Solution Models for Binary Components of Significantly Different Molecular Sizes

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Solution Models for Binary Components of SignificantlyDifferent Molecular Sizes

Wei-Nung Hung • Tsair-Fuh Lin • Cary T. Chiou

Received: 20 March 2012 / Accepted: 23 January 2013 / Published online: 1 August 2013! Springer Science+Business Media New York 2013

Abstract As a solution theory, Raoult’s law is commonly used to estimate the activitiesof solutes and solvents of comparable molecular sizes while the Flory–Huggins (F–H)model is used for the activities of small liquids in high polymers. For a great many systemswhere the solute and solvent differ only moderately in molecular size (e.g., by 4–10 times),there has been no confirmed choice of a preferred model; examples of such systems arethose of ordinary organic compounds in liquid triolein (MW = 885.4 g!mol-1) andpoly(propylene glycol) (PPG) (MW = *1,000 g!mol-1). The observed nearly athermalsolubilities of many nonpolar organic solids in these solvents provide unique experimentaldata to examine the merit of a solution model. As found, Raoult’s law underestimateswidely, and the F–H model underestimates slightly, the solid solubilities in triolein andPPG because these models underestimate the solution entropy for these solute–solventpairs. To rectify this problem, the molecular segments of a large sized liquid solvent (e.g.,triolein) are assumed to act as independent mixing units to increase the solute–solventmixing entropy. This adjustment leads to a modified F–H model in which the ‘‘ideal’’ or‘‘athermal’’ solubility of a solid in volume fraction, at a particular temperature, is equal tothe solid’s activity at that temperature. Results from other studies give further support forthe modified F–H model to interpret the partition data of compounds with organic solvents.

Keywords Solution models ! Activity ! Entropy ! Molecular size ! Raoult’slaw ! Flory–Huggins model ! Modified Flory–Huggins model

W.-N. Hung (&) ! T.-F. Lin ! C. T. Chiou (&)Department of Environmental Engineering and Sustainable Environment Research Center,National Cheng Kung University, Tainan 70101, Taiwane-mail: [email protected]

C. T. Chioue-mail: [email protected]

C. T. ChiouDenver Federal Center, U.S. Geological Survey, Denver, CO 80225, USA

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J Solution Chem (2013) 42:1438–1451DOI 10.1007/s10953-013-0041-7

Author's personal copy

1 Introduction

The distinct partition effects of given organic solutes with various organic phases (e.g.,solvents, mineral oils, lipids, and amorphous polymers) are attributed to their unique solutesolubilities in different organic phases [1]. In a solution, the compatibility of a solute with asolvent is determined by the proximity of the solution property to some reference state. Themost useful reference state appears to be the ‘‘ideal (or perfect) solution’’ state or the‘‘athermal solution’’ state, where the solute and solvent mix to yield the highest entropy (orrandomness) without absorbing or releasing heat. A rigorous account of the mixing entropyis therefore crucial to the establishment of the ideal/athermal reference states. Computationof the entropic factor for a solution becomes complicated, however, when the molecularsizes and structures of the components differ substantially from each other.

In Raoult’s law [2], the entropy of solution for a solute–solvent pair is estimated byassuming the entire solute and solvent molecules act as separate mixing entities, and hencethe activity of a component in solution is weighted by its mole fraction. On the other hand,the Flory–Huggins (F–H) model [3–6], which was formulated chiefly for the activity of anordinary (small sized) liquid with an amorphous polymer, estimates the entropic contri-bution of a polymer by treating the repeating segments of the molecule as relatively freebut not independent mixing units, i.e., the spatial configuration of the segments from apolymer molecule is restricted to a certain extent by the polymer’s molecular network. Fora small sized liquid in a high polymer, the F–H model rectifies the vastly overestimatedactivity from Raoult’s law, as exemplified by the measured activity of benzene in naturalrubber [7].

In general, Raoult’s law performs well only when the solute and solvent molecules areclosely similar in their sizes and structures [3, 4, 8, 9]. Although the F–H model is clearlysuperior to Raoult’s law for describing the activity of a common liquid in a polymersolution [3–7], the situation for systems where the solute and solvent differ only moder-ately in molecular size (e.g., by about 4–10 times) is not well settled. A good example ofsuch a system is a solution of an organic compound in the lipid triolein, the latter being aliquid at room temperature with a molar volume (0.976 L!mol-1) that is about 4–10 timesthose of common organic compounds [9]. In this case, although the F–H model accountsmore effectively than Raoult’s law for the solubilities of nonpolar solids in triolein [9], theobserved solubilities of some solids (e.g., naphthalene) fall very close to, and in some casesslightly above, the limiting athermal solubilities of the F–H model. This suggests that theactivity of an organic solute in triolein as defined by the conventional F–H model is onlyapproximately accurate despite its superior ability over Raoult’s law to reconcile thesolubility data.

Up to now, to our knowledge, no other model has been realized that is able to ade-quately handle the solute solubility data in triolein or other large sized liquid solvents.Resolution of this issue is important because accurate predictions of solute solubilities orpartition coefficients with many natural and man-made organic phases depend on properlydefined solute activity coefficients and related parameters [10]. In this respect, an analysisof the model applicability for systems involving a significant solute–solvent size disparitywas given earlier by Shinoda and Hildebrand [11, 12] utilizing solubility data for iodine(solute) in solvents of varying molecular sizes over a range of temperatures. In that study,the authors advocated that the entropies of solution of iodine with nonpolar quasi-sphericalsolvents comply well with the ideal solution law (i.e., Raoult’s law) independent of thesolvent–solute size ratio. However, for common binary component solutions of

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significantly unequal molecular sizes, more insightful data are needed for acquiring asatisfactory solution model.

