanthrene, the emission andexcitation wavelengths were 475 and 443 nm, 390 and 342nm, and 400 and 357 nm, respectively. All reported valuesare averages of two values, and all duplicates were within10% of each other with most within 5%. All numerical dataare reported in ref 25.

ResultsUNIFAC. The base-10 log solubilities (mg/L) of naphthalene(2), phenanthrene, pyrene, and perylene in alcohol-watermixtures are reported in Figure 1. Also shown on this figureare UNIFAC model predicted solubilities, calculated withgroup interaction parameters reported by Hansen et al. (26).Note, that for both UNIFAC and Margules equation calcula-tions, units on S are dimensionless mole fractions, and areconverted to milligrams per liter only for graphical presenta-tion. Also, the lines from top to bottom on each graphcorrespond to predicted solubilities in propanol, ethanol, andmethanol as the cosolvent. From v3 ) 0 to v3 ) 1, S increasesby 3.3, 4.4, 4.1, and 5.5 orders of magnitude for naphthalene,phenanthrene, pyrene, and perylene, respectively, in eachwater-alcohol system. For each solute dissolved in the purealcohol phases, S increases only slightly through the seriesmethanol < ethanol < propanol. Similarly, deviation froma straight-line log-arithmetic function increases through theseries methanol < ethanol < propanol. As the hydrophobicityof the solute increases through the series naphthalene <phenanthrene < pyrene < perylene, the deviation (curvature)also increases. Hence, the greatest curvature occurs forperylene in propanol-water mixtures. The UNIFAC predic-tions are generally within an order of magnitude of theexperimental data; however, some values deviate by 2 ordersof magnitude, especially at alcohol volume fractions near0.5. The curvature in the ethanol and propanol data are notpredicted by the UNIFAC model.

Margules Equations. Various forms of the Margulesequations were evaluated with the experimental data pre-sented in Figure 1 requiring estimates for all interactionparameters within each model. Parameters A and B, for eachPAH and for each alcohol (B only), were determined from themeasured solubilities in the pure water (v3 ) 0) and alcohol(v2 ) 0) phases, respectively, with eq 12 (i.e., volume fractionbasis). Values for parameter C were determined from alcoholvapor pressure measurements above water-alcohol mixturesreported by Butler et al. (27). Recall that C is independentof solute. The data (alcohol activity coefficients) reported byButler et al. are reproduced in Figure 2 along with activitycoefficients (the regression line) calculated with

where a3 is the activity of the alcohol, x3 is the alcohol molefraction, and other terms are as previously defined. The valueof C was obtained by regressing ln γ3 versus v2

2 for eachalcohol, forcing the regression line through the origin. Thesevalues and those for A and B are reported in Table 2. Althougheq 17 overpredicts ln γ3 at high values of v2

2 and underpre-dicts ln γ3 at low values of v2

2, the use of mole fraction units

in place of volume fraction units results in similar discrep-ancies in exactly the opposite direction. The curvature of theln γ3 measurements is accurately estimated with an alternatemodel that invokes two adjustable parameters, C1 and C2,rather than one parameter:

ln γ1 ) C1v22 + 2C2v2

2v3 (19)

v2 + v3 ) 1 (13)

ln γ1 ) Av2 + Bv3 (14)

ln S ) v2 ln S2 + v3 ln S3 (15)

log S ) log S2 + v3(log S3 - log S2) (16)

lna3

x3) Cv2

2 ) ln γ3 (17)

gE

RT) C1v2v3 + C2v2v3

2 (18)

3518 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 31, NO. 12, 1997

The dashed lines on Figure 2 are represented by this model(i.e., eq 19) with the values for C1 and C2 reported in Table2 for each alcohol.

Returning to the PAH solubilities, a straight line con-structed from the solubility in pure water to that in the purealcohol phase on a log-arithmetic scale is defined by eq 16(C ) C1 ) C2 ) 0). This line, referred to as the two-parametermodel, is shown on Figure 3 with the experimental values forperylene in propanol-water mixtures. Note that these dataexhibit the most curvature of all reported data. Includingthe term involving C within the model (eq 12) improvespredictions, as shown by the line on Figure 2 labeled threeparameters (A, B, and C), indicating that the contribution tothe total excess free energy due to nonideal alcohol and waterinteractions is significant. Pinal et al. (28) arrived at a similarconclusion after applying the straight-line log-arithmetic (two-parameter) model to other solubility data. Including the termsinvolving C1 and C2 (i.e., four parameters, A, B, C1, and C2)results in nearly the same line, suggesting that the nonlinearity

FIGURE 1. Experimental and UNIFAC calculated PAH solubilities inalcohol-water mixtures at 25 °C. The naphthalene data are fromFu and Luthy (2).

FIGURE 2. Experimental and calculated (line) activity coefficientsof methanol, ethanol, and propanol as a function of v2

2 in binarysolutions at 25 °C. The data are from Butler et al. (27).

