Solvation thermodynamics and the physical–chemical meaning of the constant in Abraham solvation equations

Chemosphere 87 (2012) 125–131

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Solvation thermodynamics and the physical–chemical meaning of the constant in Abraham solvation equations Paul C.M. van Noort ⇑ Aquatic Ecology and Water Quality Management Group, Wageningen University, P.O. Box 47, 6700 AA Wageningen, The Netherlands Deltares, P.O. Box 85467, 3508 AL Utrecht, The Netherlands

a r t i c l e

i n f o

Article history: Received 10 October 2011 Received in revised form 23 November 2011 Accepted 28 November 2011 Available online 22 December 2011 Keywords: Partitioning LFER Solvation Entropy Enthalpy

a b s t r a c t Abraham solvation equations find widespread use in environmental chemistry. Until now, the intercept in these equations was determined by fitting experimental data. To simplify the determination of the coefficients in Abraham solvation equations, this study derives theoretical expressions for the value of the intercept for various partition processes. To that end, a modification of the description of the BenNaim standard state into the van der Waals volume is proposed. Differences between predicted and fitted values of the Abraham solvation equation intercept for the enthalpy of solvation, the entropy of solvation, solvent–water partitioning, air-solvent partitioning, partitioning into micelles, partitioning into lipid membranes and lipids, and chromatographic retention indices are comparable to experimental uncertainties in these values. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction For application in, for instance, environmental chemistry and pharmaco-chemistry, the partitioning of organic compounds over two phases A and B, defined by a partition coefficient PAB, can be estimated from molecular descriptors for the various types of solute–solvent interactions by using so-called Abraham solvation equations (Eqs. (1) and (2)).

log PAB ¼ log C A ðmol=volumeÞ=C B ðmol=volumeÞ ¼ c þ eE þ sS þ aA þ bB þ v V log PAB ¼ c þ eE þ sS þ aA þ bB þ lL

ð1Þ ð2Þ

In Eqs. (1) and (2), the capital letters denote the solute descriptors. The values of the solute descriptor coefficients (given as lower case letters) depend on the phase system. Some prefer to use Eq. (1) for condensed systems, and to use Eq. (2) for gas–solvent partitioning. V is the McGowan characteristic volume [(dm3 mol1)/100], calculated from atom increments (Abraham and McGowan, 1987; Goss et al., 2006; Van Noort et al., 2011). L is the logarithm of the experimentally determined gas to n-hexadecane partition coefficient at 298 K. Both solute descriptors are some measure of the solute’s potential for van der Waals interactions and solvent cavity formation. ⇑ Address: Aquatic Ecology and Water Quality Management Group, Wageningen University, P.O. Box 47, 6700 AA Wageningen, The Netherlands. Tel.: +31 (0)623879203; fax: +31 (0)88 3357775. E-mail address: [email protected] 0045-6535/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chemosphere.2011.11.073

E is the calculated solute excess molar refractivity relative to an alkane with the same V; S is the solute dipolarity/polarizability; A and B are the solute overall hydrogen bond acidity and basicity. Values for S, A, and B are usually determined from experimental phase distributions or chromatographic retention data. For an in depth description of the background of the Abraham solvation equations the reader is referred to the recent review by Poole et al. (2009). Recently, Goss (2005a) showed that Eqs. (1) and (2) can be very well replaced by Eq. (3). For those that use Eq. (1) for condensed phases and Eq. (2) for air-solvent partitioning, the advantage of Eq. (3) is that a type 3 equation for condensed phase j and air can be directly calculated from the type 3 equations for condensed phase j-water partitioning and for air–water partitioning. An additional advantage of Eq. (3) is that it avoids the use of the E solvation parameter the calculation of which is sometimes associated with considerable errors (Bronner et al., 2010). Furthermore, for some highly fluorinated compounds, Eq. (3) yielded better air–water partition coefficients than those from Eqs. (1) and (2) (Goss, 2005a)

LogP AB ¼ c þ lL þ sS þ aA þ bB þ v V

ð3Þ

Values for the coefficients c, e, s, a, b, v, and l in Eqs. (1)–(3) are determined by fitting experimental partition coefficients to Eqs. (1)–(3). The aim of the present study is to derive expressions for the value of the coefficient c. The advantage of the a priori calculation of values for c, which is a determinant for the partitioning of all compounds concerned, is that the number of fitting parameters is reduced from 6 to 5. As a result, statistics of the fit to Eqs. (1)–(3) may be substantially improved when calculated values for c are