The immediate objective of this study is to seek an improved solution model that inaccord with the activities of solutes in liquid solvents of much greater molecular sizes,including those mentioned earlier. Toward this objective, the present study examined themeasured solubility data of 25 organic solids in two relatively large sized liquid solvents,triolein and poly(propylene glycol) (PPG) at room temperature. The molecular size ratiosof selected solute–solvent pairs are about 4–10, a range found in many binary componentsystems. The molar heats of fusion of the solid compounds needed to compute theiractivities for analyzing the solubility models are taken from the literature.

2 Theoretical Considerations

To evaluate the capabilities of different solution models to account for the observed solidsolubilities in large molecular solvents, we start with two fundamental models: Raoult’slaw and the F–H model. By comparing the observed solubility data with the theoreticalpredictions of these two models, one should gain a clear picture of the model capabilities.If the experimental data deviate significantly from the model predictions, then the mag-nitudes and directions of the deviations provide helpful guides for adjusting the currentmodel(s) to improve their performance.

In Raoult’s law, the activity of a component (relative to its pure liquid state) is weightedby its mole fraction in solution. The ideal solubility (in mole fraction) of a solid compound(xo

id) in a solvent at a temperature below the solid’s melting point is numerically equal tothe pure solid activity (as) at that temperature [2]. It is obtained by setting the activitycoefficient of the dissolved solid at saturation equal to unity to give:

ln xoid " ln as " # DHf=R

! "Tm # T$ %=TTm& ' $1%

where DHf is the molar heat of fusion of the solid at T (J!mol-1), T is the systemtemperature (K), Tm is the solid’s melting point (K), and R is the gas constant

(8.31 J!K-1!mol-1). In Eq. 1, DHf is assumed to be constant between T and Tm and thus it

may be replaced by DHf at Tm, an approximation commonly made if the differencebetween solid and liquid heat capacities (DCp) is small or if T and Tm are not far apart;

otherwise, ln as at temperature T must be obtained by correcting DHf at Tm for the

difference between Tm to T using the DCp data [2]. With known DHf , Tm, and T,the activity of the solid (as) and its ideal solubility (xo

id) at T can then be determined. If thesolution of a solid (solute) in a solvent deviates significantly from the ideal state, then

the deviation is dealt with by an activity coefficient (cmf), i.e., cmf = xoid

#xo

ob, where xoob is

the observed solute’s mole fraction solubility. In the absence of molecular associationbetween a solute and a solvent, the solute cmf is usually C1, i.e., the solution exhibits onlypositive deviations from ideality. For a solid, the xo

id value is fixed at a given temperature,independent of the solvent; thus, the ideal solubility of a solid expressed in mole (or mass)per unit volume (or weight) must decrease with increasing molecular weight (or molarvolume) of the solvent, if the model is to hold.

In systems where the solvent is an amorphous polymer, the athermal solubility of a solidcompound, expressed in volume fraction of the dissolved solid in solution (uo

at), may besimilarly derived by equating the solute activity in a polymer solution [3–5] with that of thepure solid [9] to give:

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ln uoat ( up 1# V=Vp

! "$ %( vsu

2p " # DHf=R

! "Tm # T$ %=TTm& ' $2%

where V (L!mol-1) is the molar volume of the small sized solute, Vp (L!mol-1) is the

molar volume of the polymer, up (= 1 - uoat) is the volume fraction of the polymer, and vs

is an empirical parameter correcting for the excess entropic reduction in a solute–polymersolution due to a specific nonlinear polymer structure. For a highly linear polymer, vs = 0.Again, if the solubility of a solid in a polymer is less than athermal, the deviation may be

quantified by an activity coefficient (cfh), i.e., cfh = uoat

#uo

ob, where uoob is the observed

solute’s volume fraction solubility. As noted with Eq. 2, if the molar volume of a linear

(amorphous) polymer is very large, i.e., if V=Vp & 0, then the value of uoat of a solid

compound at a temperature in any linear polymer is largely independent of the molecularsizes of the solid (solute) and polymer.

For a small molecular solute at a given concentration in an amorphous polymer or amacromolecular solvent, Raoult’s law (Eq. 1) predicts a sharply higher solute activity thanthe F–H model (Eq. 2). Therefore, the ideal solubility of a small sized solid compoundgiven by Raoult’s law is sharply lower than the corresponding athermal solubility based onthe F–H model. In this study, we examine first the abilities of Eqs. 1 and 2 to accommodatethe observed solubilities of solid organic compounds in both liquid triolein and PPG. (Nopartially miscible organic liquids were selected for investigation because their equilibriumliquid phases with these solvents would be significantly impure to complicate the solubilityanalysis). An alternative activity model is introduced later to improve over Raoult’s lawand the F–H model for analyzing solubility data. The selected PPG has a molecular weight(MW) of about 1,000 g!mol-1, which is comparable in size to that of triolein(MW = 885.4 g!mol-1), while it has a moderately higher polarity than triolein.

