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The effect of proton-acceptor sites of the solute on its solubility in proton-donor solvents
PHARMACEUTICA ACTAHELVETIAE ELSEVIER
Pharmaceutica Acta Helvetiae 68 (1993) 49-60
The effect of proton-acceptor sites of the solute on its solubility in proton-donor solvents Paul Ruelle
a . , E l i e S a r r a f a, L e e n V a n D e n B e r g e b, K a t a r i n e Ulrich W. Kesselring
a
S e g h e r s b, M i c h e l B u c h m a n n
a,
a
Institut d'Analyse Pharmaceutique, Ecole de Pharmacie, Universit~ de Lausanne, BEP, CH-1015 Lausanne, Switzerland b Department of Chemistry, University ofLeuven, Celestijnenlaan 200 F, B-3030 Heverlee, Belgium
Abstract
The solubilities of five solid ketones and two esters are predicted in common organic nonelectrolyte solvents using the solubility equation derived from the mobile order theory. In the framework of this theory, the formation of solute-solvent hydrogen bonds is treated on the basis of standard stability constants. Two different values characterizing the ketone-alcohol and the ester-alcohol hydrogen bonds, respectively 170 and 110 cm3/mol, have been determined. The formation of specific molecular interactions brings about a net increasing of the solubility without modifying the values of the other contributions relevant to the solution process. Using the predetermined values of the stability constants, the solubility equation is then successfully applied to predict the solubility of testosterone propionate in 28 solvents including alcohols and water from the sole knowledge of its solubility in hexane. Key words: Solubility; Mobile order theory; Hydrogen bond; Stability constant; Modified solubility parameter; Testosterone propionate
I. Introduction
The prediction of the solubility of a solid solute in a given solvent requires, from a thermodynamic point of view, the knowledge of its ideal solubility and activity coefficient. The ideal solubility of a solid solute is related to the energy needed to transform that solute from its solid state to its hypothetical liquid state at the considered experimental temperature. It only depends on the melting properties of the solute, i.e., the melting t e m p e r a t u r e (Tm) , the enthalpy (AmeltH) or the entropy (AmeltS) of fusion and the heat capacity difference (ACp) between the solid and the supercooled liquid forms of the solute. A m o n g all these parameters, the latter is obviously the most difficult to determine experimentally. Accordingly, two alternative approximations are commonly used, either ACp is assumed to be zero or is assumed to be equal to the entropy of fusion. These two assumptions have recently been
* Corresponding author.
tested and critically discussed by Mishra and Yalkowsky (1992). From this discussion, it appears that none of these approximations can be considered to be superior or yielding better results. Once the solute is dissolved, the activity coefficient indicates the degree of deviation of the solute-solvent system formed with respect to the ideal behaviour. An activity coefficient which differs from 1 is an expression of excess (non-ideal) free-enthalpy, and may furthermore be decomposed into a partly enthalpy-governed residual activity coefficient and an excess-entropygoverned combinatorial activity coefficient. Various models dealing with the estimation of the activity coefficients can be found in the literature (Prausnitz et al., 1986; Scott, 1987). Depending on the origin attributed to the deviation, some of these models consider exclusively entropic effects while others consider entirely enthalpic effects. The latter include the methods based on the Hildebrand regular solution theory (Scatchard, 1931; Hildebrand and Scott, 1962; Hildebrand et al., 1970) and the universal group activity coefficient (UNIFAC) method (Magnussen et al., 1981). U N I Q U A C
0031-6865/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved SSDI: 0031-6865(93)E0004-Z
50
P. Ruelle et al. / Pharmaceutica Acta Heh'etiae 68 (1993) 49-60
(Abrams and Prausnitz, 1975) is one of the models that include both the enthalpic and entropic contributions to the activity coefficients. However, no single approach has by now been completely satisfactory. The ability to predict thermodynamic activity coefficients of substances in any given environment would however be of great value for understanding solution behaviour in general, and for predicting solubility and distribution properties of the molecules in particular. As the magnitudes of the activity coefficients are largely determined by the extent and the nature of molecular interactions between the dissolved molecules and their surrounding solvent molecules, the ability to predict solubility directly depends on how well one can quantify the various solute-solute, solvent-solvent and solute-solvent interactions involved in the solution process. An important step towards the predictions of the solubilities was made around 1930 when Scatchard published an equation, based on the previous work of Hildebrand, which allows to obtain the heat of mixing of two liquids on the basis of their molar heats of vaporization and molar volumes, or more precisely on the basis of their solubility parameters. This equation still constitutes the background of practically all theoretical approaches of solubility prediction found in the literature. It is a kind of modern formulation of the empirical principle of the alchemists "Similia similibus solvuntur". As a matter of fact, according to that equation, the best solvent would be that with the same solubility parameter as the solute. The ScatchardHildebrand equation is based on two fundamental assumptions: (i) the