Fátima L Mota

António J Queimada

Simão P Pinho

Eugénia A Macedo

Solubility of Drug-like Molecules in Pure Organic Solvents with the CPA EoS

Keywords: CPA, drug-like, modelling, solubility, organic solvents

Solubility data in different solvents are an important issue for separation processes involving complex molecules such as natural products and pharmaceutical drugs. Nonetheless, solubility data are in general scarce and difficult to obtain, and so models are important tools to generate the necessary estimates.

Different correlative, statistical and thermodynamic models have been proposed to evaluate solubilities. From these, the more theoretically sound thermodynamic models allow to generate estimates at broader temperature, pressure and composition conditions while using a smaller amount of experimental information. Among these, the Cubic-plus-Association equation of state that combines the simplicity and robustness of a cubic equation of state with the Wertheim's association contribution has been under attention. In this work, this EoS is for the first time proposed to model organic phase solubilities of drug-like molecules in a wide range of temperatures.

Solubilities of acetanilide, acetylsalicylic acid, adipic acid, ascorbic acid, hydroquinone, ibuprofen, paracetamol and stearic acid were estimated in alcohols, ketones, alkanes, esters, acids, aromatics, chlorinated solvents, as well as in other common solvents. The hydrogen bonding behaviour was explicitly accounted for with each associating group being treated individually, as well as multiple group substitutions.

Accurate correlations were obtained using a single binary interaction parameter (global AAD of 24.2%), while considering the complexity of the studied systems predictions were generally also satisfactory.

A c c e p t e d M a n u s c r i p t 3

Introduction

The first step in the physicochemical characterization of a pharmaceutical drug candidate involves solubility studies. In general, solubility data in several solvents at various temperatures are important during pre-formulation steps [1].

Given the complexity of most drug and drug-like molecules, and the variety of their solvent interactions, many thermodynamic tools have been proposed for modelling such equilibrium data. Reviews on the most commonly used models in the pharmaceutical industry can be found elsewhere [2,[START_REF] Pinho | Solubility in Food, Pharmaceuticals, and Cosmetic Industries[END_REF]. However, the readily application of these theoretical methods is commonly hampered by the lack of information about the molecules under study, and therefore empirical or semiempirical models are usually used, showing good agreement between experimental and correlated data [START_REF] Maia | [END_REF][5][6][7][8][9][10][11].

One of the most widely used thermodynamic approaches in the pharmaceutical industry is the regular solution theory [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-phase Equilibria[END_REF], which is still being used [START_REF] Hojjati | [END_REF]. The solubility is a function of the solvent and solute solubility parameters, and the determination of solubility parameters has been subject of many studies [14][15][16]. As a drawback, solubilities higher than the ideal solubility cannot be described, as the minimum activity coefficient is one. In spite of some poor results were obtained and temperature dependence cannot be described accurately, this model became very popular for solvent screening due to its higher predictive character.

On the other hand, models based on the excess Gibbs energy have been widely used and proved to be able to describe properly the experimental data [6,7,[START_REF] Hojjati | [END_REF][17][18][19].

However, the application of these models is limited to the availability of interaction parameters or, when not available, to the existence of data for their fitting. In this sense, the application of predictive group-contribution methods, such as UNIFAC, has been increasing [START_REF] Hojjati | [END_REF]19,20], but the description of the solubility temperature-dependence of A c c e p t e d M a n u s c r i p t 4 high molecular weight compounds are still a weakness of this model [21], as well as the necessary group-interaction parameters can sometimes be unavailable. A successful and quite recent model is the Non-random Two Liquid Segment Activity Coefficient (NRTL-SAC) model [22,23], which has been widely applied to correlate and predict phase equilibria of non-ideal systems, like those containing complex pharmaceuticals [2,24]. One advantage is that it only requires a set of well-chosen data to fit a small number of parameters. Despite its suitability, this model has a correlative nature since the solute parameters have to be estimated from mixture data. Other models applied for pharmaceuticals are based on statistical mechanics, especially COSMO and its derivatives, where only data from quantum chemical calculations are used [20]. Some attempts were done to use these models for complex pharmaceuticals, but their capabilities for solid-liquid equilibria have not yet been proven [20,21,25].

The increasing interest on the high pressure processing of pharmaceuticals suggests the use of equations of state for modeling their phase equilibria. Some equations of state have already been applied for that purpose, not so widely to pharmaceutical compounds, but to more simpler drug-like molecules.

Ruether and Sadowiski [26] investigated the use of the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) to predict the solubility of five typical drug substances and intermediates in pure and mixed solvents. The model allowed an accurate description of the temperature dependence of the solubility applying a linear temperature-dependent binary interaction parameter per system. The solubility in mixed solvents was reliably predicted without any additional fitted parameter. Tsivintzelis et al. [27,28] applied the nonrandom hydrogen-bonding (NRHB) EoS to model the phase behaviour of several mixtures of pharmaceuticals. Very satisfactory correlations were obtained using only one optimized binary interaction parameter, even in supercritical Recently, our group successfully used the CPA EoS for modeling the aqueous solubility of drug-like molecules [29].

The aim of this work is to evaluate the CPA EoS for modeling drug-like solubilities in organic solvents in a wide temperature range. The solubilities of acetanilide, acetylsalicylic acid, adipic acid, ascorbic acid, hydroquinone, ibuprofen, paracetamol and stearic acid were studied in solvents covering several alcohols, ketones, alkanes, esters, benzene derivatives and carboxylic acids. The molecular structures of the compounds under study are presented in Figure 1.

As observed before for the aqueous solubilities [29], there is a good agreement between the experimental and calculated data using a single temperature independent binary interaction parameter.

Thermodynamic framework

The solubility of a solid in a liquid solvent is calculated by solving the thermodynamic equations of equilibrium [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-phase Equilibria[END_REF]. Assuming a pure solid phase, no significant temperature dependence on the difference between the heat capacity of the pure liquid and solid (C p ), and neglecting the effect of pressure on melting temperature (T m ), solute enthalpies of solid-solid and melting transition ( tr H), and C p , the following expression for the mole fraction solubility is obtained [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-phase Equilibria[END_REF]:

A c c e p t e d M a n u s c r i p t 6                                        T T T T R C T T R H x m m p tr tr tr liq s liq s s ln 1 1 1 exp 0   (1)
where x s is the solute mole fraction solubility, R is the ideal gas constant, T is the absolute temperature, is the fugacity coefficient and subscripts 0 refers to a pure component, tr for a pure solute phase transition and m for melting. The different solidsolid and fusion phase transitions of the solute are concerned in the summation.

The CPA-EoS [30] combines a cubic contribution accounting for physical interactions, and a chemical term from the Wertheim theory, taking into account the association due to hydrogen bonding between like molecules (self-association) and unlike molecules (cross-association or solvation). The activity of each bonding site is assumed independent of the other bonding sites on the same molecule being steric hindrance and cooperativity effects neglected. This model has already been successfully employed for mixtures of water with other associating compounds [29,[31][32]. The cubic and association contributions to the Helmholtz energy (A) are the following [31,[START_REF] Kontogeorgis | [END_REF]34]:

         b nRT b b b an A cubic               1 ln 1 1 ln 2 1 1 2 (2)              i i i A A A i i assoc X X n RT A 2 1 2 ln ( 3
)
where i is a component index, b is the co-volume parameter, a is the energy parameter,  is the molar density, n is the number of moles, and given by a Soave-type temperature dependency, while b is constant:

      2 1 0 1 1 r T c a T a    (4)
where a 0 and c 1 are pure component parameters and T r is the reduced temperature. For mixtures, classical van der Waals one-fluid mixing rules are employed, where a binary interaction parameter k ij is introduced on the cross-energy parameter.

     j j i j i B B A B j j A X x X  1 1 (5)
For cross-associating systems, the computationally simpler Elliot rule is adopted for the cross-association strength [35].

Experimental solubility data are used for the regression of the k ij parameters with the objective function (OF):

2 exp exp            NP i i calc i i x x x OF ( 6
)
where NP is the number of available solubility points, and the superscripts exp and calc refer to experimental and calculated values, respectively.

The absolute average deviations (AAD) were used to compare experimental and calculated results:

Results and Discussion

In a previous work, the nature and number of associating groups in each of the studied solutes were established [29], following the associating schemes proposed by Huang and Radosz [36]. The one-site (1A) scheme was used for carboxylic acid groups; the two-site (2B) for alcohols and the three-site (3B) and two-site (2B) schemes for primary and secondary amines, respectively [34]. Solvents like chloroform, alkanes, 1,4-dioxane, ketones, toluene, acetonitrile and esters are considered to be non-selfassociating.