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used. To derive expressions for c, solvation will be thermodynamically evaluated based on Ben-Naim’s theory of solvation. After that, predictions of the value of the coefficient c from this evaluation will be compared with fitted values of c from literature on Abraham solvation equations for various partitioning processes.

standard Gibbs energy change when calculated from molar concentrations. Vitha and Carr (2000) suggested on theoretical grounds that a correction based on the free volume of the phases on DGairsolv ent should be applied. Graziano (2003) claimed to have disproved Vitha and Carr’s suggestion and that no correction at all is needed. However, Goss (2003) found that nonane Ben-Naim standard state based free energies of solvation calculated from 2. Theory equilibrium partitioning data were slightly but systematically larger than estimates based on nine times the value for a methylene To interpret solvation at the molecular level, the use of the sogroup. He assigned this difference to an entropy difference that called Ben-Naim standard state (Ben-Naim, 1987) can be advantaarises from the difference in the accessible free volume of the solgeous for the following reasons. This standard state refers to the vent as compared to its saturated vapor phase. Graziano (2005) transfer of a molecule from a fixed position in the ideal gas phase criticized the methylene group additivity assumed by Goss, who, to a fixed position in the solvent, at constant temperature and presin his turn, refuted the criticism (Goss, 2005b). sure. The corresponding Ben-Naim standard Gibbs energy change To solve this dilemma, it is of importance to note that, up to (DGairsolv ent , where the superscript dot denotes the Ben-Naim standate, for evaluations of solvation based on Ben-Naim standard dard state) reflects only the coupling work of the solute with the states, concentrations based on the macroscopic volume were surrounding solvent molecules. This Gibbs energy change is diused. However, at the molecular level, solute–solvent interactions, rectly related to the ratio of the molar concentrations of the solute which determine the magnitude of enthalpy and entropy changes, in the gas and liquid phases at equilibrium and does not contain take place within the van der Waals volume of the solute–solvent any translational entropy contributions (Ben-Naim, 1978). Theresystem. The van der Waals volume of a molecule is the volume not fore, for aqueous solvation, for instance, the entropy change based penetrable by other molecules or atoms. Therefore, the van der on Ben-Naim standard states accounts only for the entropy change Waals volume of a solvent (or a solute–solvent system) equals of water molecules, and for possible changes in the solute internal the sum of the van der Waals volumes of all molecules considered. degrees of freedom. For standard states based on for instance mol In case of low solute concentrations, which will be the case for, for fractions, the evaluation of entropy changes is much more compliinstance, solvation partitioning data, this van der Waals volume is cated. In that case, the entropy change also includes entropy practically equal to the solvent van der Waals volume. Recall that changes due to changes in solute translational freedom among the Ben-Naim standard state refers to a fixed position of the solute others. in the solvent. By definition, the free volume of the solute–solvent A separate interpretation of enthalpy (DHairsolv ent ) and entropy   system cannot contain the solute. The fixed position of the solute is (DSairsolv ent ) contributions to DGairsolv ent (Eq. (4)) will be helpful therefore located in the van der Waals volume of the solvent–solfor the interpretation of solvation at the molecular level, especially ute system, which is practically equal to the van der Waals volume for the identification of the thermodynamic origin of the constant c of the solvent. Accordingly, on solvation evaluations based on Benin Abraham solvation equations. In Eq. (4), R and T denote the gas Naim standard states, molar concentrations based on moles per constant and the system temperature, respectively. Kair-solvent is the van der Waals solvent volume should perhaps be used rather than air-solvent partition coefficient moles per macroscopic volume. DGairsolv ent ¼ RT ln K airsolv ent ðmoles per volume=moles per volumeÞ To obtain concentrations based on van der Waals volumes instead of macroscopic volumes, solute molar concentrations on a ¼ DHairsolv ent  T DSairsolv ent ¼ DU airsolv ent þ RT  T DSairsolv ent macroscopic volume basis have to be divided by the solvent packð4Þ ing density n. The solvent packing density n is defined as the ratio of the van der Waals volume and the macroscopic volume. It can be In Eq. (4), the internal energy change for solvation (DU airsolv ent ) is readily calculated from the solvent molecular van der Waals volalso considered. That is because DU airsolv ent has a fundamental ume, Avogadro’s number, the solvent specific density, and the solphysical meaning as it measures the energy released by bringing vent molecular weight. Van der Waals volumes, needed for the molecules (or atoms) from an infinite distance, as in the gas state, calculation of n were estimated from the molecular formula using to the dissolved solute state. the Zhao et al. (2003) procedure. The van der Waals volume for To evaluate values for c, we need to consider an infinitely small water was taken to be 12.4 Å3 (Edward, 1970; Daubert and Danner, solute as will be described below. For such an infinitely small solute with infinitely small interactions with the solvent, there is no 1992). Solvent specific densities were taken from the SPARC Online energy released by bringing such a small solute from an infinite database (http://archemcalc.com/sparc). Note that, if Ben-Naim distance to the dissolved state. In other words, DU airsolv ent should standard states should be based on van der Waals volumes, rather than macroscopic volumes, the condition K 0air solvent (on a van der be equal to zero for an infinitely small solute. This infers (see Eq. (4)) that DHairsolv ent for this infinitely small solute should equal Waals volume base) equals 1/eE requires that K 0air solvent (on a macRT. What’s more, for a zero volume solute, dissolution will not reroscopic volume base) should equal n/eE. Similarly, P0solvent Asolvent B sult in an entropy change of the solvent molecules. In other words, (on a macroscopic volume base) should equal nA/nB. DSairsolv ent should be zero for the infinitely small solute. Now, it can easily be derived from Eq. (4) that K 0airsolvent for the infinitely small solute should equal 1/eE, where eE (to be distinguished from the 3. Results and discussion Abraham solvation equation coefficient e) denotes Euler’s number and the superscript 0 means that we are dealing with an infinitely 3.1. Enthalpy of solvation small solute. Besides, for solvent–solvent partitioning, it can now be easily seen that K 0solvent Asolvent B should equal 1. From these conEquation type 1 Abraham solvation equations for the enthalpy siderations, it should be in principle possible to estimate the value of solvation of organic compounds at 298 K in several organic solof c in Abraham solvation equations. vents and water were derived from experimental data collected by There is, however, a dilemma as to the choice of the type of Mintz et al. (2007a, 2007b, 2007c, 2007d, 2007e, 2008a, 2008b, volume to be used for molar concentrations. Some authors have 2008c, 2009a, 2009b), who carried out an impressive evaluation claimed that a correction should be added to the Ben-Naim of solvation enthalpies in terms of the Eq. (2) type of Abraham