hypothesis of the geometric mean (the cohesive energy per unit volume between the solute and the solvent is the geometric mean of the values of the pure components); and (ii) the interactions between molecules in the solution occur "at random" and there are no preferential contacts. If these two assumptions can be regarded as crude approximations for the dispersion and dipole-dipole cohesion forces, they however contradict the very essence of hydrogen bonds. The principle of the alchemists is entirely erroneous when H-bonding interactions are involved because hydrogen bonds do not follow the rule of the geometric mean and do not lead to random contacts. One of the most important characteristics of Hbonding in the liquid phase is the establishment of "highly preferential" contacts between the protondonor and proton-acceptor sites of the various molecules. These particular contacts confer to Hbonded liquids a higher degree of "order" and a kind of structure. The structuration produced by H-bonding
is however intermittent or labile: contrary to valence bonds, H-bonds in liquids are regularly broken, their life times lying between 10 -5 and 10 -I1 s (Hasted, 1974; Kohler, 1974). The fundamental difference with the crystalline state lies in the "mobility" of the Hbonds in the liquid phase: the disorder or the order in a liquid is a dynamic one. Both the order introduced in the liquid by the formation of the hydrogen bonds and the perpetually moving character of these bonds constitute the basic foundations of the "mobile order" theory initiated by Huyskens and Siegel (Huyskens and Siegel, 1988; Huyskens, 1990; Siegel et al., 1990). The mobile order theory is at the basis of a new thermodynamic treatment of the liquid state, the quantitative development of which led to equations describing the effect of solvent-solvent, solute-solute and solute-solvent interactions on the chemical potential of the solute. A universal predictive equation for solubility (in volume fraction) of solid and liquid substances has therefore been derived (Ruelle et al., 1991) and has already been applied with success to predict the solubility of inert systems in non-polar, polar and hydrogen-bonded solvents (Ruelle et al., 1992a,b, 1993). The aim of this paper is to further apply this new theoretical approach to predict the solubility of crystalline proton-acceptor substances able to interact by hydrogen bond with proton-donor solvents like alcohols, and to determine the stability constants characterizing these hydrogen bonds. For that purpose, five solid ketones (10-nonadecanone, 11-heneicosanone, 12-tricosanone, p-benzoquinone and benzil) and two solid esters (hexadecanoic methyl ester and octadecanoic methyl ester) have been chosen. Their solubilities are predicted in a series of common organic nonelectrolyte solvents of widely differing polarities.
2. Results and d i s c u s s i o n
The predictive solubility equation takes into account the various contributions to the free-enthalpy change accompanying the solubilization process. As far as only chemically inert solid solutes are concerned, their solubilities are predicted, whatever the nature of the solvent, by the sum of four contributions (Eq. 1): the fluidization of the solute (A-term), the placing (exchange) entropy correction resulting from the difference in the molar volumes of solvent and solute (Bterm), the changes in the non-specific cohesion forces upon mixing (D-term), and the effect of the self-association of the solvent (F-term), i.e., the hydrophobic effect. On the other hand, when proton-acceptor so-
P. RueUe et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
lutes are considered, one has to add to the previous terms an additional contribution (O-term) to the solubility, in order to account for the effect of the H-bond formation between the proton-acceptor site of the solute and the proton-donor group of the solvent. For such substances, the solubility results therefore from the sum of five terms (Eq. 2). solubility of apolar substances In ~bB = A + B + D + F ,
(1)
solubility of proton-acceptor substances In c k B = A + B + D + F + O .
(2)
Characterizing single physical phenomena related to the solubilization process, each term of the above equations corresponds to well-defined contribution favourable or unfavourable to the solubility. A = -AmeltH(1/T-
1/Tm)/R
B = O.5cks(VB/Vs - 1) + 0.5 ln(~bB + cksVB/Vs)
D = - 62Vn(¢5~ - 3~)2/(RT) F = -rsqbsVR/V s
O = ln(1.0 + K o ( c k s / V s - ckB/VB)) In these expressions, AmeltH and Tm represent the molar heat and the temperature of fusion respectively; VB, 6'B, ckB, Vs, 6~, 4~s stand for the molar volume, the modified non-specific cohesion parameter and the volume fraction respectively of the solute B and of the solvent S; r s represents the "structuration or mobile order factor" of the solvent amounting approximately to 1 for solvents associated by single H-bond chains like alcohols, 2 for diols and water, and 0 for non-associated solvents; K o is the stability constant which
Table 1 Physical properties of the solutes Solute
10-Nonadecanone ll-Heneicosanone 12-Tricosanone p-Benzoquinone Benzil Hexadecanoic methyl ester Octadecanoic methyl ester
VB (cm3/ mol)
t~ (jl/2 cm - 3/2)
(kJ/mol)
Tm (°C)
338.0 371.0 404.0 79.6 180.0 309.0 342.0
17.30 17.20 16.93 24.92 21.72 17.63 17.11
67.264 76.200 78.034 18.531 19.762 55.647 64.434
57.00 63.50 69.00 115.00 95.20 30.60 39.05
a b b c c c c
AmeltH
a b b ~ d f f
a b b e d f g
a this work; b Huyskens (1989); c determined from the experimental solubility of the solute in hexane or in heptane; d Acree Jr. (1992); Acree Jr. and Ry ning (1982a); f Davies and Kybett (1965); g Simonelli and Higuchi (1962).