The pure component parameters were obtained regressing experimental vapor pressure and liquid density data compiled from the DIPPR database [START_REF]DIPPR 801 Thermophysical Properties Database[END_REF], where the critical properties and van der Waals volume, as well as property correlations as a function of temperature, were also collected. These properties and parameters were presented by Mota et al. [29]. Regression of all the CPA parameters was performed in the reduced temperature range from 0.45 up to 0.85 where, in general, the DIPPR correlations are in agreement with the experimental data. Only for hydroquinone and paracetamol liquid density data are not available, and like that just the correlations are used. The AAD's found for vapor pressure and liquid density were 2.5 and 1.8%, respectively, which are good results given the complexity of the studied molecules.

Parameters for the various solvents are well established and were collected from the A c c e p t e d M a n u s c r i p t 9 literature [START_REF] Kontogeorgis | [END_REF][START_REF] Folas | [END_REF] or whenever unavailable were estimated for this work. The used parameters and respective associating schemes are presented in Table 1. The thermal properties, melting temperature (T m ), enthalpy of fusion ( fus H) and heat capacity difference (C p ) are also presented in Table 1 for all the drug molecules [29].

Following a previous work where the aqueous solubilities of the same drug-like molecules were studied [29], in this work the same set of CPA parameters for describing the solubility in organic solvents are evaluated. Results of the model predictions (without fitting any parameter) and correlations (with a temperature independent binary interaction parameter k ij ) are given in Tables 2 to 5.

All the investigated solutes are associating molecules, while some solvents such as ketones, esters, cyclic ethers and toluene, although not explicitly forming solutesolvent hydrogen bonds, can present increased solubilities due to stronger solvating interactions. To account for this issue, Folas et al. [START_REF] Folas | [END_REF] proposed to use a crossassociation energy half of that of the associating component and left the crossassociation volume as an adjustable parameter, to be fitted from equilibrium data. This was initially done in this work but it was found that in most of the cases, this fitting was not necessary, as the fit of a single and small k ij already provided excellent results. Whenever higher k ij values were found for a particular solute it was also observed that this also occurred for associating solvents. So to keep a higher predictive character of the model, only the k ij values were fitted from binary data.

The results for acetanilide are presented in Table 2. Except for the alkane solvents (n-hexane and cyclohexane), where CPA overpredicts the experimental data, the model is able to represent the data with small k ij 's. In the alkane solvents the solubilities are, unexpectedly, very low, considering that the solvent with the maximum solubilities is octanol. For this same solvent the model is able to correctly estimate the A c c e p t e d M a n u s c r i p t 10 solubilities. In Figure 2, the solubility of acetanilide in the studied alcohols is presented and it can be seen that as long as the molecular weight of the alcohol increases, the solubility increases, but the model reverses the order between methanol and ethanol.

The results obtained by Tsivintzelis et al. [27] with the NRHB EoS in chloroform are better (AAD of 1.5%). Only one parameter was used in both works, but these authors investigated a narrower temperature range. However, if the regression of k ij is made in the same temperature range as Tsivintzelis et al. [27], a higher value is found and the results have almost the same accuracy. In octanol, the results found in this work are better than those found by Tsivintzelis et al. [27] with the NRHB EoS.

Acetylsalicylic acid has the associating group -COOH and the ester group (-COO), which are able to cross associate with alcohol molecules. Thus, the associating behaviour in the corresponding solutions is rather complicated. The correlation results showed that all the interaction parameters are small and negative (Table 3), proving that the interactions are stronger than expected by the model. Nevertheless the model is still able to adequately correlate the data. For this solute, the solubility values are similar, except in acetone where they are slightly higher. Indeed, acetone is the solvent where the model gives the best estimates, both for prediction and correlation. The results in alcohols are in the same accuracy range of those obtained before for aqueous solubilities (AAD of 15.5%) [29], but the k ij values were significantly more pronounced for the water binaries.

Excluding the adipic acid/toluene system where the solubilities are very low, the correlation results found with CPA are very good, with small interaction parameters (Table 3). The results are in the same order of accuracy of those found before by Mao et al. [7] with the UNIFAC method, and not much worse than those found with the h and Apelbat correlative equations. Comparing with the results obtained for aqueous A c c e p t e d M a n u s c r i p t 11 solubilities in a previous work [29], only for the toluene system poorer results were obtained. As presented in Figure 3, very good correlative results were found for the systems acetylsalicylic acid/acetone and adipic acid/acetone at the expense of a small binary interaction parameter. Ascorbic acid is a complex chiral molecule containing as much oxygens as carbons in its molecular structure. It has four hydroxyl groups located in different environments that for modelling purposes were divided into two classes: ring and aliphatic, having different associating parameters [29]. The solubilities are low in all the solvents studied and probably the high melting point is a reason for that. In Table 3 it can be seen that the model results are rather satisfactory. If a comparison is made with the previous solutes, the interaction parameters are more pronounced, except in the case of acetonitrile where the model is satisfactory even in pure prediction. A great improvement is achieved after estimating a single temperature independent binary interaction parameter: in average, from 87.1 to 18.1%. In Figure 4 a comparison of the correlated solubilities in several solvents is presented. Among the solvents studied in this work and the previous results in water [29], it can be concluded that the results here are much better than for the water solubilities.

Hydroquinone is the solute showing more pronounced corrections (Table 4), except in acetic acid where the predictions are excellent. The solubility values in this solvent are the lowest, in the order of 10 -2 (mole fraction) and, like that, do not contradict some indications from the other solutes that showed larger deviations in solvents where the solubilities were very low. Excluding esters, where the correlated solubilities are overestimated, the values in the other solvents are underestimated. In Figure 5, the results for the solubility in ethyl acetate are presented. Once again, the A c c e p t e d M a n u s c r i p t 12 studied solvents in this work presented better results than the aqueous solubilities [29], but in this case the k ij were in the same order of magnitude.

Among the studied compounds, ibuprofen has in general the higher solubilities and also the lowest melting point. Figure 6 shows a comparison for the solubility of ibuprofen in ethanol and ethyl acetate. The results for ibuprofen are presented in Table 5 where it can be seen that the higher interaction parameters were found for the simplest alcohol solvents. In the systems ibuprofen/acetone, 4-methyl-2-pentanone or propylene glycol, the interaction parameters and the results found in this work are very close to those by Tsivintzelis et al. [28] with the NRHB EoS. For the other systems, the results reported by these authors are slightly better. Among the solvents studied, only in 2propanol and cyclohexane the AAD's were higher than those obtained for aqueous solubilities, but in this last case the k ij (absolute value) was higher [29].

Good correlation results were also obtained for paracetamol in 16 solvents with an average deviation of 28% (Table 2) and a maximum deviation of 65% in cyclohexane where the solubilities are especially low. However, even in this solvent, the improvement was significant between prediction and correlation, using a small interaction parameter. It must be noted that the presence of the -ACOH group renders the hydrogen-bonding behaviour of paracetamol much more complicated compared to that of acetanilide, but the results are similar, what shows the importance of explicitly considering association. A comparison between several prediction and correlation results is presented in Figure 7, showing the importance of fitting a k ij parameter. In 2propanol, the results are almost similar to those obtained by Hojjati and Rohani [START_REF] Hojjati | [END_REF] with the UNIQUAC model, what is very good since this model has two adjustable parameters from experimental binary data. The interaction parameters found in this work for alcohols are all negative, except for propylene glycol, where those found A c c e p t e d M a n u s c r i p t 13 previously with the NRHB EoS [27] are mostly positive. So, in prediction, CPA considers the solute-alcohol interactions to be weaker, while the NRHB EoS considers them to be stronger. Generally, the results found by these authors are slightly better.

The predictions of the mole fraction solubilities of stearic acid are surprisingly good (average AAD of 61.4%) and, after regressing small binary interaction parameters, the deviations are further reduced being specially good for the alkanes, as can be seen in Table 3. The solubility results for n-heptane are presented in Figure 8.

Ethanol systems were studied for all the solutes and, in all of them, CPA estimates were good. The maximum AAD found was 39% in paracetamol and the average AAD was 22%. Except for paracetamol in propylene glycol, all the binary interaction parameters for these solutes in alcohols are negative, showing that the interactions are stronger than that estimated by the model.

In acetone, the results are also good with an average AAD of 21%, as well as in the esters where the deviations did not exceed 41%. However, for the esters it must be focused that the group -COO is considered non-self-associating, and its effects to increase the solvation are not explicitly taken into account by the model.

