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J. Phys. Chem. B 2009, 113, 6446–6458
Modeling the Phase Behavior in Mixtures of Pharmaceuticals with Liquid or Supercritical Solvents Ioannis Tsivintzelis,† Ioannis G. Economou,‡ and Georgios M. Kontogeorgis*,† Center for Phase Equilibria and Separation Processes (IVC-SEP), Department of Chemical and Biochemical Engineering, Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark and Molecular Thermodynamics and Modeling of Materials Laboratory, Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, GR-15310 Aghia ParaskeVi Attikis, Greece ReceiVed: September 8, 2008; ReVised Manuscript ReceiVed: February 16, 2009
The concept of solubility parameter, which is widely used for the screening of solvents in pharmaceutical applications, is combined with a thermodynamic theory that is able to model systems with large deviations from ideal behavior. The nonrandom hydrogen-bonding (NRHB) theory is applied to model the phase behavior of mixtures of six pharmaceuticals (i.e., ibuprofen, ketoprofen, naproxen, benzoic acid, methyl paraben, and ethyl paraben). The pure fluid parameters of the studied pharmaceuticals were estimated using limited available experimental (or predicted) data on sublimation pressures, liquid densities, and Hansen’s solubility parameters. The complex hydrogen-bonding behavior was explicitly accounted for, while the corresponding parameters were adopted from simpler molecules of similar chemical structure or/and fitted to the aforementioned pure fluid properties. In this way, the solubility of the studied pharmaceuticals in liquid solvents was calculated. The average root-mean-square deviation between experimental and calculated solubilities is 0.190 and 0.037 in log10 units for prediction (calculations without a binary interaction parameter adjustment) and for correlation (calculations using one binary interaction parameter fitted to experimental data), respectively. In addition, using one temperature-independent binary interaction parameter the phase behavior of pharmaceuticals in supercritical CO2 and ethane was satisfactorily correlated. Finally, preliminary encouraging results are shown concerning two ternary mixtures where the model is able to predict accurately the solubility of pharmaceuticals in mixed solvents based on interaction parameters fitted to corresponding single solvent data. 1. Introduction Pharmaceuticals consist of substances from different chemical families, with different molecular structures. Usually, many functional groups are present in pharmaceutical molecules, rendering the prediction of their physical properties a challenging task. Due to the complexity of most such molecules and the variety of their interactions with solvents, mainly empirical or semiempirical models are used for modeling their thermodynamic properties. The most popular predictive approach for modeling their solubility in various solvents is the universal functional activity coefficient (UNIFAC)1 model. However, the model presents limited success in the prediction of the temperature dependence of solubility for high molecular compounds.1 Recently, the nonrandom two liquid segment activity coefficient model (NRTL-SAC), an extension of the NRTL model, was proposed with promising results.2 An additional popular model often used for modeling pharmaceuticals is the conductor-like screening model for real solvents (COSMO-RS).3 The main novelty of the latter is the prediction of thermodynamic properties using only data from quantum chemical calculations, which is very useful in cases of limited experimental data. Very often the screening of solvents is performed using the empirical approach of Hildebrand’s solubility parameters.1 In this direction, a more sophisticated approach is the extended Hansen’s model.5 According to this, the solubility parameter * To whom correspondence should be addressed. Phone: +45 45 25 28 59. Fax: +45 45 88 22 58. E-mail:
[email protected]. † Technical University of Denmark. ‡ National Center for Scientific Research “Demokritos”.