51
governs the H-bonding formation between the protonacceptor solute and the proton-donor solvent. All these physico-chemical properties, except K o, needed for the solubility predictions are reported in Table 1 for the solid solutes (five ketones and two esters), and in Table 2 for the solvents in which solubilities are predicted. Remember that the molar volumes of the solutes in solution are calculated by adding group contributions (Ruelle et al., 1991) for the various groups constituting the molecule. To study the effect of active proton-acceptor sites of the solute on their solubility, all the solubilities have been first calculated using Eq. 1 by considering the ketone and ester solutes as apolar substances unable to form hydrogen bonds with proton-donor solvents. The results expressed in molar fraction (Table 3) are compared to the experimental values found in the literature or determined by ourselves at the appropriate temperature by using the visual thermoturbidimetric technique. The quality of the predictions is demonstrated (Figs. 1 and 2) by the solubility ratios, x p r e d / x exp, between the predicted and experimental values. In the case of the non-associated so!vents, the ratios vary from 0.62 to 1.39 (except for 12-tricosanone in acetonitrile and nitroethane, and for octadecanoic methyl ester in acetone and acetonitrile). However, in the case of alcohols, one observes that all the predicted solubilities are substantially lower than the experimental results, with similar solubility ratios amounting to about 0.35 for ethanol to octanol, and 0.05 for methanol. Such results clearly demonstrate the power and the validity of Eq. 1 to predict the solubility of polar proton-acceptor substances in a wide range of non-associated nonelectrolyte solvents for which the whole solubility process does not differ from that of apolar substances, but also show that Eq. 1 is not sufficient when proton-donor solvents are considered. As a matter of fact, the formation of solute-solvent hydrogen bonds in solution increases the solute solubility by decreasing the free-energy of the system. This additional contribution to the solubility is treated in the frame of the mobile order theory by the O term (Ruelle et al., 1991) included in Eq. 2. Contrary to the currently used approaches based on Hildebrand (or Hansen) solubility parameters which do not take preferential contacts and entropy effects originating from H-bonding into account, the present treatment is based on stability constants, K o , and on the concentration of the active sites of the solvent, 4~s/Vs, which allows to include the important entropy effects accompanying the formation of specific molecular complexes. To make solubility predictions by means of Eq. 2,
P. Ruelle et al. / Pharmaceutica Acta Heh,etiae 68 (1993) 49-60
52
one therefore needs to know the order of magnitude of the stability constants, K o , characterizing the hydrogen bonds formed between the O H groups of the alcohols and the ketone or ester groups of the solutes. However, the stability constants, K o, appearing in Eq. 2 are influenced by the self-association of the solvent and hence do not correspond to those which one should find (by IR spectroscopy for instance) when diluting the associated solvent in an inert one so as to suppress the self-association bonds. Moreover, as these stability constants depend on the solute and on the solvent, different values should be used for each particular
system, which would make the predictions impossible. Fortunately, it appears that, for a given class of solutes, the values of K o, deduced from experimental solubilities by means of the solubility equation, do not vary very much in a given family of solvents (Table 4). Without impairing the predictive character of the model, one can therefore use mean standard values of stability constants, K o , which characterize a given type of hydrogen bond. In this work, two different values, i.e., 170 and 110 cm 3 mo1-1 as mean values of the particular stability constants reported in Table 4 (excluding the results related to methanol), have been
Table 2 Molar volume, Vs, and modified non-specific cohesion parameter, 6~, of various solvents at 25°C Solvent n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane n-Tetradecane n-Hexadecane Decaline Squalane Cyclohexane Methylcyclohexane t-Butylcyclohexane Cyclooctane Isooctane Benzene Toluene Ethylbenzene p-Cymene m-Xylene p-Xylene Mesitylene Tetraline Carbon tetrachloride Chloroform Dichloromethane 1,1-Dichloroethane