Many pharmaceutical solids and molecular crystals show polymorphism, which

is the ability to exhibit more than one crystalline form, displaying different thermodynamic properties [START_REF] Brittain | Polymorphism in Pharmaceutical Solids[END_REF]. In fact, each polymorph should be regarded as a different solid material and the control of polymorphism provides a way to tune the properties of a product. As referred in the previous work [29], some of the compounds studied in this assay were reported to show polymorphic forms. Solubility data of stearic acid polymorphs were available in n-hexane and methanol [18]. In the n-hexane system, good predictions were obtained in the case of form B (AAD 13.0%), while for form C the use of the corresponding k ij value (Table 3) A c c e p t e d M a n u s c r i p t 14 was required, leading to an AAD of 49.2%. In the case of methanol, better estimates were obtained for the solubility of form C (AAD of 19.9% with k ij = -0.037) than for form B (AAD of 46.8% with k ij = -0.031). No polymorph solubility data for the other solutes were found, and so, only the effect of the melting properties could be evaluated.

For paracetamol, the results considering the melting data of form I [START_REF] Sacchetti | [END_REF] were better (AAD of 26.0%). For example in methanol, the AAD is 0.64% with the melting data of form I, while with the DIPPR properties or form II, the AAD's are 14.9 and 33.5%, respectively. Ibuprofen is one of the most widely known chiral compounds, and also no solubility data were found for the racemate and enantiomers [41]. But, analyzing in terms of different thermal data, different values of binary interaction parameters were obtained for the (+)-enantiomer and the racemic mixture, leading to different correlation results. The use of the racemic properties gave in general better results.

Conclusions

Given the previous success of the CPA EoS to model complex molecules in aqueous systems, its performance for organic systems of complex molecules was investigated in this work. The CPA EoS was applied to model the solubility of several drug-like molecules, namely acetanilide, acetylsalicylic acid, adipic acid, ascorbic acid, hydroquinone, ibuprofen, paracetamol and stearic acid in liquid solvents: such as alcohols, alkanes, acetone, acids and esters. The complex associating behavior was explicitly accounted for in the association energy and volume parameters, where the corresponding parameters were estimated from vapor pressure and liquid density data.

Each associating group is treated individually as well as multiple group substitutions were considered in the association term.

A c c e p t e d M a n u s c r i p t 15

In the solvents studied, a rather good representation of the solubility data is obtained: AAD's of 66.7 and 24.2%, respectively for model predictions and correlations (using only one temperature independent binary interaction parameter per system). In general the results obtained are better than those for the aqueous systems of the same complex molecules. Very good results were obtained in some acetone systems where the solubilities were the highest, as well as in some alcohols. In general, the binary interaction parameters were small and, in the majority of the cases, negative, showing that the real interactions are stronger than those estimated by the model. Often the predictions are poor for very insoluble materials, but after correlating relatively small and temperature independent binary interaction parameters good estimates were obtained. Using thermal properties of different polymorphic forms, solubility results showed quite the same quality. Even the solubility of ibuprofen, considering its enantiomer or racemic properties was successfully correlated.

In general, the CPA EoS showed to be able to represent the solid-liquid equilibrium data of drug-like molecules, proving the importance of taking into account association effects. The results obtained so far in pure solvents suggest further work concerning mixed solvent systems, as well as supercritical solvents.

A c c e p t e d M a n u s c r i p t 5 CO 2

 52 and ethane. This model was also able to predict accurately the solubility in mixed solvents based on interaction parameters fitted to the corresponding single solvent data.

X

  is the mole fraction of component i not bonded at site A. As the Soave-Redlich-Kwong cubic term is used in A c c e p t e d M a n u s c r i p t 7 this work,  1 is set equal to 1 and  2 to 0. The pure-component energy parameter a is

X

  is the key element of the association term and is related to the association strength, two sites belonging to two different molecules, site A on molecule i and site B on molecule j by:

A c c e p t e d M a n u s c r i p t 16 Acknowledgement

 16 The authors are grateful for the support provided by Fundação para a Ciência e a Tecnologia (FCT, Lisboa, Portugal) and by LSRE financing by FEDER/POCI/2010. A.J. Queimada acknowledges support from POCI/N010/2006. F.L. Mota acknowledges her FCT Ph.D. scholarship SFRH/BD/32372/2006, co-financed by the Fundo Social Europeu (FSE). Nomenclature a = energy parameter a 0 = parameter in the CPA EoS cubic term (Pa.m 6 /mol 2 ) A = Helmholtz energy (J) b = co-volume parameter (m 3 /mol) c 1 = parameter in the CPA EoS cubic term k ij = binary interaction parameter n = number of moles NP = number of data points R = gas constant (8.314 J/(mol.K)) T = absolute temperature (K) x = mole fraction A c c e p t e d M a n u s c r i p t 17 i A X = mole fraction of component i not-bonded at site A Greek Symbols  = specific parameters in each cubic equation  = association volume  ij = solvation parameter j i B A  = association strength between site A on molecule i and site B on molecule j C p = difference between the heat capacity of the solute pure liquid and solid (J/mol.K)  fus H = fusion enthalpy (J/mol)  = association energy (J/mol)  = molar density (mol/m 3 )   = fugacity coefficient Subscripts 0 = pure component c = critical i, j, k = component index m = melting r = reduced s = solute A c c e p t e d M a n u s c r i p t 18 tr = transition Superscripts assoc = association term calc = calculated cubic = cubic term exp = experimental List of Abbreviations AAD =absolute average deviation (%) CPA = cubic-plus-association EoS = equation of state NRHB = Non-Random Hydrogen Bonding OF = objective function A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t 23

A c c e p t e d M a n u s c r i p t 28 Figure Captions:Figure 1 .

 281 Figure Captions:

Figure 2 .

 2 Figure2. Solubility of acetanilide in methanol[42] (◊), ethanol[42] (•) and 1-octanol[43] (■), and CPA correlations (-, methanol and 1-octanol; ---, ethanol).

Figure 3 .

 3 Figure 3. Solubility of acetylsalicylic acid [4] (•) and adipic acid [7] (■) in acetone, and CPA correlation results.

Figure 4 .

 4 Figure 4. Summarized results for ascorbic acid solubility in several solvents: experimental data [53] versus CPA correlations: ethanol (•), methanol (o), 2-propanol (♦), acetone (◊), acetonitrile (■), ethyl acetate (□), and tetrahydrofuran (▲).

Figure 5 .

 5 Figure 5. Solubilities of hydroquinone as a function of temperature in ethyl acetate [9],and CPA modelling results: correlation (-) and prediction (---).

Figure 6 .

 6 Figure 6. Solubilities of ibuprofen as a function of temperature in (a) ethanol[25] and (b) ethyl acetate[25], and CPA modelling results: correlation (-) and prediction (---).

Figure 7 .

 7 Figure 7. Summarized results for paracetamol in several solvents: experimental data versus CPA (a) predictions and (b) correlations: ethanol [46, 49] (•), methanol [46] (o), 2-propanol[46,[START_REF] Hojjati | [END_REF] (♦), acetone[46,50] (◊), acetonitrile[46] (■), ethyl acetate[46] (□), propylene glycol[47,49] (∆), 1-propanol[46] (),and 1-butanol[46] (x).

Figure 8 .

 8 Figure 8. Solubilities of stearic acid as a function of temperature in n-heptane [19] and CPA results: correlation (-) and prediction (---).