With triolein and PPG as solvents, the measured solubilities of 25 organic solids (seeTable 1) form a comprehensive database for evaluating the model performances. To ensureaccuracies of the calculated solid activities (as), and thus the ideal or athermal solidsolubilities, all solids selected have only low to moderate melting points and relatively lowmolar heats of fusion at their melting points. Solids with high melting points ([150 "C) or

high molar heats of fusion (DHf [ 29 kJ!mol-1) have been excluded so that the calculatedsolid activities at room temperature will not be highly sensitive to the imprecision of thereported solid heats of fusion.

3 Experimental Section

3.1 Materials

Solid organic compounds used as test solutes consist of both industrial chemicals andpesticides. Naphthalene, 2,6-dimethylnaphthalene, phenanthrene, fluoranthene, 1,2,3-tri-chlorobenzene, pentachlorobenzene, 4-chlorobiphenyl (4-PCB), 4,40-dichlorobiphenyl(4,40-PCB), azobenzene, dibenzothiophene, carbaryl, and aldicarb were purchased fromAldrich; acenaphthene, fluorene, 1,4-dichlorobenzene, 1,2,3,4-tetrachlorobenzene, 1,2,4,5-tetramethylbenzene, biphenyl, benzil, lindane, linuron, and chlorpyrifos were from ChemService; p,p0-dichlorodiphenyltrichloroethane (p,p0-DDT) and p,p0-dichlorodiphenyldi-chloroethylene (p,p0-DDE) were from Dr. Ehrenstorfer; and 1,5-dimethylnaphthalene wasfrom Alfa Aesar. They were all of reagent grade or analytical standard with purities of99.0–99.5 %, except for phenanthrene (98 %), pentachlorobenzene (98 %), p,p0-DDT

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(98.5 %), and p,p0-DDE (98.5 %). The related physical chemical properties of the selectedcompounds are listed in Table 1. The solvents, triolein (glyceryl trioleate) and PPG, werepurchased from Sigma and Aldrich, respectively; both are liquids at room temperature.Triolein (C57H104O6, MW = 885.4 g!mol-1), with a purity of [99 %, has a density of

0.907 g!mL-1 (at 22 "C) and a molar volume (Vgt) of 0.976 L!mol-1; PPG

(H[OCH(CH3)CH2]nOH), with a purity of [99 %, has a density of 1.005 g!mL-1 (at

25 "C), an average MW of *1,000 g!mol-1, and a molar volume (Vppg) of

*0.995 L!mol-1. All of the chemicals were used as received without further purification.

Table 1 Molecular weights (MW), estimated liquid densities (q), molar volumes (V), melting temperatures

(Tm), molar heats of fusion at melting temperatures (DHf), and activities at 26 "C (as) of the selected organicsolids

Compound MW(g!mol-1)

q(g!cm-3)

V(L!mol-1)

Tm

(K)DHf

(kJ!mol-1)aas

1,2,4,5-Tetramethylbenzene

134.2 0.838 0.160 352 21.0 0.280

Naphthalene 128.2 1.03 0.125 353 19.0 0.311

1,5-Dimethylnaphthalene 156.2 1.02 0.154 355 20.0 0.280

2,6-Dimethylnaphthalene 156.2 1.03 0.152 383 24.2 0.118

Acenaphthene 154.2 1.04 0.148 369 20.7 0.205

Fluorene 166.2 1.08 0.153 389 19.5 0.162

Phenanthrene 178.2 1.06 0.169 374 18.6 0.223

Fluoranthene 202.3 1.13 0.179 384 18.9 0.185

1,4-Dichlorobenzene 147.0 1.25 0.118 326 18.2 0.545

1,2,3-Trichlorobenzene 181.5 1.45 0.125 326 17.3 0.561

1,2,3,4-Tetrachlorobenzene

215.9 1.52 0.142 321 17.0 0.626

Pentachlorobenzene 250.3 1.51 0.166 359 20.6 0.251

Biphenyl 154.2 0.866 0.178 344 17.5 0.398

4-PCB 188.7 0.984 0.192 351 19.8 0.307

4,40-PCB 223.1 1.05 0.212 422 23.8 0.061

Azobenzene 182.2 1.09 0.167 340 22.0 0.341

Dibenzothiophene 184.3 1.25 0.147 371 15.3 0.303

Benzil 213.2 1.23 0.173 368 19.7 0.224

Lindane 290.9 1.68 0.173 386 23.6 0.118

p,p0-DDT 354.5 1.42 0.250 382 26.3 0.100

p,p0-DDE 319.0 1.35 0.236 361.5 24.2 0.185

Linuron 249.1 1.25 0.199 368 28.6 0.115

Carbaryl 201.2 1.11 0.181 415 24.2 0.065

Aldicarb 190.2 1.20 0.159 373 25.9 0.126

Chlorpyrifos 350.6 1.40 0.250 316 25.9 0.570

a Values cited in Refs. [13–16]