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solvation equations, and by Stephens et al. (2011). Here and elsewhere, Abraham solvation parameters were from Abraham et al. (1994a, 1994b), Poole et al. (2006), and van Noort et al. (2010). The values of the constant c in the type 1solvation equations are listed in Table 1. The values for all type 1 solvation equation coefficients are given in Table S-1 (Supplementary material). Note that Mintz et al. acknowledge that some listed enthalpies are in fact internal energies. This is especially the case for water (Goss, 2006). For the other solvents, many enthalpy values were calculated from liquid or solid solubilities and the standard molar enthalpy of vaporization or sublimation, and are, therefore, true enthalpies. The average of c from Table 1 is 3.3 ± 2.0 kJ/mol (not including water) which is not different from the value of RT of 2.54 kJ/mol at 298 K, as expected (see Theory Section). For water, the reported enthalpy values are in fact internal energies. Recalculation of the solvation equation for water with c set at 0 kJ/mol, as suggested by theory, resulted in a negligible increase of the standard deviation from the fit from 4.0 to 4.6 kJ/mol and an improvement of the statistics (increase of r2adj from 0.958 to 0.989). For most other solvents in Table 1, the use of c = 2.54 J/mol results in even less increased standard deviations from the fit and in substantially improved statistics (see Table S-1, Supplementary material).

Table 2 Values of c in solvation equations with standard deviations from the fit for organic compound solvation entropies based on solvent van der Waals volumes.

a

Solvent

c (J/mol.K)

n-Hexadecane Octan-1-ol Water N,N-Dimethylformamide Methanol Ethanol di-n-butyl ether Chloroform

2.8(7.3a) ±1.7 1.2(5.5) ±3.2 0.3(7.7) ±3.5 2.6(6.8) ±1.6 0.1(5.6) ±1.7 1.1(6.3) ±1.9 2.3(7.2) ±4.7 2.7(7.9) ±3.1

Based on macroscopic volume.

For the eight solvents in Table 2, values of c based on van der Waals volumes are not different from zero (within 0.1 to 1.6 standard deviations) as suggested by the theoretical considerations given above. The values of c based on macroscopic volume, however, are evidently different from zero. That supports the notion that van der Waals volumes may have to be the basis for the interpretation of partitioning in terms of Ben-Naim standard states. Table S-2 shows that the statistics of the fits are substantially better when the value of c is set at zero.