A c c e p t e d M a n u s c r i p t 30

 30

Figure 2 :A c c e p t e d M a n u s c r i p t 32

 232 Figure 2:

Figure 4

 4 Figure 4:

Figure 5 :

 5 Figure 5:

Figure 8 :u s c r i p t 39 Highlight

 839 Figure 8:

Table 1 . Pure component properties and CPA parameters. 1

 1

	Compound	T m (K)	 fus H (kJ/mol)	C p (J/mol.K)	a 0 (Pa m 6 /mol 2 )	c 1	b x 10 5 (m 3 /mol)	Group	(kJ/mol) 	
	Acetanilide [29]	388	21.65	1.92	4.37	1.17	12.25	NH	3.74	1.00 x10 -1
	Acetylsalicylic acid [29]	408	25.60	26.55	4.62	1.48	13.56	COOH	11.4	1.68 x10 -4
	Adipic acid [29]	426	34.85	108.07	4.80	1.49	12.48	COOH	18.5	3.45 x10 -4
	Ascorbic acid [29]	465	29.20	81.40	3.79	2.96	11.94	OH ring OH aliphatic	14.9 14.7	9.08 x10 -3 4.10 x10 -5
	Hydroquinone [29]	445	27.11	55.19	2.20	1.27	10.37	OH	19.2	2.38 x10 -2
	Ibuprofen [29]	349	25.61	149.74	7.33	1.37	20.47	COOH	1.10	9.61 x10 -4
	Paracetamol [29]	441	27.70	32.14	3.45	1.76	15.96	NH	6.45	5.48 x10 -2
	Stearic acid [29]	343	61.21	116.84	12.2	1.58	33.47	OH	17.9	2.06 x10 -2
	methanol [33]				0.405	0.431	3.09	OH	2.46	1.61 x10 -2
	ethanol [33]				0.867	0.737	4.91	OH	2.15	8.00 x10 -3
	1-propanol [33]				1.19	0.917	6.41	OH	2.10	8.1 x10 -3
	2-propanol [33]				1.06	0.947	6.41	OH	2.10	9.1 x10 -3
	1-butanol [33]				1.57	0.978	7.97	OH	2.10	8.2 x10 -3
	1-octanol [33]				4.16	1.15	14.9	OH	2.68	1.40 x10 -4
	propylene glycol [33]				1.38	0.937	6.75	OH	1.74	1.90 x10 -2
	acetic acid [33]				0.912	0.464	4.69	COOH	4.03	4.5 x10 -3
	n-hexane [38]				2.37	0.831	10.8
	n-heptane [38]				2.92	0.914	12.5
	cyclohexane*				2.11	0.771	9.09
	acetone*				1.39	0.788	6.14
	4-methyl-2-pentanone*				2.41	0.856	8.72
	ethyl acetate*				1.87	0.957	8.31
	butyl acetate*				2.92	1.03	11.7
	isopropyl myristate*				10.5	1.57	32.1
	1,4-dioxane*				1.85	0.825	7.39
	chloroform*				1.50	0.782	6.64
	toluene*				2.34	0.804	9.24
	acetonitrile*				1.28	0.605	4.68
	tetrahydrofuran*				1.56	0.771	6.80
	trichloroethylene*				1.85	0.771	7.60
	2	*Obtained in this work.
	3

Table 2 . CPA Modelling Results for Acetanilide and Paracetamol, and Respective Binary Interaction Parameter.

 2

	Prediction	Correlation

Table 3 . CPA Modelling Results for the Acids, and Respective Binary Interaction Parameter.

 3

	Prediction	Correlation

a Without outliers: AAD > 150%.

Table 4 . CPA Modelling Results for Hydroquinone, and Respective Binary Interaction Parameter.

 4

	Prediction	Correlation

Table 5 . CPA Modelling Results for Ibuprofen, and Respective Binary Interaction Parameter.

 5

	Prediction	Correlation


Anto ´nio

J Queimada

Fa ´tima

L Mota

Sima ˜o

P Pinho

Euge ´nia

A Macedo

Solubilities of Biologically Active Phenolic Compounds: Measurements and Modeling

Aqueous solubilities of natural phenolic compounds from different families (hydroxyphenyl, polyphenol, hydroxybenzoic, and phenylpropenoic) were experimentally obtained. Measurements were performed on tyrosol and ellagic, protocatechuic, syringic, and o-coumaric acids, at five different temperatures (from 288.2 to 323.2 K), using the standard shake-flask method, followed by compositional analysis using UV spectrophotometry. To verify the accuracy of the spectrophotometric method, some data points were measured by gravimetry, and in general, the values obtained with the two methods are in good agreement (deviations lower than 11%). To adequately understand the solubilization process, melting properties of the pure phenolics were obtained by differential scanning calorimetry (DSC), and apparent acid dissociation constants were measured by potentiometry titration. The aqueous solubilities followed the expected general exponential trend. The melting temperatures did not follow the same solubility tendency, and for tyrosol and ellagic acid, not only the size and extent of hydrogen bonding, but also the energy associated with their crystal structures, determine the solubility. For these binary systems, acid dissociation is not important. Approaches for modeling the measured data were evaluated. These included an excess Gibbs energy equation, the modified UNIQUAC model, and the cubic-plus-association (CPA) equation of state. Particularly for the CPA approach, a new methodology that explicitly takes into account the number and nature of the associating sites and the prediction of the pure-component parameters from molecular structure is proposed. The results indicate that these are appropriate tools for representing the water solubilities of these molecules.

Introduction

Phenolic compounds are a chemical family whose members have one or more hydroxyl groups attached directly to an aromatic ring. These compounds are characteristic of plants, some fruits, and vegetables, [START_REF] Vermerris | Phenolic Compound Biochemistry[END_REF] and as a group, they are usually found as esters or glycosides rather than as free compounds. [START_REF] Vermerris | Phenolic Compound Biochemistry[END_REF] Interest in phenolic compounds has grown greatly, with attention focused on finding naturally occurring antioxidants for use in foods or medical materials to replace synthetic components. [START_REF] Leopoldini | [END_REF] There is significant evidence that phenolic compounds have positive effects on human health, 3 with many recently published studies reporting anti-inflammatory and cardioprotective activities, 4 as well as antioxidant properties. [5][6][7] Recently, some of these compounds were reported to exhibit protective effects on hormone-dependent breast tumors, 3 as well as other types of cancers. [8][9][10] Perhaps the oldest medical application of phenolic compounds is the use of phenol as an antiseptic. [START_REF] Vermerris | Phenolic Compound Biochemistry[END_REF] Another very common use of phenolic compounds is in sunscreens. The presence of the aromatic ring results in the effective absorbance (and filtering) of UV-B radiation (between 280 and 315 nm) from the sun, thus preventing sunburn and other major consequences, such as skin cancer. Aside from medical applications, phenolic compounds, including flavonoids and tannins, are an integral part of human and animal diets, because they represent one of the most numerous and ubiquitous groups of plant metabolites. 10 In this work, the aqueous solubilities of some natural phenolic compounds such as tyrosol and ellagic acid, the hydroxybenzoic acids protocatechuic and syringic acids, and the phenylpropenoic o-coumaric acid are addressed. Their chemical structures are presented in Figure 1. The scarcity of data reported in the literature increases the importance of the development of correlation and prediction models. Different approaches can be found for modeling the solid solubilities of phenolic compounds.

Noubigh et al. 11,12 obtained experimental values for the solubilities in pure water and in some chloride solutions, at 298.2 K, of syringic, gallic, protocatechuic, vanillic, and ferulic acids, as well as vanillin. They represented the solubility data using an activity coefficient thermodynamic model based on the Pitzer-Simonson-Clegg formalism 12 and used van't Hoff plots 11 to obtain solution standard molar enthalpies. Apelblat et al. 13 measured the aqueous solubilities of some polycarboxylic acids, expressing the solubility with the Williamson equation. Lu and Lu 14 reported the solubilities of gallic acid and its esters in water as a function of temperature. They proposed an empirical equation to fit the data for gallic acid and octyl gallate, whereas for the other esters, the data were better represented with the simplified λ-h equation. Also for gallic acid, Daneshfar et al. 15 calculated the temperature dependence of the solubility data using the modified Apelblat model. The solubility of salicylic acid was investigated in pure solvents as a function of temperature by Nordstro ¨m and Rasmuson, 16 who observed a correlation between the solubility and van't Hoff enthalpy of solution. Shalmashi and Eliassi 17 used an empirical correlation, with two parameters, to fit the solubility data of salicylic acid.

Although the importance of phenolic compounds is wellknown, very few solubility data are available, and comprehensive experimental studies combining solubility with melting properties and acidity constants are clearly lacking in the literature. This work follows a previous one where the aqueous solubilities of other hydroxybenzoic (gallic and salicylic) and phenylpropenoic (cinnamic, ferulic, and caffeic) acids were addressed. 18 The final aim is to determine, and accurately predict, the solubilities of these molecules in different solvents in order to adequately understand the fate and separation processes of these compounds. To accomplish this, experimental measurements of solubility as a function of temperature were performed, and models were used for its estimation, namely, the modified UNIQUAC model 19,20 and an associating equation of state (EoS), the cubic-plus-association EoS. 21,22 2. Experimental Section 2.1. Materials. In all experiments, bidistilled water (∼2.5-3 µS/cm) was used. o-Coumaric acid (g98% purity; CAS no. 614-60-8), protocatechuic acid (g99% purity; CAS no. 99-50-3), syringic acid (g99% purity; CAS no. 530-57-4), ellagic acid (99% purity; CAS no. 476-66-4), and tyrosol (98% purity; CAS no. 501-94-0) were obtained from Sigma-Aldrich. All chemicals were used without any further purification.