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3.2 Solubility Measurements

The solubilities of selected organic solids in triolein and PPG were measured in an airconditioned room maintained at 26 ± 1 "C. To determine the solubility, an excess amountof a test solid (solute) was added to 0.50–1.0 mL of the solvent in a 2.0 mL glass vial tosaturate the solute in solution. Samples were prepared in triplicate for each test solid andequilibrated for 3–4 days on a rotary mixer in the temperature controlled room(26 ± 1 "C); the indicated equilibration time has been verified to be sufficient forsolid-solvent mixtures to attain equilibrium. Upon solute saturation, the solid and solutionphases were separated by centrifugation under 2,4609g for 2 min. An aliquot of thesupernatant was then withdrawn and diluted with hexane for subsequent analysis of thesolute concentration with a gas chromatograph (GC), Thermo Finnigan Trace GC System(Thermo Electron Corp., Italy), equipped with a flame ionization detector. Individualsolute concentrations were quantified by respective external standards with the injected1 lL of hexane solution using an automatic sampler. A DB-5MS capillary column (30 mlong 9 0.25 mm inner diameter with a 0.25 lm film thickness) from Agilent Technolo-gies, Inc., was used for the solute separation. The GC carrier gas used was high purityhelium (Jing-Shang Corp., Kaohsiung, Taiwan) maintained at a constant flow rate of1 mL!min-1.

3.3 Data Manipulation and Handling

For all solid solutes, the measured solubilities and standard deviations (SD), in grams ofsolid per 100 g of solvent, from triplicate samples were averaged and reported. Theobserved solid solubilities in triolein (Sgt) and PPG (Sppg) were converted to theirrespective solute mole fractions (xo

ob) and volume fractions (uoob) using the number of

moles of dissolved solutes and the molar volumes (V) of solutes and solvents. The molarvolume of a dissolved solid (solute) used in this conversion is that based on the estimatedliquid-solute density, which is assumed to equal the solid-solute density if the solid meltingpoint is\100 "C. If the melting point is higher than 100 "C, then the liquid-solute density

is assumed to be 90 % of the solid’s density [10]. The molar heats of fusion (DHf) at themelting points (MP) of compounds used to calculate the solid activities are taken from theliterature [13–16]. For all solids except azobenzene, dibenzothiophene, and benzil, the

selected DHf values were checked for internal consistency with the established correlationbetween the liquid-solute or subcooled-solute aqueous solubilities (the latter obtained fromrespective solid-solute water solubilities and calculated solid activities based on the

selected DHf values) and the octanol–water partition coefficients of the compounds [10].

The selected DHf value is usually within 10 % of other cited values.

4 Results and Discussion

The relevant properties and calculated activities (as) at 26 "C of test organic solids based

on their DHf values at their melting points are presented in Table 1. Most selected solidsare relatively nonpolar, except azobenzene, dibenzothiophene, benzil, chlorpyrifos, car-baryl, and aldicarb that display weak to moderate to high polarities. The measured solidsolubilities in triolein (Sgt) and PPG (Sppg) at 26 ± 1 "C, their respective mole fractions(xo

ob) and volume fractions (uoob), and their respective athermal volume-fraction solubilities

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(uoat) from Eq. 2 are listed in Tables 2 and 3. In calculations of uo

at, the vs in Eq. 2 is set tozero because triolein has a flexible fibrillar form [9] and PPG is made of a single flexiblechain. The present Sgt data agree well with the earlier values for the same compounds (e.g.,naphthalene, phenanthrene, and lindane) [9]. For most nonpolar solids, the measured Sppg

with PPG are generally lower than the respective Sgt with triolein by about 10–20 %; forazobenzene and dibenzothiophene, the Sgt and respective Sppg are comparable in magni-tude; for benzil and chlorpyrifos, the Sppg are moderately greater than the respective Sgt; forthe two highly polar solids (carbaryl and aldicarb), the Sppg values are sharply higher thantheir respective Sgt values. The observed solid solubilities are intrinsically consistent withthe polarities of triolein and PPG as the solvents.

We next examine the observed mole-fraction solubilites (xoob) in both triolein and PPG

against the Raoult’s law ideal solubilities (xoid = as) of the studied solids, which are listed

in Tables 1, 2 and 3. The data show clearly that, even with the triolein–solute (or PPG–solute) size disparity as small as about 4:1, as with p,p0-DDT, the observed xo

ob values of all

Table 2 Solid solubilities in triolein at 26 "C (Sgt, in g solid per 100 g solvent), respective mole-fractionsolubilities (xo

ob), volume-fraction solubilities (uoob), theoretical athermal volume-fraction solubilities (uo

at),and calculated solute activity coefficients by Raoult’s law (cmf), the F–H model (cfh), and the modified F–Hmodel (cvf)

Compound Sgt ± SD(g!100 g-1)