3.2. Entropy of solvation 3.3. Solvent–water partitioning Graziano (2009) recently evaluated the hydration entropy for 83 non-polar and polar organic compounds in terms of their hard-sphere volumes which are close to the van der Waals volumes. He found that the data fell on a single curve, although with some (considerable) scatter of up to about 25 J/mol K. For the present study, entropies of solvation were fitted to Eq. (1). Entropy of solvation values were calculated from solvation enthalpies used in the previous section and from air to solvent partitioning data from Goss (2005a), Mintz et al. (2007a), and Abraham et al. (1998a, 1998b, 1999a, 2001a, 2001b, 2009). In this way, hydration entropies for a larger number of compounds (227) were obtained than used previously by Graziano, along with solvation entropy data for seven organic solvents for 23–98 compounds. Partitioning data used for the calculation of the solvation entropy were based on van der Waals volumes as well as on macroscopic volumes. The values of c in the solvation equations are given in Table 2. Values for c based on macroscopic volumes are listed between brackets in Table 2. Values for all coefficients along with the statistics of the fits are given in Table S-2 (Supplementary material).

Table 1 Values of the constant c (standard error between brackets) in type 1 Abraham solvation equations for the enthalpy of organic compound solvation at 298 K. Solvent

c (kJ/mol)

Solvent

c (kJ/mol)

n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tetradecane n-Hexadecane Cyclohexane Benzene Toluene Methanol Ethanol 1-Butanol

0.48(0.78) 4.23(0.77) 3.09(0.58) 5.79(0.73) 7.12(1.50) 5.41(0.85) 5.98(0.93) 6.50(1.01) 4.52(0.61) 3.04(0.54) 4.34(0.62) 4.20(0.76) 3.94(0.95) 4.41(0.82) 3.21(0.74)

1-Propanol t-Butanol 1-Octanol Tetrachloromethane Chloroform 1,2-dichloroethane Propylene carbonate Ethyl acetate Dimethyl sulfoxide N,N-dimethylformamide Acetonitrile Acetone di-n-Butyl ether Tetrahydrofuran Water

0.09(0.81) 3.32(1.49) 3.74(1.04) 3.15(0.65) 0.44 (0.82) 3.62(1.00) 1.53(0.97) 0.68(0.91) 2.26(0.83) 2.18(0.87) 2.40(1.11) 3.23(1.04) 0.92 (1.21) 4.90(0.69) 6.62 (0.65)

For 86 solvent–water systems, Table S-3 (Supplementary material) lists the fitted value of the constant c in Eq. (1) type solvation equations from Abraham et al. (2010). Note that these values are positive for 80% of the 86 solvent–water systems and are only slightly larger than zero. The average value of c for all solvent– water systems is 0.13. That means that, for an infinitely small solute, the ratio of the molar concentrations in the solvent and water is 1.4 on average. On theoretical grounds, this ratio should be 1. This deviation can be minimized if concentrations would have been based on van der Waals volumes instead, as seems to be the case for the entropy of solvation. Therefore, for comparison, Table S-3 lists the log of the solvent–water ratio of the volume packing densities n. The average difference between log nsolvent/nwater and c in Table S-3 is 0.02 ± 0.22, which is very close to theoretical expectations. The absolute average difference is 0.17 ± 0.14. The correlation between log nsolvent/nwater and c is very poor because of this small difference and the small variation in nsolvent values. Furthermore, fitted values of c are generally associated with some uncertainty, which is reflected by the absolute difference between theoretical and fitted values of c. Nevertheless, the systematic deviation of fitted c values from zero along with the removal of this deviation by using volume packing densities indicates the applicability of the van der Waals volume concept. 3.4. Air-solvent partitioning Type 1 solvation equations were derived from air-solvent partitioning data compilations in Goss (2005a), Abraham et al. (1993, 1998a, 1998b, 1999b, 2001a, 2001b, 2003, 2009), and in Abraham and Acree (2008). The solvation equation coefficients are listed in Table S-4. According to theory, the value of c based on van der Waals volumes should equal log n/eE. Values for c and log n/eE are given in Table 3. The average of log n/eE  c is 0.34 ± 0.34. For log 1/eE  c, it is 0.59 ± 0.32, which is relatively less close to zero than for the van der Waals volume based values. That suggests that values of c for air-solvent partitioning Abraham equations are better predicted based on van der Waals volumes. All in all, this and the findings for the entropy of solvation, the enthalpy of solvation,

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Table 3 Values for c in solvation equations for air-solvent partitioning along with values for log n/eE.