Vs
6~
(cm3/mol)
(j1/2 cm-3/2)
116.1 131.6 147.5 163.5 179.6 195.9 213.3 228.6 260.3 294.1 156.9 525.0 108.8 128.3 173.9 134.9 166.1 89.4 106.9 123.1 156.0 123.2 123.9 139.6 137.1 97.1 80.7 64.5 84.8 1,2-Dichloroethane 78.8 1,1-Dibromoethane 92.9 1,2-Dibromoethane 97.0 n-Chlorobutane 105.0 1,4-Dichlorobutane 112.1 Chlorobenzene 102.1 o-Dichlorobenzene 113.1 Bromobenzene 105.3 trans-l,2-Dichloroethylene 78.0 cis-l,2-Dichloroethylene 76.0 Nitrobenzene 102.7
14.18 14.56 14.66 14.85 15.07 15.14 15.25 15.34 15.49 15.61 18.90 16.25 14.82 15.00 15.50 15.40 14.30 18.95 18.10 18.02 20.20 17.20 17.30 17.00 19.43 17.04 18.77 20.53 18.51 20.99 18.77 20.75 17.12 19.78 19.48 18.77 21.22 18.41 18.61 21.77
Solvent Nitroethane Pyridine N,N-Dimet hylformamide Acetonitrile Carbon Disulfide Thiophene Acetone Methyl ethyl ketone Diethyl ketone Methyl isobutyl ketone Dipropyl ketone Anisole Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Dipropyl ether Dibutyl ether Dipentyl ether Dioxane Methyl formate Methyl acetate Ethyl acetate Butyl acetate Hexyl acetate Ethyl propionate Diethyl adipate Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol Isobutanol t-Butanol 1-Pentanol 1-Octanol Cyclohexanol Furfuryl alcohol Water
Vs
~
(cm3/mol)
(j1/2 cm-3/2)
71.8 80.9 77.0 52.9 60.0 79.6 74.0 90.2 106.4 125.8 140.4 119.1 104.8 120.4 124.3 141.8 170.3 204.0 85.8 62.1 79.8 98.5 132.5 165.8 115.5 202.2 40.7 58.7 75.1 76.9 92.0 92.4 92.8 94.3 108.6 158.3 106.0 86.5 18.1
22.44 20.94 22.15 23.62 20.50 18.70 21.91 20.90 20.13 19.95 19.49 20.20 18.78 15.70 16.12 17.96 17.45 17.30 20.89 22.96 21.67 20.79 19.66 19.15 20.05 18.17 19.25 17.81 17.29 16.80 17.16 16.60 16.14 15.78 16.85 16.38 17.88 18.99 20.50
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
53
Table 3
Experimental, X exp, and predicted (Eq. 1), X pred, solubilities of the ketone and ester solutes and A, B, D, F contributions to their solubility Solute/Solvent
X exp
10.Nonadecanone (25°C): A = - 2.630 CCI 4 0.110 Benzene 0.109 Cyclohexane 0.0635 Hexyl acetate 0.0532 Butyl acetate 0.0448 Hexadecane 0.0431 Tetradecane 0.0429 Diethyl ketone 0.0467 Methyl isobutyl ketone 0.0448 Decane 0.0445 Hexane 0.0474 Methyl ethyl ketone 0.0333 Ethyl acetate 0.0317 Acetone 0.0152 Nitroethane 0.00704 Acetonitrile 0.00221 Octanol 0.0217 Butanol 0.0109 Isopropanol 0.00766 Propanol 0.00726 Methanol 0.00207 l l - H e n e i c o s a n o n e (25°C): A = - 3.515 GEl 4 0.0739 p-Xylene 0.0606 Benzene 0.0575 Butyl acetate 0.0135 Methylisobutyl ketone 0.0172 Ethyl acetate 0.0107 Methyl acetate 0.00416 Butanol 0.00403 Propanol 0.00269 Ethanol 0.00109 12-Tricosanone (30°C): A = - 3.529 CC14 0.0505 Benzene 0.0447 Cyclohexane 0.0306 Hexane 0.0204 Butyl acetate 0.0125 Methyl ethyl ketone 0.0719 Ethyl acetate 0.00569 Acetone 0.00188 Nitroethane 0.00195 Acetonitrile 0.000485 Butanol 0.00262 Isopropanol 0.00142 Methanol 0.000473 Benzil(25°C):A = - 1.519 CH2CI 2 0.277 1,2-Dichloroethane 0.260 CHC13 0.297 Benzene 0.179 Toluene 0.149 CCI 4 0.0790 Cyclooctane 0.0145 Cyclohexane 0.0107 Octane 0.00726 Heptane 0.00657 Hexane 0.00573 Isooctane 0.00587
X pred
B
D
0.104 0.0934 0.0581 0.0570 0.0516 0.0513 0.0495 0.0483 1.01 0.0453 0.0410 0.0327 0.0314 0.0144 0.00691 0.00203 0.00956 0.00376 0.00231 0.00224 0.000143
1.391 1.550 1.393 0.790 1.111 0.136 0.263 1.465 1.205 0.592 1.147 1.837 1.674 2.406 2.557 3.582 0.930 1.964 2.417 2.481 4.706
-0.005 -0.192 -0.591 -0.370 -0.586 -0.345 - 0.392 - 0.810 - 0.753 -0.544 -0.831 -1.392 - 1.345 - 2.546 - 3.377 - 5.307 - 0.111 - 0.003 - 0.033 -0.000 -0.517
0.0521 0.0431 0.0424 0.0188 0.92 0.00981 0.00474 0.00121 0.000693 0.000290
1.767 1.386 1.979 1.352 1.456 1.983 2.545 2.204 2.761 3.576
-0.003 -0.001 -0.327 -0.816 - 1.031 -1.793 -2.862 -0.000 - 0.001 -0.055
0.0555 0.0407 0.0301 0.0204 0.0152 0.0704 0.00690 0.00236 0.00100 0.000228 0.000949 0.000566 0.0000167
1.902 2.164 1.833 1.513 1.520 2.424 2.203 3.045 3.162 4.328 2.427 2.949 5.610
-0.001 - 0.461 -0.572 -0.794 -1.091 -2.376
0.244 0.238 0.185 0.183 0.124 0.0708 0.0136 0.00810 0.00750 0.00623 0.00573 0.00415
0.802 0.655 0.707 0.615 0.496 0.653 0.306 0.572 0.098 0.208 0.338 0.082
-0.029 -0.013 -0.280 -0.266 -0.621 - 1.224 -2.801 -3.370 - 3.376 -3.570 -3.670 -3.967
- 3.878 - 4.815 - 7.153 - 0.009 -0.003 -0.864
F
Ref.