2.2. Experimental Procedure. 2.2.1. Solubility Measurements. Aqueous solubilities were determined for each phenolic compound using the standard analytical shake-flask method, with constant-temperature jacketed glass cells, at five different temperatures ranging from 288.2 to 323.2 K. Saturated solutions were generated using an excess of solid and bidistilled water. These mixtures were then stirred on a plate stirrer for a period of time determined by continuously sampling the aqueous phase until no concentration change was observed. After equilibrium had been attained, the solutions were kept still. Maximum stirring times of 140 and 120 h were found for the lower and higher temperatures, respectively; settling times between 24 and 40 h were sufficient to guarantee thermodynamic saturation. Thermostatization was ensured by a constant-temperature ((0.10 K) circulating water bath (Grant LTC1) and monitored with four-wire platinum resistance probes [Pt(100)] placed in the thermostatic jackets and connected to an Agilent 34970A data acquisition unit ((0.01 K), previously certified with a maximum deviation of 0.06 at 303.67 K. Samples of the saturated solutions were taken using thermostatized plastic syringes coupled with syringe filters (0.45 µm) and analyzed by spectrophotometry (Thermo Electron Corporation UV1). The wavelengths used for analysis were 290 nm for o-coumaric, ellagic, protocatechuic, and syringic acids and 272 nm for tyrosol. The analysis was not done at the maximum absorption wavelength because of linearity problems. Further details about the solubility experimental procedure can be found elsewhere. 18 To verify the accuracy of the data measured using the spectrophotometric method, some analyses were carried out using the gravimetric method, with the same solubilization and sampling procedure. For each determination, three samples were withdrawn and inserted into preweighed glass vessels; the vessels were left to cool and were then weighed. The samples were then placed in the freezer (Christ lyophilizer, model Alpha 1-4 equipped with an FTS system vacuum pump) for 24 h and afterward were lyophilized for 48 h, to evaporate water. Finally, the vessels were again weighed. Each experimental value reported is an average of at least three different measurements.

DSC Measurements.

Thermograms of the solid phenolics were obtained with a Netzsch DSC 200 F3 Maia differential scanning calorimeter. Samples of 4-6 mg ((0.1) were sealed in aluminum crucibles and heated under nitrogen, while using an empty crucible as reference. Initial estimates of the melting temperatures were obtained using a 10 K/min heating rate over a larger temperature range. Afterward, several runs at a rate of 1 K/min around the expected melting temperature were averaged. To ascertain the precision of the instrument, an indium standard was run, and the obtained melting temperatures and heats of fusion were compared with the reported values. All experiments were conducted at least in triplicate.

Potentiometric Measurements.

Phenolic compounds are typically acidic, so evaluate the extent of solubilization due to acid dissociation, the pH values of the saturated aqueous solutions and the acid dissociation constants (pK a ) were measured by potentiometry, using a glass electrode and a VWR SympHony SB70P meter. Apparent acid dissociation constants were obtained by potentiometric titration of aqueous solutions of the phenolic acids with a sodium hydroxide solution.

Thermodynamic Properties of Dissolution

The most common way to evaluate the effect of temperature on solubility is the calculation of solubility products at different temperatures. Alternatively, the thermodynamic properties of dissolution, namely, the molar Gibbs energy (∆ sol G), enthalpy (∆ sol H), and entropy (∆ sol S) of dissolution can be calculated using experimental solubility data. These parameters reflect the modification of the solution properties due to the presence of the solute at infinite dilution, at a given temperature. Assuming that the activity coefficient of the solute is equal to 1 in the hypothetical dilute ideal solution and starting from the Gibbs-Helmholtz equation and the thermodynamic definition of the equilibrium constant (eq 2), the following equation can be obtained [START_REF] Adkins | Equilibrium Thermodynamics[END_REF] ∆ sol H ) RT 2

( d ln x s dT ) P (1)
where the molar enthalpy of dissolution, ∆ sol H, is the difference between the partial molar enthalpy of the compound in solution and the pure molar enthalpy at temperature T, x s is the mole fraction solubility of the solute, R is the ideal gas constant, and P is the pressure. By plotting ln x versus temperature and considering this plot as a straight line, ∆ sol H can be obtained. The remaining thermodynamic functions are obtained from the following expressions

∆ sol G ) -RT ln(x s ) P ( 2
)
∆ sol S ) ∆ sol H -∆ sol G T (3)

Modeling

Considering a pure solid phase with no solid-solid phase transitions, assuming the difference in the heat capacities between the liquid and solid phases of the solute (∆C p ) to be constant in the temperature range [T, T m ], and neglecting the effect of pressure (melting temperature and enthalpy, heat capacities, and Poynting term), the solubility of a solute s can be calculated from the following generalized expression that relates the reference-state fugacities 24 ln

[ f s liq (T, P) f s sol (T, P) ] ) ∆ fus H R ( 1 T - 1 T m ) - ∆C p R [ T m T -ln ( T m T ) -1 ] (4)
where ∆ fus H is the enthalpy of fusion, T is the absolute temperature, T m is the melting temperature, ∆C p is the difference of the liquid and solid molar heat capacities, and R is the ideal gas constant.

Using an activity coefficient model, the following expression for the mole fraction solubility (x s ) is obtained [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-Phase Equilibria[END_REF] x s ) 5) where γ s is the solute activity coefficient. When an equation of state is selected, eq 6 is used instead 6) where φ is the fugacity coefficient and the subscript 0 refers to a pure component. In this work, both an activity coefficient model (the modified UNIQUAC model 19,20 ) and an equation of state (the cubic-plusassociation EoS 21,22 ) were used to represent the measured data. Both models have already been successfully employed for mixtures of water with other associating compounds. 18,20,22,[START_REF] Derawi | [END_REF] 4.1. Modified UNIQUAC Model. Abrahams and Prausnitz 26 derived an equation that in a sense extends the quasichemical theory of Guggenheim for nonrandom mixtures to solutions containing molecules of different sizes. This extension was therefore called the UNIversal QUAsiChemical theory (UNI-QUAC). [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-Phase Equilibria[END_REF] The UNIQUAC equation consists of two parts: a combinatorial term that attempts to describe the dominant entropic contribution (that depends on the composition and the sizes and shapes of the molecules) and a residual term that is due mainly to the intermolecular forces responsible for the enthalpy of mixing.

1 γ s exp { -∆ fus H R ( 1 T - 1 T m ) + ∆C p R [ T m T -ln ( T m T ) -1 ] } (
x s ) φ s liq 0 φ s liq exp { -∆ fus H s R ( 1 T - 1 T m ) + ∆C p R [ T m T -ln ( T m T ) -1 ] } (
Larsen et al. 19 proposed a modified version of this model that was successfully used to study the solubility of highly associating compounds in water and also in mixed solvents. 20 The results achieved were very good and consistent over broad temperature and composition ranges.

The combinatorial part is given by the Flory-Huggins expression, and the residual part is given by the original Abrams and Prausnitz expression. 26 Although a temperature dependence was suggested for the binary interaction parameter (a ij ), this is not considered here because of the reduced number of experimental data available.

The volume and area parameters of the molecules were calculated from the volume and area parameters of the groups, using the UNIFAC parameter tables presented elsewhere. 27 4.2. Cubic-Plus-Association (CPA) Equation of State. In this work, we are considering water and phenolic compounds, which means that both the solute and the solvent can associate. Traditional cubic equations of state have proved to have good predictive capabilities for nonassociating fluids, but they typically fail in describing these particular interactions. This handicap of cubic equations of state was solved with the cubicplus-association equation of state (CPA EoS). 21,22 The CPA EoS combines a cubic contribution [from the Soave-Redlich-Kwong (SRK) or the Peng-Robinson (PR) equation of state] with an association contribution, 28 originally proposed by Wertheim and used in other associating equations of state such as the statistical associating fluid theory (SAFT), 29 accounting for intermolecular hydrogen bonding and solvation effects. Using a generalized cubic term [START_REF] Michelsen | Thermodynamic Models: Fundamentals & Computational Aspects[END_REF] (for the SRK approach, δ 1 ) 1, δ 2 ) 0; for the PR approach, δ 1 ) 1 + 2, δ 2 ) 1 -

2), the cubic and association contributions to the Helmholtz energy (A) are the following