ð7Þ and

Solvent

log n/eE

c

Water Methanol Ethanol Dry propan-1-ol Dry octan-1-ol Wet decan-1-ol Diethyl ether di-n-Butyl ether Tetrahydrofuran 1,4-Dioxan Chloroform Methylene iodide N,N-dimethylformamide

0.82 0.72 0.71 0.69 0.66 0.66 0.73 0.69 0.64 0.62 0.71 0.68 0.65

1.00(0.03) 0.97(0.07) 0.83(0.13) 0.55(0.10) 1.22(0.08) 1.14(0.12) 0.80(0.09) 0.89(0.09) 1.08(0.11) 1.22(0.12) 0.58(0.09) 1.70(0.12) 1.39(0.05)

and solvent–water partitioning suggests that Ben-Naim standard states are better based on van de Waals volumes and that the value of c for the entropy of solvation, solvent–water partitioning, and air-solvent partitioning can be calculated from n. In this respect it is noteworthy that, for the Ben-Naim standard state based free energies of n-nonane solvation in 66 organic solvents, Goss (2003) calculated that the difference (D) between Gibbs free energy values calculated from equilibrium partitioning data and estimates based on nine times the value for a methylene group, was different from zero and positive. The average value of exp(D/RT), calculated from the individual data in Goss (2003) for the 66 organic solvents (see Table S-5, Supplementary material) is 0.56 ± 0.16, which compares favorably with the range of 0.50– 0.75 for the value of n for these solvents. Water as a solvent falls outside these ranges, as well as to exp(D/RT) (=0.12) as to n (=0.41). Exp(D/RT) seems a bit (r2 = 0.26) negatively correlated with n for unknown reasons. Largely, the close similarity of exp(D/RT) and n values further confirms the need for a correction on Ben-Naim standard state concentrations when expressed in terms of moles per macroscopic volume. 3.5. Application to other types of Abraham solvation equations With the correction n on the macroscopic volume based concentration at hand, it now is easy to derive expressions for c in Eqs. (2) and (3) type Abraham equations. For an infinitely small solute, the L and V (or L and E) based solvation equation for partitioning between solvent A and B is: 0

log P0 ðmol=volumeÞ=ðmol=volumeÞ ¼ c þ lL ¼ logðnsolvent A =nsolvent B Þ ð5Þ From the evaluation of c values for air-solvent partitioning given above, it follows that L0 = log (nn-hexadecane/eE). Rearrangement of Eq. (5) and substitution of the values for nn-hexadecane and eE gives:

c ¼ logðnsolvent A =nsolvent B Þ þ 0:67l

log P ¼ logðnsolvent A =nsolvent B Þ þ eE þ sS þ aA þ bB þ lðL þ 0:67Þ

ð6Þ

Note that Eq. (6) implies that the value of c in a V and E based solvation equation is related to the values of c and l in L and V (or E) based solvation equations. For example, for water–air, n-octanol–water, di-n-butyl ether–water, and chloroform–water partitioning solvation equations from Goss (2005c), the average of the difference between values for c from the E and V based solvation equation and c from the L and V based solvation equation in Goss (2005c) when divided by l is 0.68 ± 0.06. This is equal to the value of 0.67 for log (nn-hexadecane/eE). Elimination of c from Eqs. (2) and (3) yields for solvent–solvent partitioning:

log P ¼ logðnsolvent A =nsolvent B Þ þ sS þ aA þ bB þ lðL þ 0:67Þ þ v V ð8Þ Similarly, for solvent-air partitioning, it can be derived that:

log K airsolvent ¼ logðnsolvent =eE Þ þ eE þ sS þ aA þ bB þ lðL þ 0:67Þ ð9Þ and

log K airsolvent ¼ logðnsolvent =eE Þ þ sS þ aA þ bB þ lðL þ 0:67Þ þ v V ð10Þ Eq. (9) can be applied to compare fitted values of c for 86 air-solvent partitioning Abraham type 2 equations from Abraham et al. (2010) with estimated values. Individual data are given in Table S6 (Supporting material). The average difference between fitted and estimated c values is 0.043; the average absolute difference is 0.166. Similarly, for solvation into 1-methyl-3-butylimidazolium bis(trifluoromethylsulfonyl)imide, the predicted value of c for the type 2 solvation equation is 0.15 (from a density of 1440 g/L (Fredlake et al. (2004)), which yields n = 0.66) which is reasonably close to the fitted value of 0.41 from Abraham and Acree (2006). 3.6. Application to partitioning into micelles and lipids In the previous sections, for simple solvent systems, it was shown that values for c can be successfully predicted from theory. For more highly organized phases, such as micelles and lipid bilayers, the solvent packing density may perhaps be less easily calculated. We will now explore this based on data for partitioning into sodium dodecylsulfonate (SDS) and on data for partitioning into lipids and membranes. Sprunger et al. (2007a, 2007b) derived a solvation equation for SDS micelle-water partitioning. They used partitioning data on a mole fraction basis. Recalculation of their data on a mol per volume base, as required for the Ben-Naim standard state, gave c = 0.09 ± 0.05. This value is hardly different from the value of 0.194 for log ndodecylsulfonic acid/nwater (dodecylsulfonic acid density is 1021 g/L, yielding n = 0.64). Note that ndodecylsulfonic acid is used as a reasonable approximation for nSDS micelle, which is practically impossible to calculate because SDS is a solid. The statistics for the solvation equation substantially improved when the regression was based on van der Waals volumes (and therefore taking c = 0): r2 increased from 0.967 to 0.993, F increased from 1188 to 5803, while the standard error of 0.21 did not increase. Fitting SDS-gas partitioning data from Sprunger et al. (2007a) to Eq. (9) gave a fitted value for log nSDS micelle/eE of 0.767 ± 0.068, which is hardly different from the calculated value of 0.627 for log ndodecylsulfonic acid/eE. Using this calculated value for the fit resulted in substantially improved statistics: r2 increased from 0.9915 to 0.9987, F increased from 3170 to 20557, while the standard error of 0.23 did not increase. For membrane lipid–water partitioning (Kmw), Endo et al. (2011a) found:

log K mw ¼ 0:26ð0:08Þ þ 0:85ð0:05ÞE  0:75ð0:08ÞS þ 0:29ð0:09Þ A  3:84ð0:10ÞB þ 3:35ð0:09ÞV; N ¼ 131; SD ¼ 0:28; r2 ¼ 0:979

ð11Þ

As required by theory, the value for c of 0.26 ± 0.08 is not different from the average log nliposome/nwater value of 0.198 ± 0.003 for ditetradecanoyl, dihexadecanoyl, and dioctadecanoyl phosphatidylcholine. For bioconcentration into organisms, Park and Lee (1993) derived an Abraham solvation equation from 51 bioconcentration

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factors (BCF), which predicted values with a standard deviation of 0.45. Unfortunately, the log BCF values used by them were not lipid content normalized. That may explain the relatively large standard deviation. Therefore, for the present study, 182 BCF values were collected from the literature (data given in Table S-7, Supplementary material). BCF data were not used when indications were present either for non-linear relations with log Kow in case of high BCF values (due to experimental artifacts (Jonker and van der Heijden, 2007) or when indications for metabolization were present. The dimensions of the BCF data were L/kg and not L/L as required by the use of the Ben-Naim standard state. Nevertheless, data were used as such because lipid and liposome density at 298 K can be expected to be close to 1 L/kg, based on ditetradecanoyl, dihexadecanoyl, and dioctadecanoyl phosphatidylcholine bilayer densities of 0.936–0.941 L/kg (Nagle and Wilkinson, 1978). Assuming that lipid density equals 1, may lead to a negligible bias of about 0.03 log units. The BCF data fitted to:

log BCF ¼ 0:45ð0:13Þ þ 0:92ð0:06ÞE  0:99ð0:10ÞS þ 0:23ð0:18ÞA  3:63ð0:10ÞB þ 3:37ð0:11ÞV;

ð12Þ

2

N ¼ 182; standard error ¼ 0:33; r ¼ 0:956 The value of c in Eq. (12) is only slightly larger than the theoretical value of 0.20. Note the similarity of the coefficient values in Eqs. (11) and (12). That is not surprising because membrane lipid-water partitioning may be considered to mimic lipid bioconcentration of non-metabolizable compounds. Eq. (13) gives the solvation equation for the combined membrane lipid–water and BCF data set with c set at the theoretical value of 0.20

log K mw or BCF ¼ 0:20 þ 0:87ð0:04ÞE  0:93ð0:07ÞS þ 0:22ð0:07ÞA  3:71ð0:07ÞB þ 3:545ð0:055ÞV; N ¼ 313; standard error ¼ 0:31; r2 ¼ 0:996 ð13Þ Eq. (13) predicts log Kmw and log BCF with average deviations of 0.02 and 0.01 log units, respectively, which does not indicate differences in the fit of these two partition coefficients to the solvation equation. The substantially improved statistics for solvation equations for partitioning into SDS, liposomes, and lipids when based on solvent van der Waals volumes further illustrates the applicability of the solvent packing density n correction. 3.7. Application to partitioning into polymeric materials For polymeric materials, calculation of n may be problematic for several reasons. In crystalline materials, solute dissolution may only occur at sufficiently high concentrations to locally induce transition to the rubbery state. Hale et al. (2011) proposed that the relatively large negative value of c for the cellulose–water partition Abraham equation suggests a large crystalline cellulose fraction. For polymers, neither the van der Waals volume nor the molar volume, which are both needed for the calculation of n, can be calculated. However, by analogy with low molecular weight solvents, amorphous, rubbery polymer n values can be expected to be in the range of 0.5–0.75. Therefore, in this section, for some amorphous (rubbery) polymeric materials, values for n will be derived from fitted values for c for a comparison with non-polymeric solvent n values. Sprunger et al. (2007b) derived an Eq. (2) type solvation equation for polydimethyl siloxane(PDMS)–air partitioning. Fitted values for c and l were 0.045 and 0.856, respectively. These values suggest that nPDMS = 0.65. For the type 1 solvation equation for PDMS–water partitioning, they obtained c = 0.246. That suggests nPDMS = 0.72. The average of 0.69 ± 0.05 for these two nPDMS values is at the high end of the range of 0.5–0.75 for organic solvents. This