-2.092 -3.624 -4.351 -4.456 -8.295
Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) this work Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Ruelle et al. (1991) Ruelle et al. (1991) Garland et al. (1943) Ruelle et al. (1991) Garland et al. (1943)
-4.013 -4.923 -6.309
Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens(1989) Huyskens (1989) Huyskens(1989) Huyskens (1989) Huyskens(1989)
-4.373 -5.238 -9.925
Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland
et et et et et et et et et et et et et
al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943)
Jimenez and Ordonez (1986) Jimenez and Ordonez (1986) Jimenez and Ordonez (1986) Acree and Procyck (1988) Acree and Procyck (1988) Acree and Procyck (1988) Acree and Rytting (1982b) Acree and Bertrand (1983) Acree and Rytting (1982b) Acree and Procyck (1988) Acree and Procyck (1988) Acree and Rytting (1982b)
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
54 Table 3 (continued) Solute/Solvent
X exp
X pred
(25°C): A = - 1.733 0.0344 0.0264 0.00872 0.0116 0.00800 0.0105 0.00749 0.00771 0.00652 0.00704 0.00664 0.00664 0.00665 0.00534 Hexadecanoic m e t h y l e s t e r (20°C): A = - 0.797 CHCI 3 0.561 0.479 CCI 4 0.544 0.476 Benzene 0.499 0.475 Toluene 0.500 0.474 1,2-Dichloroethane 0.500 0.452 Cyclohexane 0.438 0.434 Butyl acetate 0.387 0.441 Acetone 0.349 0.435 Ethyl acetate 0.412 0.430 Hexane 0.394 0.394 Octane 0.396 0.377 Butanol 0.117 0.0362 Propanol 0.0961 0.0218 Methanol 0.0330 0.00171
B
D
F
Ref.
p-Benzoquinone
CC14 Dodecane Cyclooctane Octane Cyclohexane Heptane Isooctane
Octadecanoic
-
0.185 0.848 0.465 0.614 0.289 0.536 0.626
-
1.910 2.926 2.876 3.234 3.244 3.359 3.606
Acree Acree Acree Acree Acree Acree Acree
and and and and and and and
Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a)
0.555 0.503 0.532 0.474 0.607 0.518 0.427 0.657 0.558 0.480 0.381 1.612 2.101 4.262
-
0.008 0.003 0.013 0.002 0.080 0.099 0.075 0.130 0.110 0.187 0.213 0.022 0.012 0.325
- 2.982 - 3.769 - 7.495
Sedgwick et al. (1952) Sedgwick et al. (1952) Sedgwick et al. (1952) Bailey et al. (1970) Bailey et al. (1970) Sedgwick et al. (1952) Sedgwick et al. (1952) Sedgwick et al. (1952) Sedgwick et al. (1952) this work this work this work this work this work
0.981 1.126 1.085 1.009 0.905 0.690 0.900 1.349 2.060 3.586 1.947 2.474 3.250 4.754
-
0.000 0.071 0.105 0.239 0.415 0.380
-
0.900 1.916 5.650 0.000 0.004 0.067 0.639
-3.571 - 4.426 - 5.736 - 8.379
Sedgwick et Sedgwick et Sedgwick et Sedgwick et this work this work Sedgwick et Sedgwick et Sedgwick et Sedgwick et this work this work this work Sedgwick et
methyl ester (20°C): A = - 1.613
CCI 4 CHCI 3 Benzene Cyclohexane Hexane Octane Butyl acetate Ethyl acetate Acetone Acetonitrile Butanol Propanol Ethanol Methanol
0.271 0.324 0.237 0.167 0.156 0.144 0.140 0.110 0.0373 0.00151 0.0248 0.0144 0.00640 0.000643
0.244 0.240 0.228 0.194 0.156 0.151 0.156 0.115 0.0607 0.00400 0.0109 0.00633 0.00270 0.000334
used to characterize respectively the ketone-alcohol a n d e s t e r - a l c o h o l H - b o n d e q u i l i b r i u m in s o l u t i o n .
al. al. al. al.