TABLE 2. Interaction Parameters for PAH, Water, and AlcoholMixtures

solvent orcosolvent

solute orcosolvent A B D (w/water)

water perylene 20.74pyrene 16.29phenanthrene 15.13naphthalene 11.09

methanol perylene 7.38 -2.58pyrene 6.00 -0.60phenanthrene 4.20 1.28naphthalene 2.61 1.96

ethanol perylene 6.77 -5.48pyrene 5.42 -4.12phenanthrene 3.72 -1.57naphthalene 2.00 1.11

propanol perylene 6.31 -8.49pyrene 4.70 -7.80phenanthrene 3.68 -5.58naphthalene 1.36 -1.83

solvent orcosolvent

solute orcosolvent C C1 C2

water methanol 0.69 0.44 0.52ethanol 2.02 1.10 1.71propanol 3.77 2.30 2.50

VOL. 31, NO. 12, 1997 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 3519

of ln γ3 versus v22 is of secondary importance in estimating

these solubilities. In either case, the predictions remain lessthan satisfactory, suggesting that the additional excess freeenergy must be contributed at least partially by the solute.A simple alteration to the model that invokes an additionalsolute contribution is the inclusion of an interaction terminvolving all components:

where D is a solute-solvent-cosolvent interaction parameter,which is different for each system. Equations 20 and 21 area simple form of the three-suffix Margules equations. Becausethe binary alcohol-water interaction term involving Cprovides approximately the same result as those involving C1

and C2 in all cases, this term is used for simplicity (the limitingcase is shown subsequently). Again, it is reasonable to assumethat v1 is negligible:

Combining eqs 4 and 22 results in

The experimental data again are reported in Figure 4, wherereported also are solubilities calculated with eq 23 (i.e., asimplified three-suffix model) and those interaction param-eters reported in Table 2. Note again, that for all calculations,units on S are dimensionless mole fractions and are convertedto milligrams per liter only for graphical presentation. A valuefor D was determined for each data set with eq 23 by nonlinearleast squares analysis, with the values of A, B, and C previouslydetermined. The addition of parameter D makes it possibleto capture the curvature in the experimental results with aconsistent set of interaction parameters. Inclusion of D inthe model is most significant for the solute, perylene, inpropanol-water mixtures: This line (four parameters: A, B,C, and D) is shown also on Figure 3.

Calculations made with C versus C1 and C2 should displaythe largest discrepancies for the case of naphthalene in

propanol-water mixtures, as the least excess free energy (i.e.,highest solubility) exists for naphthalene and the nonlinearityof ln γ3 versus v2

2 is the largest for propanol.

FIGURE 3. Experimental and calculated solubilities of perylene inpropanol-water mixture.

gE

RT) Av1v2 + Bv1v3 + Cv2v3 + Dv1v2v3 (20)

ln γ1 ) Av22 + Bv3

2 + (A + B - C + D - 2Dv1)v2v3 (21)

ln γ1 ) Av22 + Bv3

2 + (A + B - C + D)v2v3 (22)

Av22 + Bv3

2 + (A + B - C + D)v2v3 )

-ln S -∆Smelt,Tm

R (Tm

T- 1) (23)

FIGURE 4. Experimental and simplified three-suffix model calculatedPAH solubilities in alcohol-water mixtures at 25 °C. The naphthalenedata are from Fu and Luthy (2).

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The naphthalene-propanol data are again shown onFigure 5 with lines calculated with those values of A, B, C, andD and A, B, C1, C2, and D reported in Table 2. As with perylene’ssolubility in proponal-water mixtures, predictions with C1

and C2 are only slightly higher than those with C at alcoholvolume fractions near 0.5.

DiscussionSolute-(Co)Solvent Interactions (A and B). A and B aredirectly proportional to -ln S (eqs 12 or 23) for each of therespective pure phases. Therefore, as A and B increasethrough the series naphthalene < phenanthrene < pyrene <perylene, solubility decreases in water and each alcohol,respectively (i.e., excess Gibbs free energy, gE, increases). AsB increases through the series (for each solute) propanol <ethanol < methanol solubility again decreases (i.e., gE

increases). Also, -log S is proportional to a solute’s octanol-water partition coefficient, Kow (29):

where Vh*o is octanol’s molar volume corrected for octanol-water mutual saturation; γ*o and γ*w, are the solute’s activitycoefficients in mutually saturated octanol and water phases,respectively; and γw is the solute’s activity coefficient in purewater. The middle three terms are independent or nearlyindependent of solute. Hence, an equation relating A to log

Kow may be constructed:

Regression of A versus log Kow results in (R 2 ) 0.98)

Because log solubilities for PAHs in liquid alcohols correlateto their log water solubilities, a similar relationship may beproposed for B:

where the calculated values of R and â are provided in Table3 for each alcohol.