can be expressed as a combination of three partial parameters that characterize the dispersive, polar, and hydrogen-bonding interactions, respectively. However, this approach can only give an estimation of the ability of a solvent to dissolve a specific substance. Successful prediction of the solubility and its temperature dependence are often not possible.1,4 Despite the limitations of the extended Hansen’s approach, which originate from crucial simplifications, the concept of solubility parameter is well defined as the square root of the cohesive energy density.5 It became very popular in the screening of solvents for pharmaceuticals mainly due to the lack of experimental data or other successful predictive approaches. Consequently, the determination of the solubility parameters of various pharmaceutical substances has been the subject of many studies.6,7 Furthermore, an extended database5 with experimental and predicted values of Hansen’s solubility parameters for various chemicals exists, and appropriate group contribution methods to predict them have been developed.8,9 The aforementioned activity coefficient models that include UNIFAC and NRTL-SAC, regular solution theory based models (that use the solubility parameter approach), and more sophisticated models, such as COSMO-RS, cannot be used for highpressure phase equilibrium calculations without being coupled with another model that accounts for pressure effects, namely, an equation of state. Despite their success for fluid mixtures, other statistical thermodynamic models, such as advanced lattice and statistically associating fluid theory (SAFT) type models, are not often used for modeling pharmaceuticals, primarily due to the lack of a sufficient amount of experimental data. All models of this family
10.1021/jp807952v CCC: $40.75 2009 American Chemical Society Published on Web 04/15/2009
Modeling the Phase Behavior use fluid-specific parameters that usually are fitted to pure fluid experimental data. In common fluids, experimental vapor pressures and liquid densities are typically used. However, for the majority of pharmaceutical complex chemicals such extended experimental data do not exist. The physicochemical properties of the majority of the pharmaceuticals that are widely used in therapeutics are not known, and subsequently, the prediction of their thermodynamic properties is of great importance. Particular challenges in mixtures with pharmaceuticals are the modeling of their complex hydrogen-bonding behavior, their interactions with supercritical fluids, as well as the successful prediction of the different phase behavior of chiral molecules. Chirality is an important characteristic of biological systems, and more than one-half of the marketed drugs are chiral. Enantiomers can exhibit differences in the nature and degree of their pharmacological and toxicological profiles.10,11 The aim of this study is to test a methodology for modeling mixtures of complex chemicals, such as pharmaceuticals for which there is not enough available experimental data, with an equation of state theory. According to this approach, the complex hydrogen-bonding behavior is explicitly accounted for. The obstacle of the limited available experimental data, which are necessary for proper optimization of pure fluid parameters, is overcome by obtaining the hydrogen-bonding parameters for the various intermolecular interactions from simpler, but similar, molecules and/or by fitting them to pure fluid properties including Hansen’s solubility parameters. This approach provided promising results for the modeling of the solubility of three pharmaceutical substances (i.e., acetanilide, paracetamol, and phenacetin) in various solvents.12 The solubility of six pharmaceutical molecules, which have various functional groups, in liquid or supercritical solvents is investigated. The solubility of ibuprofen, ketoprofen, naproxen, benzoic acid, methyl paraben, and ethyl paraben in various solvents is modeled using the nonrandom hydrogen bonding (NRHB) theory. Ibuprofen, ketoprofen, and naproxen are nonsteroidal anti-inflammatory drugs derived from propionic acid. They are widely used due to their analgetic and antipyretic properties.13-15 On the other hand, benzoic acid and parabens are mainly used as an intermediate in the synthesis of various pharmaceutical substances.16 NRHB is also applied to predict accurately the solubility of pharmaceuticals in mixed solvents using binary interaction parameters fitted to single solvent data. The model is further applied to correlate the solubility of the aforementioned pharmaceuticals in supercritical fluids. Particularly, the phase behavior of the S-enantiomer and the racemic ibuprofen in supercritical CO2 is discussed. 