xoob uo

ob uoat cmf cfh cvf

1,2,4,5-Tetramethylbenzene 17.2 ± 0.46 0.532 0.157 0.136 0.527 0.865 1.78

Naphthalene 23.9 ± 1.58 0.622 0.174 0.148 0.499 0.848 1.78

1,5-Dimethylnaphthalene 16.5 ± 0.70 0.483 0.128 0.135 0.580 1.05 2.18


Acenaphthene 12.3 ± 2.02 0.414 0.097 0.095 0.496 0.985 2.12

Fluorene 9.09 ± 1.43 0.326 0.071 0.074 0.498 1.05 2.29

Phenanthrene 13.3 ± 0.15 0.399 0.103 0.106 0.559 1.03 2.17

Fluoranthene 8.77 ± 0.54 0.277 0.066 0.088 0.667 1.33 2.81

1,4-Dichlorobenzene 61.9 ± 1.54 0.788 0.311 0.293 0.692 0.944 1.76

1,2,3-Trichlorobenzene 60.2 ± 0.71 0.746 0.273 0.306 0.752 1.12 2.05

1,2,3,4-Tetrachlorobenzene 75.0 ± 0.06 0.755 0.309 0.364 0.830 1.18 2.03

Pentachlorobenzene 15.4 ± 0.57 0.353 0.085 0.121 0.710 1.42 2.96

Biphenyl 24.5 ± 0.35 0.584 0.204 0.209 0.682 1.02 1.95

4-PCB 27.7 ± 0.59 0.566 0.204 0.156 0.543 0.764 1.51

4,40-PCB 3.18 ± 0.03 0.112 0.027 0.029 0.546 1.07 2.30

Azobenzene 21.1 ± 0.22 0.507 0.150 0.172 0.674 1.15 2.28

Dibenzothiophene 9.11 ± 0.11 0.305 0.062 0.147 0.995 2.37 4.89

Benzil 3.98 ± 0.16 0.142 0.029 0.107 1.58 3.77 7.85

Lindane 10.3 ± 0.76 0.240 0.053 0.054 0.492 1.02 2.23

p,p0-DDT 10.7 ± 0.01 0.210 0.064 0.049 0.475 0.772 1.57

p,p0-DDE 16.6 ± 0.38 0.315 0.100 0.093 0.586 0.927 1.84

Linuron 6.29 ± 0.93 0.183 0.044 0.054 0.630 1.24 2.64

Carbaryl 0.686 ± 0.04 0.029 0.006 0.030 2.23 5.31 11.7

Aldicarb 0.772 ± 0.03 0.035 0.006 0.057 3.64 9.83 21.7

Chlorpyrifos 64.7 ± 1.05 0.620 0.295 0.353 0.92 1.19 1.93

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nonpolar solids unequivocally exceed the respective ideal solubilities, xoid. That is, the

solutes display negative deviations from Raoult’s law. Thus, the activity coefficients byRaoult’s law, i.e. cmf = xo

id=xoob, are \1, ranging between 0.47 and 0.70 for the nonpolar

solutes. In principle, the fractional solute activity coefficients occur only in rare caseswhere the solute and solvent are associated to each other via specific molecular forces.Since there are no active polar or ionic groups in the present solutes and solvents topromote such specific interactions, the results manifest merely the failure of Raoult’s lawto account for the solute activities or solid solubilities because of the solute–solvent sizedisparity, even though this size disparity is only moderate.

The observed cmf \ 1 is not a result of the inaccuracy of the temperature-uncorrected

DHf values used to calculate the solid activity (as) because it occurs across all nonpolarsolutes with both triolein and PPG (see Tables 2, 3). To further substantiate this point, theavailable solid and liquid heat capacity data (i.e., the DCp data) for some of the present

nonpolar solids given by Neau and Flynn [17] were used to obtain the corrected DHf at T.

Table 3 Solid solubilities in PPG at 26 "C (Sppg, in g solid per 100 g solvent), respective mole-fractionsolubilities (xo

ob), volume-fraction solubilities (uoob), theoretical athermal volume-fraction solubilities (uo

at),and calculated solute activity coefficients by Raoult’s law (cmf), the F–H model (cfh), and the modified F–Hmodel (cvf)

Compound Sppg ± SD(g!100 g-1)