A cubic ) an b(δ 2 -δ 1 ) ln ( 1 + bFδ 1 1 + bFδ 2 ) - nRT ln(1 -bF) (7) A assoc ) RT ∑ i n i ∑ A i [ ln(X A i ) -X A i 2 + 1 2 ] ( 8
)
where i is a component index, b is the covolume parameter, a the energy parameter, F is the molar density, n i is the number of moles of molecules of component i, n is the total number of moles, g is a simplified radial distribution function, [START_REF] Kontogeorgis | [END_REF] and X A i is the mole fraction of component i not bonded at site A. The pure-component energy parameter is given by

a(T) ) a 0 [1 + c 1 (1 -√ T r )] 2 (9)
where a 0 and c 1 are constants to be calculated and T r is the reduced temperature. Only a binary interaction parameter, k ij , on the cross energy parameter a ij is introduced in the cubic contribution, as in previous works. 32,33 X A i is related to the association strength, ∆ A i B j , between two sites belonging to two different molecules and is calculated by solving a set of equations, involving all nonbonded mole fractions

X A i ) 1 1 + F ∑ j x j ∑ B j X B j ∆ A i B j ( 10
)
For cross-associating systems, such as those considered in this work, both self-and cross-association ∆ functions need to be obtained. Additionally, for multifunctional associating molecules, such as the phenolic compounds under study (Figure 1), different associating groups can be present on the same molecule, in some cases with multiple substitutions. Applications of associating equations of state to these complex multifunctional molecules have traditionally been performed with a single association energy and volume for the entire molecule. 22,[START_REF] Derawi | [END_REF] Only the group-contribution-association (GCA) 34 and, very recently, the CPA equations of state 18 have been adopted for multifunctional associating molecules with explicit association energies and volumes for the different associating groups.

The procedure followed in this work is the same as that recently proposed by the authors. 18 The cubic-term parameters are calculated from the critical temperature (T c ), critical pressure (P c ), and van der Waals volume (vdWV)

where T c is in K, P c is in Pa, and the van der Waals volume is in m 3 /mol. The corresponding association energies and volumes to be used in the association term are obtained from a set of model molecules for which vapor pressure and liquid density data are available using the cubic-term parameters from eqs 11-13. This is further discussed in the following section.

Results and Discussion

Solubility results, in grams per liter, and the corresponding pH of each saturated solution are reported in Tables 1 and2. As can be observed for all compounds, the solubility follows a general exponential trend with temperature. In a previous work, 18 we reported the solubilities of other phenolic compounds, and the results were in good agreement with most of the available literature data.

Among the hydroxybenzoic acids (Table 1), the extent of hydrogen bonding with water seems to be an important variable during the solubilization process. Protocatechuic acid is more soluble than syringic acid (and salicylic acid, 18 both having one hydroxyl group), but less soluble than gallic acid (which has three hydroxyl groups in the aromatic ring 18 ). Literature data for these two hydroxybenzoic acids were found only in the works of Noubigh et al., 11,35,36 but the disagreement with our data is relevant (Figure 2). Such disagreement was also found before for gallic acid, where the data from these authors were in considerable disagreement with our data and those of Lu and Lu. 14 For all of these hydroxybenzoic acids and for ferulic acid, Noubigh et al.'s data do not increase significantly with temperature, in contrast to what would be expected.

For o-coumaric acid (Table 2), no literature data were found to compare with our results, but comparison with previous data for other phenylpropenoic acids such as trans-cinnamic, ferulic, and caffeic acids 18 shows that the obtained results follow the expected trend in solubility (trans-cinnamic acid < o-coumaric acid < ferulic acid < caffeic acid) considering the extent of hydrogen bonding with water. As found before, for other hydroxybenzoic and phenylpropenoic acids, the latter usually present lower solubilities, as was also found in this work.

Among the studied phenolic compounds, ellagic acid (Table 2) is the one with the lowest solubilities. Only one value obtained by Bala et al. 37 at 310 K (9.7 mg/L) was found for comparison with our results at the nearest temperatures (4.4 and 5.9 mg/L for 303.2 and 313.2 K, respectively). The deviation from our results is significant, but the standard deviation found by Bala et al. 37 is so high ((3.2 mg/L) that the value measured in this work shows consistency with their findings.

Tyrosol (Table 2) is the most soluble phenolic, and no solubility data for it were found, except for a single point at 293.2 K available in the Merck Index (124 g/L). [START_REF] O'neil | The Merck Index: an Encyclopedia of Chemicals, Drugs and Biologicals[END_REF] Our results compare well with that value. The density of the saturated solution of tyrosol at 298.2 K was measured and found to be 998.86 g/L. As this compound has high solubilities, for the calculation of mole fractions for modeling purposes, the mass of the solute cannot be considered negligible in comparison to the solvent mass, and this density value was used.

As mentioned previously, to verify the accuracy of the spectroscopic method, some measurements at the higher temperatures were also done by the gravimetric method. The analytical gravimetric method is a quite simple technique, very accurate and highly reproducible, and it has already been applied to a large range of compounds. For o-coumaric acid, the difference between the results from the two methods is 1.5%; for protocatechuic acid, 7%; and for tyrosol, 11%. These differences are not significant, meaning that the spectrophotometric method is reliable.

Relative differences in solubility can also be understood with the help of the thermodynamic properties of solvation presented in Table 3. These were obtained by considering the plots of ln x vs T to be linear functions, so that the derivatives in eq 1 are the slopes of the corresponding linear fits.

In terms of enthalpy, we can see that ellagic acid would be more favorably solubilized than the other phenolics, but this is not the case, as there is also a very strong entropic effect. This can be explained by the considerable number of water hydrogen bonds that must be broken to solubilize such a large molecule, even though the molecule has four hydroxyl groups that can hydrogen bond with water.

It is also curious to note that tyrosol, although not expected because of its highest dissolution enthalpy, is very soluble because it has the highest dissolution enthalpy. Still, opposite to ellagic acid, tyrosol has a very high entropic contribution favoring dissolution. Protocatechuic and o-coumaric acids have very similar enthalpic contributions for dissolution, but protocatechuic acid is more soluble because of a more favorable entropic term. Finally, syringic acid is slightly more soluble than o-coumaric acid because of a more favorable enthalpic contribution.

For all of the compounds, it was found that the solubilization process is endergonic (∆ sol G > 0) and, therefore, nonspontaneous.

Other important information about the solubilization of solids in liquids is related to fusion enthalpies and melting tempera- 4, together with the numbers of measurements and standard deviations. For all of the phenolic compounds, except o-coumaric and ellagic acids, a linear baseline and a single peak were observed. The melting peak of o-coumaric acid is not symmetric, with other peaks continuously connected to the melting peak, indicating that decomposition processes might occur upon melting. For this reason, the fusion enthalpy of this acid could not be determined, so it was calculated using a third-order group-contribution approach proposed by Marrero and Gani 39 that was found to give accurate estimates for both melting temperatures and enthalpies of fusion of different phenolics. The same was done for ellagic acid, as the apparent melting peak was quite large and not symmetrical. All DSC crucibles were weighed before and after the DSC measurement to check for mass loss. Literature data were found only for the melting temperature of tyrosol, 40 with our data deviating by less than 0.3%. For the three compounds for which we measured and predicted the melting data, higher disagreements between the experimental values and the Marrero and Gani method were found for the melting temperature (3%) and enthalpy of fusion (25%) of syringic acid, which can be considered as good predictive results. From the measured solubility and the measured and predicted melting data, it can be seen that only for tyrosol and ellagic acid did these variables seem to have a significant influence on the relative solubility: the most soluble, tyrosol, is the phenolic compound with the lowest melting temperature and enthalpy of fusion, whereas ellagic acid is the least soluble and the one requiring the highest temperature and energy to break the crystal structure. For the other three compounds, these data do not seem to be a determining factor in solubility.

Liquid and solid experimental heat capacities are available for a limited number of compounds. From the DSC thermograms, we could determine these values for protocatechuic and syringic acids and tyrosol (Table 4), but no literature data were found for the other studied phenolics. Therefore, a three-level group-contribution method reported by Marrero and Gani [START_REF] Marrero | [END_REF] and extended by Kolska ´et al. 41 was used for the estimation of the heat capacities of the liquids. These authors presented groupcontribution parameters that were determined by both nonhierarchic and hierarchic approaches and tested the model performance in terms of extrapolation features, predictive capability, and consistency. Although the nonhierarchic approach proved to be slightly superior, the heat capacity of ellagic acid was estimated using the hierarchic approach, because the other parameters led to a too-high heat capacity value. For the estimation of the heat capacities of the solids, a correlation based on molecular structure was used. 42 This model was tested by the authors with an average accuracy of approximately 13%. For the three compounds with experimental values, the results were in good agreement (5%, 8%, and 4% deviations, respectively, for protocatechuic acid, syringic acid, and tyrosol).