(1952) (1952) (1952) (1952)
al. al. al. al.
(1952) (1952) (1952) (1952)
al. (1952)
s o l v e n t s r e s u l t s t h e r e f o r e in a p o s i t i v e e f f e c t b y i n c r e a s ing t h e solubility.
On the basis of the above given values, the solubili-
To gain deeper understanding of the driving forces
t i e s o f t h e five s o l i d k e t o n e s a n d o f t h e t w o e s t e r s a r e
o f t h e s o l u b i l i t y , it is i n t e r e s t i n g t o c o m p a r e , f o r a g i v e n solute, the relative contribution of each term appearing
r e c a l c u l a t e d in a l c o h o l s u s i n g E q . 2, a n d t h e r e s u l t s , c o n v e r t e d i n t o m o l a r f r a c t i o n , a r e r e p o r t e d i n T a b l e 5. Compared
i n E q s . 1 o r 2 t o t h e s o l u b i l i t y , i.e., t h e f l u i d i z a t i o n
t o t h e v a l u e s g i v e n in T a b l e 3, t h e a c t u a l
constant (A), the hydrophobic effect (F), the correc-
s o l u b i l i t i e s a r e e n h a n c e d a n d a r e in b e t t e r a g r e e m e n t with the experimental results; the agreement between t h e o b s e r v e d a n d p r e d i c t e d solubilities, i n d i c a t e d by
t i o n f a c t o r f o r t h e p l a c i n g e n t r o p y ( B ) , t h e c h a n g e in the non-specific cohesion forces upon mixing (D), and
the
ratio
xpred/x
o f p r e d i c t e d vs e x p e r i m e n t a l s o l u b i l i t y , exp, is n o w o f t h e s a m e o r d e r o f m a g n i t u d e
than that observed previously for the non-associated solvents. The presence of proton-acceptor sites on the s o l u t e s a b l e t o f o r m h y d r o g e n b o n d s in p r o t o n - d o n o r
the effect of hydrogen-bond formation (O). The values o f t h e s e c o n t r i b u t i o n s a r e r e p o r t e d in T a b l e 3 w h e n n o s o l u t e - s o l v e n t h y d r o g e n b o n d is c o n s i d e r e d , a n d in T a b l e 5 w h e n h y d r o g e n b o n d i n g is t a k e n i n t o a c c o u n t , e.g., f o r t h e s o l u b i l i t y p r e d i c t i o n s in a l c o h o l s . D e s c r i b ing u n f a v o u r a b l e p r o c e s s e s to t h e solubility, t h e contri-
P. Ruelle et aL / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
55
and the solvent respectively becomes large. In associated solvents like alcohols, the solubility value is mainly governed by the hydrophobic effect (term F) which varies according to the following relationship: the smaller the size of the solvent, the larger the hydrophobic effect (the decrease of the mobile order entropy) and the lower the resulting solubility. Finally, it can be observed that, when a hydrogen bond is formed between a solute and a solvent, the corresponding contribution (term O) is just added to the previous effects, without modifying significantly their values (cf. the values of the contributions B, D and F to the solubilities in alcohols reported in Tables 3 and 5). To investigate the transferability of the stability constants between molecules forming the same type of hydrogen bond with the solvent, we propose to predict the solubility of testosterone propionate, a pharmaceutical substance for which numerous experimental solubilities are given in the literature. Testosterone propionate displays two active sites, the ester and the ketone
Diethyl Methyl isobutyl Methyl ethyl Hexyl Butyl Ethyl
l~No~de~n~e
0.0
0.2
0.4
0.6
0.8
1.0
1.2
solubility ratio X"d/X °~
Fig. 1. Predicted versus experimental solubilityratio (xPred/x exp) of 10-nonadecanoneat 25°C.
butions A, D and F appear with negative signs, the relative magnitudes of which influence the order of magnitude of the solubility. For instance, in the same solvents, the solubility of 12-tricosanone remains always lower than that of octadecanoic methyl ester because of its lower fluidization constant (-3.529 versus -1.613), while they have similar values of their modified non-specific cohesion parameters. In contrast, the contributions B and O are positive and favour solubilization, except when the molar volume of the solvent is greater than that of the solute in which case the term B becomes negative. This is in particular the case with p-benzoquinone for which, in non-associated solvents, all the contributions tend to decrease its solubility. From a general point of view, the solubility of a given solute in non-associated solvents will depend on the balance between the relative opposite effects stemming from B and D, which are as great as the differences of the molar volumes and of the modified non-specific cohesion parameters between the solute
Eth~ 1.2-Dichl¢
Hexadecanoic methyl ester 0.0
0.2
0.4
0.6
0.8
1.0
1.2
solubility ratio XP"~/X °~ (xpred/x exp) of hecadecanoicmethylester at 20°C.