xoob uo

ob uoat cmf cfh cvf

1,2,4,5-Tetramethylbenzene 10.6 ± 0.15 0.441 0.113 0.136 0.635 1.20 2.48

Naphthalene 18.6 ± 0.37 0.592 0.154 0.148 0.525 0.955 2.01



Acenaphthene 9.45 ± 0.23 0.380 0.084 0.095 0.540 1.14 2.46

Fluorene 7.05 ± 0.13 0.298 0.061 0.074 0.545 1.21 2.64

Phenanthrene 11.0 ± 0.36 0.382 0.095 0.106 0.583 1.12 2.35

Fluoranthene 7.75 ± 0.11 0.277 0.065 0.088 0.668 1.35 2.86

1,4-Dichlorobenzene 50.9 ± 0.63 0.776 0.291 0.292 0.703 1.01 1.87

1,2,3-Trichlorobenzene 52.9 ± 0.08 0.745 0.268 0.306 0.753 1.14 2.09

1,2,3,4-Tetrachlorobenzene 62.4 ± 0.71 0.743 0.292 0.363 0.843 1.24 2.14

Pentachlorobenzene 8.42 ± 0.05 0.252 0.053 0.120 0.996 2.27 4.72

Biphenyl 22.5 ± 0.06 0.593 0.207 0.208 0.672 1.01 1.93

4-PCB 25.4 ± 0.14 0.572 0.205 0.155 0.536 0.757 1.50

4,40-PCB 2.53 ± 0.11 0.102 0.024 0.028 0.601 1.21 2.60

Azobenzene 19.9 ± 0.31 0.523 0.155 0.171 0.653 1.10 2.20

Dibenzothiophene 8.63 ± 0.16 0.319 0.065 0.146 0.950 2.26 4.68

Benzil 7.66 ± 0.05 0.264 0.059 0.107 0.847 1.82 3.80

Lindane 9.20 ± 0.61 0.240 0.052 0.054 0.490 1.03 2.26

p,p0-DDT 7.33 ± 0.50 0.171 0.049 0.049 0.583 0.994 2.03

p,p0-DDE 12.3 ± 0.05 0.278 0.084 0.093 0.665 1.10 2.21

Linuron 8.89 ± 1.01 0.263 0.067 0.054 0.438 0.809 1.73

Carbaryl 3.82 ± 0.29 0.160 0.033 0.030 0.409 0.883 1.95

Aldicarb 5.11 ± 0.12 0.212 0.041 0.057 0.596 1.39 3.06

Chlorpyrifos 72.6 ± 0.60 0.674 0.343 0.351 0.846 1.02 1.67

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It is found that the temperature correction reduces the listed DHf in Table 1 by 2.6 % fornaphthalene, 1.4 % for acenaphthene, \0.1 % for fluorene, 6.4 % for phenanthrene, and10 % for biphenyl. The corresponding increase in solid as or in solute cmf is 1.9 % fornaphthalene, 1.5 % for acenaphthene, \0.1 % for fluorene, 5.4 % for phenanthrene, and5.3 % for biphenyl. The magnitudes of these corrections are not large enough to negate theinability of Raoult’s law to accommodate the observed solid solubilities.

As noted, in contrast to the behavior of nonpolar solids in triolein, the cmf values of afew polar solids (dibenzothiophene, benzil, carbaryl, and aldicarb) in triolein are C1,ranging between 1 and 4, in superficial compliance with Raoult’s law. With the recognizedfailure of Raoult’s law for nonpolar solids in triolein, it can be reasonably assumed that theapparent fit of the polar solid solubility data with Raoult’s law is an artifact that the largesolute–solvent incompatibility equals or exceeds the counter size disparity effect on thesolid solubility. Hence, if solubility data of nonpolar solids are not available, the solubilitydata of polar solids in triolein, or the respective cmf values, might well be mistakenlyconstrued as a confirmation of Raoult’s law. Not surprisingly, the calculated cmf values forthe same polar solids in moderately polar PPG are \1 because of the improved solidsolubilities, such that the size disparity effect outweighs the solute–solvent incompatibility,as is the case for nonpolar solids with triolein. For these distinct results, the solubility dataof nonpolar and polar solids with triolein and PPG serve as a simple diagnostic tool for therigor of a solution model on the basis of whether the model-defined solubility limit isobeyed or violated. If the measured solid solubilities are far below the theoretical limits setby the models, a different approach must be taken to validate the individual models.

Comparing the measured volume-fraction solubilities of the solids in triolein and PPG(uo

ob) with the respective uoat values by the F–H model, one notices that the uo

ob values ofmost nonpolar solids lie in close proximity to the limiting uo

at values; the calculated solute

activity coefficients, i.e., cfh = uoat

#uo

ob, fall between 0.80 and 1.3 for most nonpolar

solutes (see Tables 2, 3). This shows that the F–H model rectifies to a large extent theoverestimated solute activities in triolein and PPG by Raoult’s law when the soluteentropies are estimated by the F–H model. However, the nearly equal uo

ob and uoat values

for most nonpolar solids, including uoob [ uo

at for a few solids, must be viewed as suspect,considering certain molecular dissimilarities between these solutes and the two solvents.Given that there is no evidence for molecular association between any of the dissolvednonpolar solids and the solvent (triolein or PPG), the theoretical athermal solubilities (uo

at)of the solids with triolein and PPG, based on the F–H model, are thus somewhat too smallto be accurate. Again, the possibility that lower than expected uo

at values result from the

use of temperature-uncorrected DHf can be ruled out; this problem is common to a largenumber of nonpolar solids, including naphthalene, acenaphthene, fluorene, phenanthrene,and biphenyl mentioned previously (where the errors in uo

at and cfh will not be substantial)as well as for 1,4-dichlorobenzene, 1,2,3-trichlorobenzene, and 1,2,3,4-tetrachlorobenzeneof very low melting points (48–53 "C), where the errors in calculated uo

at or cfh by Eq. 2

with assumed 10 % error in DHf will be \6 %. These corrections are not high enough tochange the view on the inadequacy of the F–H model. Again, for appreciably polar solids(e.g., dibenzothiophene, benzil, carbaryl, and aldicarb) in triolein, the calculated cfh valuesof much higher than 1 are attributed to the enhanced solute–solvent incompatibility thatprovides a superficial confirmation of the F–H model.

To explain why the F–H model may underestimate the solid solubility in liquid trioleinor PPG, one is reminded that the F–H model is intended for the solutions of ordinaryliquids in amorphous polymers [3–6], where the spatial configuration of the sequential

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segments of a polymer molecule is subject to a certain restriction. This restriction isattributed to the fact that an amorphous polymer exhibits limited molecular (translationaland rotational) motion in solution. By contrast, triolein and PPG are mobile liquids withhigh molecular motion that allows their relatively flexible segments (or fragments) to act asif they were fully dissociated (or independent) mixing units, increasing the entropy ofsolution for the entire system. As such, the conventional F–H model tends to underestimatethe solid solubilities in liquid triolein and PPG or other large sized liquid solvents.