Solvent-Cosolvent Interactions (C). The work of Butleret al. (26) and the calculated values of C indicate that theexcess Gibbs free energy in alcohol-water mixtures increasesthrough the series: methanol < ethanol < propanol. Theincreasing aliphatic chain length likely results in a decreasingexcess entropy and increasing excess enthalpy; however, therelative contribution of each to the total excess free energyat each alcohol-water ratio is unknown.

Solute-Solvent-Cosolvent Interactions (D). D increasesthrough the series: perylene < pyrene < phenanthrene <naphthalene. Equations containing alternate terms (e.g.,Dv1

2v3) may capture the trends in the experimental dataequally well; however, we have not evaluated such terms, asthe suggested form accurately represents the data. Althoughno theoretical basis is proposed, the values of D were regressedversus solute log Kow values:

where R and â are provided in Table 4 for each alcohol. Thedata for methanol and ethanol appear quite linear (R 2 ) 0.95in both cases); however, the propanol data are nearly parabolicrather than linear with an exponential increase in D withdecreasing log Kow, resulting in a weak correlation (R 2 ) 0.88).For propanol, eq 28 underestimates D at extreme values oflog Kow (<3.5 and >6.0) and overetimates between thesevalues.

Model Evaluation. The basic model (eq 23) and associatedalgorithms were tested with solubility data not used in theirconstruction. Solubilities of anthracene in ethanol-watermixtures and of chrysene in methanol-water mixtures werecalculated with eqs 23 and 26-28, those values of C reportedin Table 2, and the respective log Kow values of 4.45 and 5.61.These values are compared to measured values reported byYalkowsky (30) in Figure 5, indicating that for these similarPAH compounds quite accurate predictions result. For bothsolutes, all estimated interaction parameters are within therange of those used in the construction of the log Kow

regression lines.

FIGURE 5. Experimental and calculated solubilities of several PAHsin binary solvents. The anthracene, chrysene, and toluene data arefrom Yalkowsky (30), and the naphthalene data are from Fu andLuthy (2), all determined at 25 °C.

log Kow ) -log S - log Vh*o - log γ*o + log(γ*2γw

) +

∆Smelt,Tm

R (1 -Tm

T ) (24)

TABLE 3. Regression Coefficients for Interaction Parameter B

cosolvent r â R 2

methanol 1.56 -2.67 0.97ethanol 1.55 -3.20 0.98propanol 1.44 -3.00 0.99

TABLE 4. Regression Coefficients for Interaction Parameter D

cosolvent r â R 2

methanol -1.51 7.42 0.95ethanol -2.16 8.03 0.95propanol -2.14 4.53 0.88

A ) 2.303 log Kow + constant (25)

A ) 2.74 log Kow + 2.63 (26)

B ) R log Kow + â (27)

D ) R log Kow + â (28)

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Also shown on Figure 5 are measured solubilities of toluenein methanol-water mixtures from Yalkowsky (30). Toluenesolubilities estimated with the value of D () 3.36) calculatedwith eq 28 do not agree with the experimental data; however,this value lies outside the range of values used to constructeq 28. Therefore, D was used as a fitting parameter; A wascalculated directly from aqueous solubility; and becausetoluene and methanol are completely miscible, B was usedalso as a fitting parameter. The solid line on Figure 5 is theresult where A ) 9.12, B ) 0.86, C ) 0.69, and D ) 6.27. Alsoshown as a dashed line are calculated solubilities replacingthe C term with those containing C1 and C2. In this case, Aand B remain the same and D ) 6.17, indicating again thatthe nonlinearity of ln γ3 with v2

2 is of secondary importancein estimating these solubilities. The final value of D fortoluene-methanol-water suggests that all plots of D versuslog Kow are parobolic and not linear as implied by eq 28.Additional data are necessary to confirm this.

In summary, we have shown that all the data presentedin this study can be estimated with some accuracy with thesimplified three-suffix model (eq 23), invoking the value ofD as the only fitting parameter. Parameters A and B arederived directly from thermodynamic data (i.e., pure phasesolubilities, melting points, and entropies of fusion) or maybe estimated from Kow. Parameters C or C1 and C2 arecalculated direction from solvent-cosolvent informationalone (i.e., partial pressures). Furthermore, the calculatedvalues of D for each solute decrease as solute hydrophobicityincreases, suggesting that reasonable estimates of D can bemade for PAHs whose solubilities are within the range ofthose studied herein.

AcknowledgmentsFunding for this research was provided by the U.S. NationalScience Foundation (Grant BCS-9223656) to Purdue Uni-versity. The authors wish to thank Linda S. Lee and LarryNies for helpful discussions regarding this work.

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Received for review March 21, 1997. Revised manuscriptreceived August 5, 1997. Accepted August 7, 1997.X

ES970261H

X Abstract published in Advance ACS Abstracts, October 1, 1997.

3522 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 31, NO. 12, 1997