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is caused perhaps by the relatively high PDMS density because the high molecular weight PDMS density at 293 K is about 960 g/L (Dee et al., 1992) and because n values for non-halogenated solvents tend to increase on increasing molecular weight and density (see Table S-4). Goss (2005a) derived Eqs. (2) and (3) type solvation equations for capacity factors (k0 ) on 3 megabore columns (DB-1, DB-225, and DB-WAX). The capacity factor is related to the stationary phase-gas partition coefficient (Kd) by k0 = KdVs/V0 (Scott and Kucera, 1977), where Vs and V0 are the volume of the adsorption layer and the dead volume, respectively. For a zero volume solute, Kd = n/ eE. Therefore, log k0 0 = log n4d/eED, where d and D are the thickness of the adsorption layer and the internal column diameter, respectively. Based on this relation, with adsorption layer thicknesses from Li (1992), taking D = 0.53 mm (Jianjun Li, personal communication), and using the fitted c and l values obtained by Goss, the n values for DB-1, DB-225, and DB-WAX are calculated to be 1.04 ± 0.02, 0.60 ± 0.01, and 0.73 ± 0.04, respectively. The fitted DB-225, and DB-WAX n values are comparable to those for PDMS (0.69), dodecylsulfonic acid (0.64), and decanoyl phosphatidylcholines (0.65). The n value for DB-1 (1.04) seems to be an outlier for unknown reasons. For rubbery polymers, a first impression is that a default n value of 0.68 seems reasonable. This will be explored below for two other polymers. For polyoxymethylene (POM)–water partitioning at 25 °C, Endo et al. (2011) obtained c = 0.37 ± 0.11 for the type 1 solvation equation. POM is partly (75%) crystalline (also indicated by a high density of 1.452 g/cm3) (Linton and Goodman, 1959). The difference of 0.58 log units between the fitted c value of 0.37 and the theoretical value of 0.21 for rubbery polymers suggests a cristallinity of 73%, which is comparable to the literature value of 75%. For polyacrylate–water partitioning at 20–25 °C, the fitted value of c for the type 1 solvation equation is 0.12 ± 0.11 (Endo et al., 2011b), which is less than the expected value of 0.21 for rubbery polymers. The polyacrylate density of about 1.03 (Sham and Walsh, 1987) does not indicate substantial cristallinity. Fitting the original data to a type 1 equation with c set at 0.21 resulted in a negligible increase of the standard error of the fit from 0.23 to 0.24 while r2 increased from 0.97 to 0.994. The fitted value of 0.12 ± 0.11 as obtained by Endo et al. may therefore well be a statistical artifact. These results suggest that, for many polymers, n = 0.68 is useful when corrections are made if cristallinity is substantial. These corrections can be done on the basis of polymer density. Further work is needed to derive density–cristallinity relations. 3.8. Practical limitations For chemically very poorly defined materials, estimation of c becomes even more complicated. For instance, values for c will be impossible to derive for air-to-blood solvation equations. For Pahokee peat log Koc values, Bronner and Goss (2011) recently derived solvation equations. Fitting their log Koc data (assuming that the peat density is 1 g/cm3) to S, A, B, V and (L + 0.67) and to E, S, A, B, and (L + 0.67) gave values for c (=log (nOC/nwater)) of 0.35 ± 0.11 and 0.26 ± 0.13, respectively. The average nOC value of 0.14 is much smaller than for organic solvents. That suggests that a substantial part, roughly 80%, of the peat is not accessible to organic contaminants. Pahokee peat density is rather high at 1.52 g/cm3 (Dinar et al., 2006). At this density, and taking C4.70H3.82O2.33N0.26S0.022 as the Pahokee peat formula from http://www.ihss.gatech.edu/elements.html, n for Pahokee peat would be 0.93. In other words, 7% of the Pahokee peat volume is free volume. In organic solvents, 25–50% of the volume is free volume, which is 4–7 times more than in Pahokee peat. This suggests (again, see above) that 75–85% of the Pahokee peat is very densely packed and may not be accessible to contaminants. This densely packed material is probably not crystal-