2. Theory The NRHB theory is a compressible lattice model, where holes are used to account for density variation as a result of temperature and pressure changes. NRHB accounts explicitly for the nonrandom distribution of molecular sites, while Veytsman’s statistics is used to calculate the contribution of hydrogen bonding to the thermodynamics of the system.17 Thus, the model is suitable for property calculations of highly nonideal fluids. According to the model, N molecules are assumed to be arranged on a quasi-lattice of Nr sites, N0 of which are empty, with a lattice coordination number, z. Each molecule of type i in the system occupies ri sites of the quasi-lattice. It is characterized by three scaling (pure fluid) parameters and one geometric, or surface-to-volume-ratio, factor s. The first two scaling parameters, ε*h and ε*, s are used for calculation of the
J. Phys. Chem. B, Vol. 113, No. 18, 2009 6447 mean interaction energy per molecular segment, ε*, according to the following equation
ε* ) ε*h + (T - 298.15)ε*s
(1)
while the third scaling parameter,V*sp,0 , is used for calculation of the close packed density, F*) 1/V*sp, as described by the following equation
V*sp ) V*sp,0 + (T - 298.15)V*sp,1
(2)
The parameter V*sp,1 in eq 2 is treated as a constant for a given homologous series.17,18 The hard-core volume per segment, V*, is constant and equal to 9.75 cm3 mol-1 for all fluids. The number of sites, r, is given by
r)
MwV*sp V*
(3)
where Mw is the molecular weight. Finally, the shape factor is defined as the ratio of molecular surface to molecular volume, s ) zq/zr ) q/r, and is calculated from UNIFAC group contribution method,19 while the product zq denotes the average number of external contacts per molecule. In the case of mixtures the following mixing and combining rules are used
∑ xiri
(4)
∑ ∑ θiθjε*ij
(5)
ε*ij ) √ε*i ε*j (1 - kij)
(6)
r)
i
ε* )
i
j
where θ is the surface (contact) fraction and kij a binary interaction parameter. The equation of state for a fluid mixture assumes the following form17
(∑
[
P˜ + T˜ ln(1 - F˜ ) - F˜
i
φi
)
li - νhb ri
]
z q z ln 1 - F˜ + F˜ + ln Γ00 ) 0 (7) 2 r 2
(
)
while the chemical potential for the component i is given by
φjlj + ln F˜ + ri(V˜ - 1)ln(1 - F˜ )rj j qi q z ln 1 - F˜ + F˜ + r V˜ - 1 + 2 i ri r ri µi,hb zqi qi P˜V˜ ln Γii + (V˜ - 1)ln Γ00 +ri + 2 qi ˜T ˜Ti RT (8)
µi φi - ri ) 1n RT ωiri
[
[
∑
][
]
]
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where φi is the site fraction of component i while li and ωi are characteristic quantities for each fluid. Also, νhb is the average number of hydrogen bonds per molecular segment, and µi,hb is the contribution in the chemical potential of component i due to hydrogen bonding. Parameters Γoo and Γii are nonrandom factors for the distribution of empty sites around an empty site and of molecular segments of component i around a molecular segment of component i, respectively. Finally, the parameters T˜ ) T/T*, P˜ ) P/P*, and V˜ () 1/F˜ ) F*/F) are the reduced temperature, pressure, and specific volume, respectively. The characteristic temperature, T*, and pressure, P*, are related to the mean intersegmental energy by
ε* ) RT* ) P*V*
(9)
Detailed expressions for the calculation of all these parameters can be found elsewhere.17 The NRHB equation of state can model properties of hydrogen-bonded fluids of any number of donor and acceptor groups. Each hydrogen bond is characterized by three parameters hb hb hb , volume, VRβ , and entropy change, SRβ , that are the energy, ERβ for the formation of hydrogen bonds between proton donors of type R and proton acceptors of type β in different molecules. However, usually the volume change for the formation of a hb , is set equal to zero, so the number of hydrogen bond, VRβ hydrogen-bonding parameters are reduced to two without compromising the performance of the model.12,17,18 In the NRHB model, polar and dispersive interactions are treated together and characterized as physical interactions, in contrast to the hydrogen-bonding or chemical interactions. Subsequently, it is not possible to estimate separately the polar and the dispersive partial solubility parameters. However, the partial hydrogen-bonding and total solubility parameter can be directly calculated. The corresponding equation for the cohesive energy is