As most of the studied compounds are organic acids, measurements of apparent equilibrium constants, K a,app , were carried out at 298.2 K, and the results are presented in Table 5. These are apparent dissociation constants, because only the H + activity is considered, as given by the pH measuring system

K a,app ) [A -](H + ) [HA] ( 14
)
where activities are represented by parentheses and concentrations are represented by square brackets.

For the purposes of this work, this is a consistent procedure, as we are mostly interested in the ratio of the basic to the acidic form, in order to evaluate the extent of acid dissociation on the solubility. The obtained pK a,app results and the reported saturated pH values presented in Tables 1 and2 lead to the conclusion that the extent of dissociation at equilibrium is very small and, thus, acid dissociation is not an important factor for determining the solubilities of the studied phenolic compounds in pure water. Tyrosol is the least acidic phenolic compound, with a pK a very close to that of phenol (9.89), as expected, followed by ellagic acid, o-coumaric acid, and the hydroxybenzoic acids.

Finally, the measured data were used to evaluate the performance of the modified UNIQUAC and CPA EoS for their estimation. For the modified UNIQUAC model, the residual term binary interaction parameters were fitted to the measured data. Two temperature-independent interaction parameters were regressed per each binary water + phenolic system. The values are presented in Table 6. Table 6 also contains the volume and area parameters.

In Figure 3, the experimental and calculated solubilities are compared. An excellent description is obtained, using the melting data presented in Table 4. The absolute average deviations (AADs) are good: 5% for syringic and protocatechuic acids, 6% for coumaric acid, 8% for tyrosol, and 27% for ellagic acid.

The CPA EoS was used with the SRK cubic term (eq 7). In the traditional way, the CPA parameters would be obtained from pure-component vapor pressure and liquid density data, but because these data are not available for the studied molecules, the cubic-term parameters were calculated as proposed before for other phenolics (eqs 11-13) using T c and P c obtained from the Marrero and Gani group-contribution procedure [START_REF] Marrero | [END_REF] and the van der Waals volumes from the Bondi group-contribution approach. 43 These parameters are presented in Table 7. Water parameters from a previous work were used 18 (a 0 ) 0.12 Pa m 6 /mol 2 , c 1 ) 0.67, b ) 1.45 × 10 -5 m 3 /mol, ε ) 16555 J/mol, β ) 0.069, four-site model). For the association contribution, values of the association energy and volume for the different associating groups involved in each molecule are presented in Table 8. These values were obtained from a previous work where the parameters were estimated from pure-component vapor pressure and liquid density data of model molecules. 18 For the case of the hydroxyl group in a phenylethanol molecule (Table 8, last line), the association energy and volume were regressed from data for 2-phenylethanol obtained from the DIPPR database. 44 For acid groups, both the association energies and volumes are considerably dependent on whether there is ring substitution with or without intramolecular association or phenylpropenoic substitution. For hydroxyl groups, the values depend on whether there are intramolecular hydrogen bonds. From this methodology, all of the cubic and associating parameters of the molecules under study can be estimated, without requiring experimental pure-component data. In the case of syringic and ellagic acids, the ether and ester groups are not considered to cross-associate with water.

The CPA modeling results are presented in Figure 3. A good description of the data can be obtained using a single temper- ature-independent parameter k ij for each binary system. A maximum average absolute deviation (in mole fraction solubility) of 66% was obtained for ellagic acid, whereas for the other phenolics, the average deviations did not exceed 39%, which can be considered as good results, given that these water + organic systems are frequently difficult to model accurately. 32 It is important to focus on the fact that the ellagic acid solubilities are too low, and therefore, a deviation of 60% in the prediction is not so bad. We also consider that this is a quite complex molecule for which we do not know whether the predictivity of the proposed CPA model and the correlations for the melting properties and ∆C p are or not sufficiently accurate. All of the correlated k ij values are negative (tyrosol, k ij ) -0.14; protocatechuic acid, k ij ) -0.06; syringic acid, k ij ) -0.13; o-coumaric acid, k ij ) -0.03; ellagic acid, k ij ) -0.12), indicating that the water-phenolic interactions are stronger than expected. This is particularly the case for ellagic and syringic acids, where the presence of the ester and ether groups, respectively, was not considered to increase solvation. As shown, the predictive ability of the proposed procedure for the pure-component parameters leads to good results provided that a single k ij value is employed.

Conclusions

In this work, aqueous solubility data were measured for some phenolic compounds, namely, tyrosol and ellagic, o-coumaric, protocatechuic, and syringic acids, in pure water, in the temperature range from 288.2 to 323.2 K, using the shake-flask method coupled with spectrophotometric analysis. All solubilities were found to increase with temperature, as expected. From the aqueous solubility data, thermodynamic molar properties of dissolution were derived. The studied compounds can be divided into two groups: substituted phenols (tyrosol), which have higher solubilities, and hydroxybenzoic acids (protocatechuic and syringic acids), phenylpropenoic acids (o-coumaric acid), and tannins (ellagic acid), which have lower solubilities. In addition to solubility data, melting temperatures, enthalpies of fusion, and apparent dissociation constants were also determined, providing broader knowledge about the solubilization processes of these molecules. From all of the measurements, it was concluded that the extent of solubilization is mostly governed by the degree of hydrogen-bond formation with water, but the energetics of the solid state also play an important role for some of the studied molecules.

The modified UNIQUAC model, with two adjustable parameters from solubility data, provided an excellent description of the solubilities of these phenolic compounds. The CPA EoS, with a single binary interaction parameter, can also be used to estimate the solubilities of these molecules, although with a worse description compared to that of the modified UNIQUAC. It was also demonstrated that the CPA parameters of these molecules can be obtained solely from their molecular structure, avoiding the need for pure-component data. The methodology proposed in this work is expected to be an important tool for the modeling of the phase equilibria of multifunctional associating molecules.

Acknowledgment. The authors are grateful for the support provided by Fundac ¸a ˜o para a Cie ˆncia e a Tecnologia (FCT) through LSRE financing (FEDER/POCI/2010 and REEQ/1164/ [START_REF] Marrero | [END_REF] c Calculated using a group-contribution method for the estimation of the heat capacities of liquids 41 and a correlation based on molecular structure for the heat capacity of solids. 42

Greek Symbols

) association energy (J/mol) β ) association volume γ ) activity coefficient ∆ A i B j ) association strength between site A on molecule i and site B on molecule j ∆C p ) difference between the liquid and solid molar heat capacities ∆ sol G ) molar Gibbs energy of dissolution ∆ sol H ) molar enthalpy of dissolution ∆ sol S ) molar entropy of dissolution ∆ fus H ) fusion enthalpy (J/mol) F ) mole density (mol/m 3 ) σ ) standard deviation φ ) fugacity coefficient Subscripts c ) critical i, j ) component indexes m ) melting r ) reduced s ) solute Superscripts assoc ) association term cubic ) cubic term liq ) liquid phase sol ) solid phase 0 ) pure component

Figure 1 .

 1 Figure 1. Structures of the five studied phenolic compounds.

  Solubilities of Biologically Active Phenolic Compounds J. Phys. Chem. B, Vol. xxx, No. xx, XXXX C -yjp/yjp-yjp/yjp99907/yjp6917d07z xppws 23:ver.3 2/10/09 10:32 Msc: jp-2008-08683y TEID: dmadmin BATID: jp4b108

c 1 )

 1 -3.557 + (6.289 × 10 -3 )T c (12) b ) -2.328 × 10 -5 + 1.884vdWV

Figure 2 .

 2 Figure 2. Aqueous solubilities of hydroxybenzoic acids as a function of temperature: this work (x, protocatechuic acid; 0, syringic acid) and results obtained by Noubigh et al. 11 (b, protocatechuic acid; 9, syringic acid).

a

  Standard deviation.

Figure 3 .

 3 Figure 3. Solubilities of phenolic compounds in water as a function of temperature: experimental results (9, protocatechuic acid; [, syringic acid; ∆, o-coumaric acid; x, tyrosol; b, ellagic acid) and modeling results (s, CPA; ---, UNIQUAC).