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line because, for a Leonardite humic acid sample, 3.3% of the total material (wt.%) is represented by crystalline components (Chilom and Rice, 2005), which is comparable to the 2.5–4% of crystalline material in two humins and one humic acid derived by using quantitative solid-state 13C NMR Hu et al., 2000). Presumably, the densely packed Pahokee peat fraction consists of highly cross-linked material. 4. Conclusions For the Ben-Naim standard state based analysis of solvation, the present analysis suggests that solute concentrations are better based on the van der Waals volume of the solvent than on the macroscopic volume. Furthermore, for partitioning into molecularly well-defined solvents, the value of the constant c in Abraham solvation equations can be calculated from the solvent packing density, which can be directly obtained from the solvent molecular weight, the van der Waals volume, and the solvent density. For many less well-defined solvents, the solvent packing density can be estimated by analogy reasoning. Acknowledgements Hans Peter Arp is thanked for stimulating comments at the start of the work. Discussions with Arieh Ben-Naim on the van der Waals volume issue were very helpful. Kai-Uwe Goss is thanked for many discussions and suggestions during the development of the work and for comments on the manuscript. Satoshi Endo is thanked for comments on the manuscript and for prepublication copies of his work on polyparameter LFERs for membrane-water and POM-water partitioning. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.chemosphere.2011.11.073. References Abraham, M.H., Acree Jr., W.E., 2006. Comparative analysis of solvation and selectivity in room temperature ionic liquids using the Abraham linear free energy relationship. Green Chem. 8, 906–915. Abraham, M.H., Acree Jr., W.E., 2008. Comparison of solubility of gases and vapours in wet and dry alcohols, especially octan-1-ol. J. Phys. Org. Chem. 21, 823–832. Abraham, M.H., McGowan, J.C., 1987. The use of characteristic volumes to measure cavity terms in reversed phase liquid chromatography. Chromatographia 23, 243–246. Abraham, M.H., Andonian-Haftvan, J., Osei-Owusu, J.P., Sakellariou, P., Urieta, J.S., Lopez, M.C., Fuchs, R., 1993. Hydrogen bonding. Part 25. The solvation properties of methylene iodide. J. Chem. Soc. Perkin Trans. 2, 299–304. Abraham, M.H., Andonian-Haftvan, J., Whiting, G.S., Leo, A., Taft, R.S., 1994a. Hydrogen bonding. Part 34. The factors that influence the solubility of gases and vapours into water at 298 K, and a new method for its determination. J. Chem. Soc. Perkin Trans. 2, 1771–1791. Abraham, M.H., Chadha, H.S., Whiting, G.S., Mitchell, R.C., 1994b. Hydrogen bonding. Part 32. An analysis of water–octanol and water–alkane partitioning and the D log P parameter of seiler. J. Pharm. Sci. 83, 1085–1100. Abraham, M.H., Whiting, G.S., Carr, P.W., Ouyang, H., 1998a. Hydrogen bonding. Part 45. The solubility of gases and vapours in methanol at 298 K: an LFER analysis. J. Chem. Soc. Perkin Trans. 2, 1385–1390. Abraham, M.H., Whiting, G.S., Shuely, W.J., Doherty, R.M., 1998b. The solubility of gases and vapours in ethanol – the connection between gaseous solubility and water–solvent partition. Can. J. Chem. 76, 703–709. Abraham, M.H., Platts, J.A., Hersey, A., Leo, A.J., Taft, R.W., 1999a. Correlation and estimation of gas–chloroform and water–chloroform partition coefficients by a linear free energy relationship method. J. Pharm. Sci. 88, 670–679. Abraham, M.H., Le, J., Acree Jr., W.E., Carr, P.W., 1999b. Solubility of gases and vapours in propan-1-ol at 298 K. J. Phys. Org. Chem. 12, 675–680. Abraham, M.H., Zissimos, A.M., Acree Jr., W.E., 2001a. Partition of solutes from the gas phase and from water to wet and dry di-n-butyl ether: a linear free energy relationship analysis. Phys. Chem. Chem. Phys. 3, 3732–3736. Abraham, M.H., Le, J., Acree Jr., W.E., Carr, P.W., Dallas, A.J., 2001b. The solubility of gases and vapours in dry octan-1-ol at 298 K. Chemosphere 44, 855–863.

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