Ecoh ) Eph + Ehb
(10)
where Eph is the potential energy due to physical (dispersive and polar interactions) and Ehb is the contribution due to hydrogen bonding. For pure fluids, these quantities are calculated from the following equations
Eph )
∑ Niiεii ) Γ11qNΘrε*
(11)
i
Ehb ) -
∑∑ R
hb hb NRβ ERβ ) -rN
∑∑ R
β
hb νRβERβ
(12)
β
where Nii is the number of intersegmental interactions, Θr is the surface (contact) fraction of molecular segments with respect to the total segments (empty and occupied) of lattice, ε* denotes the average intersegmental interaction energy, Nhb Rβ is the number of hydrogen bonds of type R-β in the system, while νRβ is the number of hydrogen bonds of type R-β per molecular segment, hb and ERβ is the energy change upon the formation of a hydrogen bond of the same type. The volume of the system is given by
V ) rNV˜ V* +
∑ ∑ NRβhb VRβhb R
β
(13)
As it was mentioned before, usually the volume change for the hb , is set equal to zero. With formation of a hydrogen bond, VRβ the above definitions the total and partial hydrogen-bonding solubility parameters are
δ ) √δd + δp + δhb )
Ecoh ) V
Γ11qΘrε* - r
∑ ∑ νRβERβhb R
β
rV˜ V*
(14)
and
δhb )
Ehb ) V
-r
∑ ∑ νRβEaβhb R
β
(15)
rV˜ V*
2.3. Solid-Liquid Equilibrium Calculations. All the studied pharmaceuticals are crystalline solids at ambient conditions and exhibit solid-liquid equilibrium (SLE) with various liquid solvents. If we assume that we have pure solute (pharmaceutical) in the solid phase and if as a reference state we select the pure subcooled liquid solute at the same temperature, T, and pressure, P, the equation that determines the phase equilibrium is
µl2(T, P) - µl02(T, P) ) µs02(T, P) - µl02(T, P) ) -∆Gs-l (16) where µ is the chemical potential, while subscripts 0 and 2 stand for the pure component and solute, respectively. ∆Gs-l is the Gibbs free energy difference of pure solute converted from crystalline solid to subcooled liquid at the same temperature, T, and pressure, P. Superscripts l and s stand for liquid and solid, respectively. If we assume that the solute volume does not change with small changes in temperature and pressure (υs, νL constant) and that the triple point is very close to the normal melting point, then20
(
)
∆Hm Tm ∆Gs-l υl - υs ) (P - Ptr) + -1 + RT RT RTm T 1 T 1 T Cpl - Cps (C C )dT dT (17) pl ps RT Tm R Tm T
∫
∫
where υ is the molar volume, ∆Hm and Tm are the fusion enthalpy and temperature, respectively, while Cp is the isobaric heat capacity. Usually a simplification of eq 17, which includes only the second term of the right side, is used in phase equilibrium calculations.21 However, this simplification was shown to have a strong impact on phase equilibrium calculations, especially in cases where the difference of the heat capacities, ∆Cp ) Cpl s Cps, of the solute in the liquid and solid state is high.12,22 Subsequently, in this study, the full thermodynamically correct eq 17 for the chemical potential of the solid was used. 3. Pure Fluid Parameters For the application of most equations of state, such as the NRHB model, physical and hydrogen-bonding fluid-specific parameters are optimized using pure fluid properties. For common fluids, such as alkanols, amines, and others, vapor
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Figure 1. Molecular structure of (a) propionic acid, (b) ibuprofen, (c) naproxen, (d) ketoprofen, (e) benzoic acid, (f) methyl paraben, and (g) ethyl paraben.
TABLE 1: Types of Self-Associating Interactions Present in the Pharmaceuticals Examined hydrogen-bonding interactions
ibuprofen
naproxen
ketoprofen
benzoic acid
methyl paraben
ethyl paraben
-COOH · · · HOOC-COOH · · · OdC< -COOH · · · -OsOH · · · OHs sOH · · · OdC-O-
s s s s
s s s
s s s
s s s s
s s s
s s s