Following the preceding analysis, we look for an alternative solution model that pro-vides a higher solubility limit than given by the F–H model (Eq. 2) so that the resultingsolute activity coefficients will be C1 for all dissolved solids. Since, as mentioned, the highmolecular motion of a large sized liquid enables its segments to act like n independentmixing units (where n is assumed to equal the solvent-to-solute size ratio), the ‘‘ideal’’ or‘‘athermal’’ solubility of a solid in triolein or PPG should thus exceed that in an amorphouspolymer (Eq. 2). In this case, with the component activity weighted by the volume fractionand with essentially no restriction on the component spatial orientation, one arrives at asimple volume-fraction-based solution model, which may be viewed as a limiting ormodified F–H model, in which a solid’s ‘‘ideal’’ or ‘‘athermal’’ solubility is given by:

ln Uos " ln as " # DHf=R

! "Tm # T$ %=TTm& ' $3%

where Uos is the ‘ideal’ or ‘athermal’ solubility in volume fraction of the solid, which

replaces the ideal mole-fraction solubility (xoid) in Eq. 1. As for xo

id, Uos is independent of the

solute and solvent molecular sizes. With the molecular size of the solvent being consid-erably greater than that of the solid solute, the limiting Uo

s in Eq. 3 is higher than either xoid

(in Eq. 1) or uoat (in Eq. 2) on a mass basis. Using calculated as values of the 25 solids

listed in Table 1, the Uos values so determined for the solids can be compared with the

respective solid solubilities (uoob) in order to evaluate the performance of Eq. 3 relative to

that of Eq. 2.As noted in Tables 2 and 3, the uo

obvalues of all nonpolar solids with triolein and PPGare now all consistently lower to a reasonable extent than the respective Uo

s values. The

calculated solute activity coefficients, cvf = Uos

#uo

ob, being all higher than 1, fall between

1.7 and 2.5 for most nonpolar solutes and appear more convincing than the lower cfh valuesgiven by Eq. 2. This improvement is illustrated in Fig. 1 for some nonpolar solutes withtriolein. The higher cvf values for appreciably polar solids (e.g., dibenzothiophene, benzil,carbaryl, and aldicarb) with triolein are well anticipated. Thus, to the extent that nomolecular association occurs between the solute and solvent and that all molecular moi-eties (or segments) of the solute and solvent are fully accessible, the simple volume-fraction activity model (Eq. 3) should be generally applicable for common solutes in anyliquid solvent irrespective of the solute–solvent size ratio. For the present solids with bothtriolein and PPG, the predicted Uo

s values are higher than the corresponding uoat values only

by about a factor of 2. Even with such a small difference between Uos and uo

at, thesuperiority of Eq. 3 over Eq. 2 for the present solid-solvent systems is clearly demon-strated. In other systems, however, the small difference in prediction by Eqs. 2 and 3 maywell escape notice because most solutions can easily deviate from the ‘‘ideal ‘‘or ‘‘ather-mal’’ solution state with c [ 2 for the solute.

We now consider other solubility or related data involving a similar solute–solvent sizedisparity. As indicated earlier, Shinoda and Hildebrand [11, 12] maintained that iodineforms ‘‘regular solutions’’ [2] with relatively nonpolar compact shaped solvents in that themixing entropies of iodine conform to those given by Raoult’s law. The solvents known to

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form ‘‘regular solutions’’ with iodine span a large range of molecular sizes. The data withtwo of these solvents, i.e., octamethylcyclotetrasiloxane, or (CH3)8Si4O4, and tetraperflu-orobutyric ester of pentaerythritol, or (C3F7COOCH2)4C, are of special interest for scrutinybecause the ratios of their molar volumes (312 and 540 mL!mol-1, respectively) to themolar volume of liquid (dissolved) iodine (59 mL!mol-1) are relatively high (5.3 and 9.1,respectively) and thus render significantly different estimates of the solution entropies byRaoult’s law (Eq. 1) and the two other models (Eqs. 2, 3). Here, as the measured solu-bilities of solid iodine in most low polarity solvents at room temperature are substantiallybelow the ideal solubility or athermal solubility limits, the solubility data alone are notsufficient for judging the model performance. Thus, following the approach taken byShinoda and Hildebrand, we compare the measured entropies of solution of iodine againstthose estimated by the alternative models.

In a regular solution [2], where the deviation from ideality is caused entirely by athermal effect, the (partial) molar entropy of solution for the solute according to Raoult’slaw is given by

DSl " #R ln xoob $4%

where R is the gas constant and xoob is the solute’s mole-fraction solubility at T, as defined.

Similarly, if the thermal effect is the cause of imperfection for a solute–polymer solution,the molar entropy of solution for the solute according to the F–H model is given as

DSl " #R ln uoob ( up 1# V

#V l

! "$ %$5%

where uoob was defined before in Eq. 2, up = 1 –uo

ob, V is the molar volume of the solute

(iodine), and V l the molar volume of the large sized solvent. The corresponding partialmolar entropy based on the modified F–H model (Eq. 3) is

DSl " #R ln uoob $6%

0.10 0.15 0.20 0.25 0.30

Act

ivity

Coe

ffic

ient

0.5

1.0

1.5

2.0

2.5

eq 1eq 2eq 3

1

gt/VV

1

1

2

2

2

3

3

3

4

4

4

5

5

5

6

6

6

7

7

7

Ideal/Athermal Reference Line

Fig. 1 The activity coefficients of selected nonpolar solutes in triolein, based on three alternative modelsversus the solute–triolein molar volume ratios. The solutes are designated by numerals: 1 1,4-dichlorobenze;2 naphthalene; 3 acenaphthene; 4 1,2,4,5-tetramethylbenzene; 5 4-PCB; 6 p,p0-DDE; and 7 p,p0-DDT

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The estimated entropies of solution for iodine in the two solvents by Eqs. 4–6 may becompared with respective experimental values. For the latter case, one measures first the

entropy of solution for solid iodine (DSs) and then subtracts from it the entropy of fusion of

iodine (DSf ) to arrive at DSl " DSs # DSf for dissolved (liquid) iodine. To obtain DSs, one

plots lnxoob (or ln uo

ob) against ln T to find DSs " R d ln xoob

#d ln T

! "at a desired temperature.