Fig. 2. Predicted versus experimental solubility ratio
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
56 Table 4
CH 3 0
Stability constants, K o , of s o l u t e - a l c o h o l H-bonds in solution, derived from the experimental solubilities, ~b~xp (volume fraction)
K o (cm3/
Solute
Solvent
6~xp
10-Nonadecanone (25°C)
Methanol Propanol Isopropanol Butanol Octanol Ethanol Propanol Butanol Methanol Isopropanol Butanol
0.01697 0.03186 0.03280 0.03898 0.04522 0.00680 0.01310 0.01600 0.004674 0.007403 0.01140
Methanol Propanol Butanol
0.2065 0.3036 0.3069
310.31 152.11 141.47
Methanol Ethanol Propanol Butanol
0.005376 0.03615 0.06240 0.08640
36.49 72.50 84.62 103.83
mol)
11-Heneicosanone (25°C)
12-Tricosanone (30°C)
Hexadecanoic methyl ester (20°C)
Octadecanoic methyl ester (20°C)
504.44 158.24 166.97 165.65 196.13 157.00 209.07 207.17 1081.08 113.78 158.56
Of
V
V
Scheme 1. Testosterone propionate.
groups (Scheme 1). While the solubility predictions in non-associated solvents require only the knowledge of the melting properties and of the modified non-specific cohesion p a r a m e t e r of testosterone propionate, the predictions in alcohols or in water require the knowledge of the stability constants, K o i , characteristic for the ester. • • alcohol, ketone • • • alcohol, ester. • • water and k e t o n e . . , water associations. Moreover, as the two active sites of testosterone propionate are positioned far away from each other and are not connected via delocalized electrons, we assume independency of those groups and additivity of their interactions with the solvent. Accordingly, the O term of the solubility equation has then to be written as: O = ln(1.O + K o l ( q b s / V s - 4~B/VB))
Table 5 Experimental, X ex°, and predicted (Eq. 2), X pred, solubilities of the ketone and ester solutes in alcohols and A, B, D, F, 0 contributions to their solubility S o l u t e / S o l v e n t X exp
X pred
B
D
F
O
- 2.046 - 3.528 -4.249 -4.350 - 8.253
0.697 1.013 1.138 1.154 1.639
1 0 - N o n a d e c a n o n e (25°C): A = - 2.630
Octanol 0.0217 Butanol 0.0109 Isopropanol 0.00766 Propanol 0.00726 Methanol 0.00207 l l - H e n e i c o s a n o n e (25°C): Butanol 0.00403 Propanol 0.00269 Eth anol 0.00109 1 2 - T r i c o s a n o n e (30°C): A Butanol 0.00262 Isopropanol 0.00142 Methanol 0.000473
0.0201 0.912 - 0 . 1 0 6 0.0111 1.920 - 0.002 0.00776 2.369 - 0 . 0 3 2 0.00765 2.431 - 0.000 0.000760 4.685 - 0 . 5 1 2 A = - 3.515 0.00351 2.187 - 0 . 0 0 0 0.00230 2.742 - 0.001 0.00115 3.560 - 0 . 0 5 5 = - 3.529 0.00274 2.411 - 0.008 0.00185 2.932 - 0 . 0 0 3 0.0000866 5.607 - 0 . 8 6 3 H e x a d e c a n o i c m e t h y l e s t e r (20°C): A = - 0.797 Butanol 0.117 0.0964 1.372 - 0 . 0 1 5 Propanol 0.0961 0.0704 1.796 - 0 . 0 0 8 Methanol 0.0330 0.00759 4.104 - 0 . 2 9 8 O c t a d e c a n o i c m e t h y l e s t e r (20°C): A = - 1.613 Butanol 0.0248 0.0257 1.860 - 0.000 Propanol 0.0144 0.0170 2.375 - 0.004 Ethanol 0.00640 0.00849 3.160 - 0 . 0 6 2 Methanol 0.000643 0.00129 4.721 - 0 . 6 2 9
- 3 . 9 7 6 1.035 - 4 . 8 8 4 1.173 - 6 2 7 5 1.354 - 4 . 3 3 9 1.037 - 5 . 2 0 3 1.159 - 9.918 1.643 - 2.473 0.580 - 3.138 0.709 - 7 . 1 7 6 1.263 - 3.385 -4.221 - 5.549 - 8.313
0.723 0.848 1.019 1.300
+ ln(1.0 + Ko2(dPs/Vs - d~B/VB) ) where Kol and Ko2 stand respectively for the interaction of the ester and the ketone groups. For the predictions in alcohols, we take the standard values of the stability constants Kketone... alcohol = 170 cm 3 m o l - 1 and Ke~ter...alcoho1 = 110 cm 3 mo1-1 determined previously. For predictions in water, we use for both Kketone...wate r and Kester...wate r the value of 3500 cm 3 m o l - t which corresponds to the mean association constant determined between water and the ester group (Ruelle et al., 1991). The melting point of testosterone propionate is 393.0 K. Its molar heat of fusion, determined by differential scanning calorimetry (James and Roberts, 1968), is 22100 J m o l - 1. These p a r a m e t e r s used together with a AC o value of 169.58 J K -1, leads to a fluidization constant A of - 1 . 3 0 at 25°C. Combined with the molar volume of 294 cm 3 mol-1, reported by James et al. (1976), one deduces the modified non-specific cohesion p a r a m e t e r of testosterone propionate, i.e., ~ = 2 0 . 5 8 4 j 1 / 2 cm-3/2, from its experimental solubility in hexane. From this unique experimental solubility value and using the stability constants established on a basis that