TABLE 1 : Experimental Solubilities (S) of Hydroxybenzoic Acids in Pure Water and the Corresponding Saturation pH Values

 1

	ohio2/yjp-yjp/yjp-yjp/yjp99907/yjp6917d07z xppws 23:ver.3 2/10/09 10:32 Msc: jp-2008-08683y TEID: dmadmin BATID: jp4b108
		protocatechuic acid	syringic acid
	T (K)	S ( σ a (g L -1 )	pH	S ( σ a (g L -1 )	pH
	323	49.3 ( 0.5	2.41	5.8 ( 0.2	2.90
		52.8 ( 0.2 b
	313	28.1 ( 0.9	2.61	4.4 ( 0.1	3.07
	303	17.4 ( 0.1	2.97	2.7 ( 0.0	3.31
	298	12.7 ( 0.2	3.06	2.3 ( 0.0	3.42
	288	7.6 ( 0.6	3.08	1.1 ( 0.0	3.52
	282
	283
	284
	285
	286
	287
	288
	289
	290
	291
	292
	293
	294
	295
	296
	297
	298
	299
	300
	301
	302
	303
	304
	305
	306
	307
	308
	309
	310
	311
	312
	313
	314
	315
	316
	317
	318
	319
	320
	321
	322
	323
	324
	325
	326

a Standard deviation. b Value measured by gravimetry. D J. Phys. Chem. B, Vol. xxx, No. xx, XXXX Queimada et al. tures, with lower values of both properties favoring solubility (eqs 5 and 6). The melting properties of the compounds measured in this work are compiled in Table

TABLE 2 : Experimental Solubilities (S) of Tyrosol, o-Coumaric Acid, and Ellagic Acid in Pure Water and the Corresponding Saturation pH Values

 2

		tyrosol		o-coumaric acid		ellagic acid
	T (K)	S ( σ a (g L -1 )	p H	S ( σ a (g L -1 )	p H	S ( σ a (g L -1 )	p H
	323	647.2 ( 7.8	4.34	1.3 ( 0.0	3.37	7.2 ( 0.3	4.38
		572.9 ( 6.7 b		1.3 ( 0.1 b
	313	300.8 ( 3.9	4.97	0.8 ( 0.0	3.45	5.9 ( 0.2	4.51
	303	154.9 ( 2.5	5.19	0.5 ( 0.0	3.56	4.4 ( 0.1	4.65
	298	106.9 ( 1.5	5.49	0.3 ( 0.0	3.73	3.8 ( 0.1	4.84
	288	63.2 ( 0.5	5.50	0.2 ( 0.0	3.79	2.7 ( 0.1	4.89

a Standard deviation. b Value measured by gravimetry.

TABLE 3 : Thermodynamic Properties for the Dissolution of the Phenolic Compounds in Pure Water: ∆ sol G (kJ/mol), ∆ sol H (kJ/mol), and ∆ sol S [J/(mol K)]

 3

		∆solS (	σ a	222.9 ( 0.7	204.9 ( 1.0	190.7 ( 0.7	183.6 ( 1.4
	tyrosol	∆solG (	σ a	4.4 ( 0.3	7.6 ( 0.3	9.5 ( 0.2	10.4 ( 0.4
		∆solH (	σ a	76.5 ( 7.3	71.8 ( 6.9	67.3 ( 6.4	65.1 ( 6.2
		∆solS (	σ a	78.7 ( 0.1	69.8 ( 0.1	60.8 ( 0.3	53.7 ( 0.4
	o-coumaric acid	∆solG (	σ a	23.7 ( 0.0	24.3 ( 0.0	24.8 ( 0.1	25.8 ( 0.1
		∆solH (	σ a	49.1 ( 4.4	46.2 ( 4.1	43.3 ( 3.9	41.8 ( 3.7
		∆solS (	σ a	-48.9 ( 0.4	-52.8 ( 0.2	-57.4 ( 0.1	-59.7 ( 0.7
	ellagic acid	∆solG (	σ a	39.4 ( 0.1	38.7 ( 0.1	38.2 ( 0.0	37.9 ( 0.2
		∆solH (	σ a	23.61 ( 1.3	22.2 ( 1.2	20.8 ( 1.1	20.1 ( 1.1
		∆solS (	σ a	56.9 ( 0.3	50.8 ( 0.2	43.1 ( 0.1	39.8 ( 0.1
	syringic acid	∆solG (	σ a	20.2 ( 0.1	20.4 ( 0.1	20.9 ( 0.0	21.0 ( 0.0
		∆solH (	σ a	2.9 38.6 (	2.7 36.3 (	2.6 34.0 (	2.5 32.9 (	30.7 (
		∆solS (	σ a	100.6 ( 0.7	91.5 ( 0.5	83.1 ( 0.6	78.2 ( 0.1	69.5 ( 0.7
	protocatechuic acid	∆solG (	σ a	13.9 ( 0.2	14.9 ( 0.2	15.6 ( 0.2	16.2 ( 0.0	16.8 ( 0.2
		∆solH (	σ a	46.4 ( 0.8	43.5 ( 0.8	40.8 ( 0.7	39.5 ( 0.7	36.9 ( 0.6
		T	(K)	323	313	303	298	288

TABLE 4 : Average Melting Points, Enthalpies of Fusion, and Differential Heat Capacities of the Studied Phenolic Acids scans

 4 T m (K) ∆ fus H ( σ a (kJ/mol) ∆C p ( σ a [J/(mol K)]

	protocatechuic acid	7	472.3 ( 1.6	31.2 ( 1.6	108.0 ( 6.2
		-	469.3 b	34.2 b	102.4	c
	syringic acid	7	480.3 ( 0.6	33.7 ( 1.8	76.8 ( 3.4
		-	463.8 b	25.0 b	83.0	c
	tyrosol	6	364.0 ( 0.2	25.9 ( 1.6	125.6 ( 21.4
		-	372.8 b	20.3 b	120.5	c
	ellagic acid	-	560.4 b	59.7 b	394.0	c
	o-coumaric acid	-	429.9 b	32.2 b	117.5

c a Standard deviation. b Calculated using a group-contribution method.

TABLE 5 : Apparent Acid Dissociation Constants of the Studied Phenolic Acids at 298.15 K

 5

	phenolic acid	pK a,app
	protocatechuic acid	4.63
	syringic acid	4.71
	o-coumaric acid	4.82
	ellagic acid	6.54
	tyrosol	10.0

TABLE 6 : Modified UNIQUAC Model Parameters in Water (j) and Volume and Area Parameters for Each Compound a

 6

	i	a ij (K)	a ji (K)	r i	q i
	protocatechuic acid	-309.81	913.32	5.0508	3.9040
	syringic acid	-10.75	363.18	6.6447	5.2400
	o-coumaric acid	-11.39	574.56	5.8036	4.4910
	ellagic acid	54.91	1056.37	8.8387	6.0000
	tyrosol	-86.13	377.71	5.7344	4.6800

a Water: r j ) 0.92; q j ) 1.40.

TABLE 7 : Critical Properties and van der Waals Volumes Used in the CPA EoS

 7

	compound	T c (K)	P c (MPa) vdWV (10 -5 m 3 /mol)
	protocatechuic acid	855.0	6.47	7.85
	syringic acid	856.6	3.77	10.53
	o-coumaric acid	831.8	4.26	8.82
	ellagic acid	1056	13.75	14.9
	tyrosol	762.3	4.71	8.03
	2-phenylethanol	684.0	3.92	7.43

TABLE 8 : Association Parameters in the CPA

 8 EQu/2005). F.L.M. acknowledges her FCT Ph.D. scholarship (SFRH/BD/32372/2006), and A.J.Q. acknowledges financial support through Grant POCI/N010/2006. ) parameter in the cubic term of CPA (Pa m 6 /mol 2 ) b ) covolume parameter in the CPA (m 3 /mol) c 1 ) parameter in the cubic term of the CPA CPA ) cubic plus association EoS ) equation of state f ) fugacity n ) number of moles P ) pressure (Pa) q ) UNIQUAC area parameter r ) UNIQUAC volume parameter K a,app ) apparent acid dissociation constant R ) gas constant [8.314 J/(mol K)] S ) solubility T ) absolute temperature (K) vdWV ) van der Waals volume (m 3 /mol) x ) mole fraction X A i ) fraction of molecule i not bonded at site A

	position	(J mol -1 )	β
	COOH Group
	benzoic acid	32010	9.820 × 10 -6
	phenylpropenoic acid	27560	3.698 × 10 -3
	OH Group
	ring, single OH	18370	1.185 × 10 -2
	ring, orto with OH	11340	6.225 × 10 -2
	aliphatic OH in phenylethanol	23367	1.417 × 10 -2

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