pressures and liquid densities are often used. Furthermore, the hydrogen-bonding parameters can be determined using calorimetric or spectroscopic data. However, for most pharmaceuticals, extended experimental data on physical properties do not exist. Recently, Panayiotou and co-workers suggested that the association energy, i.e., the energy difference due to formation of a hydrogen bond, Ehb, can be estimated using data for the partial hydrogen-bonding parameter.23-25 In this work, we apply this suggestion to complex multifunctional pharmaceutical substances, for which extensive experimental data do not exist. According to the suggested methodology, the hydrogen-bonding behavior of fluids is explicitly accounted for. The partial hydrogen bonding and the total solubility parameters (δhb and δ, respectively) are used for the regression of pure fluid parameters, while in some cases, hydrogen-bonding parameters are adopted from simpler but similar molecules. All of the pharmaceuticals examined here, i.e., ibuprofen, naproxen, ketoprofen, benzoic acid, methyl paraben, and ethyl paraben, are able to form various types of hydrogen bonds. Their molecular structure is shown in Figure 1, while the types of self-associating interactions for each molecule are presented in Table 1. The molecular structure of propionic acid, a carboxylic acid containing some of the functional groups of the pharmaceuticals examined, is shown in Figure 1 also. Initially, the pure fluid parameters for each pharmaceutical were estimated. Simultaneous regression of parameters was performed by fitting the predictions of the theory to experimental (or predicted) data for fluid properties, such as vapor (or sublimation) pressures and liquid densities as well as partial hydrogen-bonding and total solubility parameters. Pure fluid parameters for all of the solvents used in this study were adopted from literature.18,30,31 First, the pure fluid parameters, which are three physical, εh*, εs* and Vsp,0* scaling constants and two hydrogen bonding ones, Ehb and Shb for the -COOH · · · HOOC- interaction, of propionic acid were estimated by fitting the model to vapor pressures and saturated liquid densities obtained from the DIPRR database
(based on extended experimental data).26 Available experimental values for the partial hydrogen-bonding and total solubility parameters were also used.5 The hydrogen bonding parameters for the -COOH · · · HOOCinteraction, which were estimated for propionic acid, were also used for ibuprofen. The remaining three pure fluid parameters (εh*, εs*, and Vsp,0*) for this pharmaceutical were estimated by fitting the model to experimental data on sublimation pressures,11 liquid densities (from DIPPR database26 based on limited available experimental data), and partial hydrogen-bonding and total solubility parameters, which were predicted using van Krevelen’s group contribution method.8 For modeling ketoprofen and naproxen, the previously estimated parameters for the -COOH · · · HOOC- were also used. Subsequently, five parameters were regressed for each fluid: three physical (εh*, εs*, and Vsp,0*) and two hydrogenbonding (Ehb, Shb) parameters for the -COOH · · · OdC< and -COOH · · · -O- interaction in ketoprofen and naproxen, respectively. The regression was performed by fitting the model to limited available experimental data on sublimation pressures,14,15 partial hydrogen-bonding and total solubility parameters, which were predicted using van Krevelen’s group contribution method,8 and predicted liquid densities. GCVOL27,28 was used to predict liquid densities of naproxen, while the group contribution method proposed by Constantinou et al.29 was used to predict the liquid density at 298.15 K for ketoprofen. Contrary to the other pharmaceuticals examined, vapor pressure and liquid density data exist for benzoic acid. The pure fluid parameters (three physical, εh*, εs*, and Vsp,0*, scaling constants and two hydrogen bonding parameters, Ehb and Shb, for the AC-COOH · · · HOOC-AC interaction) for this molecule were estimated by fitting the model to experimental data on sublimation pressures,26 liquid densities (from DIPPR compilation26 based on limited available experimental data), and experimental values for partial hydrogen-bonding and total solubility parameters.5 For the modeling of methyl paraben, parameters for the -OH · · · OH- hydrogen-bonding interaction were adopted from
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TABLE 2: Melting Temperature (Tm), Enthalpy of Fusion (∆Hf), Difference between the Heat Capacities in Liquid and Solid State (∆Cp) and Solid Molar Volume (Vsolid) for the Studied Pharmaceuticals pharmaceutical
Tm (K)
∆Hf (J mol-1)
∆Cp (J mol-1 K-1)
Vsolid (cm3/mol)
racemic ibupofen S-(+)-ibuprofen racemic ketoprofen (+)-naproxen benzoic acid methyl paraben ethyl paraben
347.2a 323.5a 367.6d 427.6g 395.5j 399.0k 388.9k
23 100a 15 400a 28 226d 31 500g 18 070j 25 300k 26 400k