The DSf of iodine was estimated to be 33.4 J!K-1!mol-1 at 25 "C by Shinoda and Hil-debrand [11]. Table 4 summarizes the experimental and estimated entropies of solution foriodine with the two large sized compact solvents at 25 "C. All data are from the original

reports of these authors, except for the estimated DSl of iodine using the volume-fractionsolubilities that were computed from the reported iodine solubilities. Note that the solu-bilities of solid iodine in these two solvents are very low, and one finds that

DSs " R d ln xoob

#d ln T

! "" R d ln uo

ob

#d ln T

! ".

As shown in Table 4, the measured DSl values for dissolved iodine agree remarkablybetter with those estimated by the modified F–H model (Eq. 6) than by Raoult’s law(Eq. 4). Although the conventional F–H model (Eq. 5) also shows an improvement overRaoult’s law in prediction, the result is not as good as with Eq. 6. This observation iscontrary to the contention of Shinoda and Hildebrand that the entropies of solution ofiodine in large nonpolar compact solvents obey the ideal solution law (i.e., Raoult’s law).The present analysis indicates that the molecular fragments of these large sized compactsolvents are quite accessible to iodine molecules, and thus a large excess in the entropy ofmixing is gained over that estimated by the iodine mole fraction. This finding is enlight-ening in that the deviation from ideality of a solution caused by a size disparity usuallyoccurs in systems where either the solvent or the solute is a large sized linear molecule.The entropic data indicate that a similar deviation occurs even with a large sized compactmolecule as long as its fragments are accessible to the other components (solute or sol-vent). It thus appears that the entropy of mixing for a component in a regular solutionshould generally be more accurately estimated in terms of its volume fraction than its molefraction.

From the data presented, the two volume-fraction-based solution models (Eqs. 2, 3) areclearly superior to the mole-fraction-based solution model (Eq. 1) to account for the solidsolubility in those systems that exhibit a significant component size disparity and nomolecular association. The viewpoint is also supported by observations in earlier studies,such as: (i) the success of the F–H model (Eq. 2) over Raoult’s law (Eq. 1) to describe thepartitioning of organic compounds into (amorphous) soil organic matter [1, 18, 19], and (ii)the closely similar partition coefficients observed for given organic solutes in octanol–water and triolein–water mixtures, where the two liquid organic solvents exhibit compa-rable polarities but sharply different molecular sizes (by about 7 times), which is in

Table 4 Experimental and model-estimated molar entropies of solution of dissolved iodine (DSl) with twolarge-sized solvents at 25 "C

Solvent DSl (expt.) DSl (Eq. 4) DSl (Eq. 5) DSl (Eq. 6)

(CH3)8Si4O4 59.8 40.0 47.2 53.9

(C3F7COOCH2)4C 76.5 56.0 66.9 74.4

Listed below: experimental DSl (exp.) from Shinoda and Hildebrand [11, 12], DSl (Eq. 4) estimated by

Raoult’s law [11, 12], DSl (Eq. 5) estimated by the F–H model, and DSl (Eq. 6) estimated by the modified

F–H model. Units for all DS terms are J!K-1!mol-1

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keeping with the modified volume-fraction model (Eq. 3) [10]. The relative merit of Eq. 2versus Eq. 3 presumably depends on whether the organic solvent involved is a liquid or anamorphous solid, as it affects the entropy of mixing with the small sized solute.

5 Conclusions

The measured solubilities of 25 organic solids in triolein and PPG (Tables 2, 3) and thesolution entropies of iodine in two large sized liquid solvents (Table 4), where the solventand solute display significant size ratios, are well reconciled with the modified F–H model(Eq. 3). In comparison, Raoult’s law (Eq. 1) greatly underestimates the solid solubilitywhile the conventional F–H model (Eq. 2) gives a less satisfactory account of the data thanthe modified F–H model (Eq. 3) for these systems. As noted, the units of a large sizedliquid solvent act as independent mixing entities to promote the entropy of solution overthat estimated by the F–H model for a polymer solution, presumably because ordinarypolymers have less freedom in their molecular motion. Thus, to the extent that a givenvolume element of the solution is equally accessible to all molecular fragments, themodified F–H model is expected to give an improved account of the solid activities invarious liquid solvents.

Acknowledgments This work was supported in part by National Cheng Kung University (D101-33B01)and by National Science Council of Taiwan (NSC 99-2221-E-006-053). We thank Prof. Milton Manes,Professor Emeritus, Kent State University (Kent, OH, USA) and Prof. Jen-Feng Kuo, National Cheng KungUniversity (Tainan, Taiwan) for valuable comments and discussion.

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