50.3b 50.3c 115.5e 108.6h 58.8j 93.7l 92.1m
175a 175a 208f 193i 92j 115l 140n
a Perlovic et al.11 b Gracin and Rasmuson.33 c The value for the racemic compound was used. d Espitalier et al.34 e Kommuru et al.10 Macnaughton et al.35 g Perlovic et al.15 h Neau et al.22 i Kim et al.36 j DIPPR correlation.26 k Perlovich et al.16 l Adopted from 2-hydroxy benzoate.26 m Predicted with a GC method.8 n Predicted with a GC method.27,28 f
TABLE 3: NRHB Pure Component Scaling Parameters, Percentage Average Absolute Deviation (% AAD) between Experimental (or predicted) Data, and NRHB Correlation and Temperature Range Used in Parameter Regressiona ε*h (J mol-1)
ε*s (J mol-1 K-1)
V*sp,0 (cm3 g-1)
s
% AAD in Ps
%AAD in Fliq
T/K
ref
propionic acid ibuprofen
5753.7 4837.1
1.4422 -7.6536
0.9627 0.8236
0.902 0.784
1.9
0.8 3.2
this work this work
ketoprofen
6111.6
5.0892
0.8320
0.757
320-552 350-400 303-343 343-365 298 343-413 298-530 283.2-388.8 416.3-647.4 300-395 422-578 300.0 - 376.0 410.0 - 610.0 256.8 - 494.7 292.7 - 538.8 292.2 - 547.5 289.9 - 577.3 265.9 - 499.3 278.4 - 499.9 295.6 - 521.8 283.1 - 499.3 311.6 - 547.3 370.5 - 633.7 396.1 - 664.3 362.6 - 522.7 298.4 - 646.0 277.5 - 522.5 249.9 - 495.7 263.1 - 521.9 218.0 - 301.0
Component
1.0 1.6 0.1
naproxen benzoic acid
5553.5 7264.0
0.0925 -3.4723
1.0741
0.742
1.5 0.9
0.8561
0.774
3.3 4.5
methyl paraben
6246.0
1.5792
0.8510
0.781
1.2 2.0
ethyl paraben n-hexane cyclohexane benzene toluene methanol ethanol 1-propanol 2-propanol 1-butanol 1-octanol 1-decanol propylene glycol water chloroform acetone ethyl acetate carbon dioxide
5886.7 3957.1 4469.2 5148.5 5097.2 4202.3 4378.5 4425.6 4103.8 4463.1 4532.1 4573.4 5088.8 5336.5 5072.0 4909.0 6055.0 3468.4
2.3364 1.6580 1.8391 -0.2889 0.0768 1.5269 0.7510 0.8724 0.9568 1.1911 1.8686 1.8834 1.0526 -6.5057 -0.7373 -1.1500 -1.7559 -4.5855
0.8840 1.27753 1.19596 1.06697 1.06205 1.15899 1.15867 1.13923 1.12381 1.13403 1.12094 1.11028 0.84239 0.97034 0.59291 1.14300 1.31150 0.79641
0.783 0.857 0.801 0.753 0.757 0.941 0.903 0.881 0.881 0.867 0.839 0.833 0.903 0.861 0.840 0.908 0.922 0.909
2.0 1.1 0.6 1.7 1.5 2.2 1.8 0.2 0.7 0.2 1.2 1.7 5.0 1.3 1.6 1.4 1.1 1.0
3.1 0.5 1.9 0.5 1.1 2.6 1.1 0.6 0.5 0.5 0.5 0.9 0.7 2.0 3.0 0.9 5.0 1.7
this work this work this work this work this work 18 18 18 18 30 30 30 30 30 31 30 31 30 30 30 30 18
a s P and Fliq stand for sublimation or vapor pressure and liquid densities, respectively. % AAD ) (1)/(n)∑i |(Xical - Xiexp)/(Xiexp)| × 100, where X stands for Ps or Fliq and n is the number of experimental data points.
alkanols.30,31 Subsequently, five parameters were regressed again: the three physical (εh*, εs*, and Vsp,0*) and two hydrogen bonding (Ehb, Shb) for the -OH · · · OdC-O- interaction. The regression was performed by fitting the model to limited available experimental data on sublimation pressures16 and partial hydrogenbonding and total solubility parameters, which were predicted using a group contribution method.9 Also, because of the lack of available experimental data, liquid densities from DIPPR correlation26 for the 2-hydroxy methyl benzoate isomer were used. Finally, in ethyl paraben the same hydrogen-bonding parameters were used as in methyl paraben. Subsequently, only three pure fluid parameters (εh*, εs*, and Vsp,0*) for this pharmaceutical were estimated by fitting the predictions of the theory to experimental data on sublimation pressures,16 liquid densities, which were predicted using GCVOL27,28 group contribution
method, and partial hydrogen-bonding and total solubility parameters, which were predicted using a group contribution method.9 For the correlation of sublimation pressures, SLE calculations were performed using eq 17 for the evaluation of the chemical potential in the solid state. The values used for the fusion properties are presented in Table 2. The three optimized physical parameters and the shape factor s (calculated from the UNIFAC group contribution method32) for all fluids are presented in Table 3. Also, in Table 3 parameters for all solvents, which were taken from previous studies,18,30,31 are shown. The experimental or predicted (with group contribution methods8,9) solubility parameters values and those calculated by the model (see eqs 14 and 15) are presented in Table 4. Finally, the hydrogen-bonding sites for each functional group and the parameters for the various hydrogen-bonding interactions are shown in Tables 5 and 6, respectively.
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TABLE 4: Partial Hydrogen-Bonding and Total Solubility Parameters for Pharmaceuticals δhb (MPa1/2) pharmaceutical
literature (experimental or predicted)
propionic acid ibuprofen ketoprofen naproxen benzoic acid methyl paraben ethyl paraben e
δ (MPa1/2) calculated with NRHB
a
12.4 7.2b 7.4c/8.2d 7.2c/8.5d 9.8a 13.4d/14.4c 13.0d/13.9c
literature (experimental or predicted)
calculated with NRHB
a
12.4 7.2 7.2 7.2 10.1 13.0 12.1
20.0 19.5b 19.9c/22.3d 17.9c/22.2d 21.8a 22.3d/23.5c/24.8e 24.0d/22.0c/24.4e
25.1 19.4 22.0 20.8 25.3 25.1 24.1
a Experimental.5 b Experimental.6 c Predicted using van Krevelen’s group contribution method.8 d Predicted using group contribution method.9 Experimental.7
TABLE 5: Hydrogen-Bonding (HB) Sites in Each Functional Group HB sites group
proton donors
proton acceptors
sCOOH >CdO (in pharmaceuticals) >CdO (in acetone) sOs sOsCdO HOH sOH
1 0 0 0 0 2 1
1 1 2 2 2 2 1
The parameter V*sp,1 that is used in eq 2 is a characteristic parameter of a given homologous series. As previously discussed,17,18,30 it was set equal to -0.412 ×10-3 cm3 g-1 K-1 for non-aromatic hydrocarbons, -0.310 × 10-3 cm3 g-1 K-1 for alcohols, -0.240 × 10-3 cm3 g-1 K-1 for acetates, -0.300 × 10-3 cm3 g-1 K-1 for water, and 0.150 × 10-3 cm3 g-1 K-1 for all the other fluids.
and the parameters for the -COOH · · · OdC< self-associating interactions of ketoprofen were also used to model the crossassociating interactions between acids and acetone. In some cases, for which no parameters were available, they were taken from previous studies (for example, the parameters for HOH · · · OdC< were taken from Grenner et al., and they were based on VLE data for water-cyclohexanone mixture30). When no available parameters existed in the literature for cross association, appropriate combining rules were used. The following combining rules were used for the cross association between two self-associating groups30
Eijhb )
[
1/3
]
3
(18)
while for the cross association between one self- and one nonassociating group, the combining rules were30
4. Binary Mixtures The pure fluid parameters of Table 3 were used for the calculation of mixtures phase equilibria. Hydrogen bonding was explicitly accounted for, and the parameters for the various types of self- and cross-association are presented in Table 6. When possible, the estimated hydrogen-bonding parameters for solute’s self-association were also used for the cross hydrogen-bonding interactions with solvent molecules. For example, the parameters for the self-associating interactions of type -OH · · · OdC-Oof methyl paraben were also used to describe the crossassociating interactions between methyl paraben and alkanols,
1/3
Eihb + Ejhb hb Sihb + Sjhb , Sij ) 2 2
Eijhb )
Eihb hb Sihb , Sij ) 2 2
(19)
The investigated pharmaceuticals are in solid crystalline phase at ambient temperature and have relatively high melting points. Subsequently, in all cases SLE calculations were performed assuming pure pharmaceutical in the solid phase. In all cases, eq 17 was used for calculation of the chemical potential of the solid solute. All calculations were performed using the fusion properties of Table 2, while the difference of the heat capacities, ∆Cp ) Cpl - Cps, of liquid and solid solute was assumed
TABLE 6: Parameters for Hydrogen-Bonding (HB) Interactions HB groups sCOOH · · · HOOCs AC-COOH · · · HOOC-AC sOH · · · OHsa,b sCOOH · · · OdC< sCOOH · · · sOs sCOOH · · · OdCsOs sOH · · · sOs sOH · · · OdCsOs sOH · · · HOOCs HOH · · · HOH HOH · · · OHs HOH · · · HOOCs HOH · · · HOOCsAC HOH · · · OdCsOs HOH · · · OdC