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
has nothing to do with testosterone propionate itself, its solubility is predicted in 28 solvents of known modified non-specific cohesion parameters, 6~, including alcohols and water (Table 6). It must be pointed out that the reported results are really predictions. They are not obtained by fitting experimental values with adjustable parameters. A remarkable agreement is observed between experimental and predicted solubilities. The worst predictions yield solubilities which are only two times greater or lower than the experimental results. For any solvent, the solubility equation yields a correct prediction of the order of magnitude of the solute solubility spanning over seven orders of magnitude. Like for any other solute, the solubility of testosterone propionate results from the balance of elementary processes favourable or unfavourable to the solution process. The analysis of the relative contributions to solute solubility (Table 7) and of their variations through the solvents brings therefore useful information concerning the origin of solubility. Based on the main contribution and neglecting the fluidization con-
57
tribution (which is constant and independent of the solvents), four categories of solvents may be distinguished (Fig. 3): (i) in aliphatic hydrocarbons, the sum of the contributions D and B is always negative and the absolute value of the ratio B/D is lower than one. In these solvents, the testosterone propionate solubility is mainly governed by nonspecific cohesion forces (D term) owing to the large difference of the modified non-specific cohesion parameters between the solvents (14.0 < 6~ < 16.0) and the solute (8~ = 20.58). The D term playing against solubilization, it can easily be predicted that the solute solubilities in aliphatic hydrocarbons will be lower than in solvents of the second class; (ii) in the aromatic hydrocarbons and in the polar non-associated solvents, the sum of the contributions B and D is always positive and the absolute value of the ratio B/D is greater than one. In these solvents, the solubility of testosterone propionate depends essentially on the correction term for the placing entropy (B term) which favours dissolution. In this class of solvents, solute solubilities are 10 to 100 times greateI than in class (i). However, looking at the absolute
Table 6
Experimental, X exp, and predicted (Eq. 2), X pred, solubilities (in molar fraction) of testosterone propionate at 25°C Solvent
X exp
X ored
X pred//X exp
Ref.
n-Pentane n-Hexane n -Heptane n-Octane n-Nonane n -Decane n-Undecane n-Dodecane n-Hexadecane Cyclohexane Methylcyclohexane t-butylcyclohexane Cyclooctane Benzene Toluene Tetraline CCI 4 CHCI 3 1,2-Dichloroethane
0.00300 0.00500 0.00580 0.00810 0.00740 0.00950 0.00850 0.0112 0.0100 0.0120 0.0145 0.0153 0.0181 0.260 0.200 0.200 0.160 0.354 0.294 0.320 0.300 0.270 0.220 0.180 0.251 0.0279 0.0401 0.0489 0.774.10- 7
0.00305 0.00500 0.00538 0.00672 0.00880 0.00936 0.0106 0.0117 0.0139 0.00898 0.00994 0.0162 0.0171 0.297 0.255 0.279 0.206 0.302 0.292 0.297 0.303 0.297 0.296 0.330 0.297 0.0571 0.0453 0.0457 0.406.10- 7
1.02 1.00 0.93 0.83 1.19 0.98 1.25 1.04 1.39 0.75 0.69 1.06 0.94 1.14 1.28 1.39 1.29 0.85 0.99 0.93 1.0l 1.10 1.34 1.83 1.18 2.05 1.13 0.94 0.52
James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James and Roberts James and Roberts James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James and Roberts this work this work James and Roberts
tram-l,2-Dichloroethylene cis- 1,2-Dichloroethylene Chlorobenzene Nitrobenzene CS 2 Anisole Ethanol Propanol Butanol Water
(1968) (1968)
(1968)
(1968)
P. Ruelle et al. /Pharmaceutica Acta Heluetiae 68 (1993) 49-60
58
values of the B and D contributions, it appears that the very reason of the greater solubilities in the present class is not due to an increase of the B term (with respect to the first class), but to the important decrease of the D contributions which is related to the smaller differences between the modified non-specific cohesion parameters of the solvents (17.0 < 3~ < 22.0) and the solute ( ~ = 20.58); (iii) in solvents associated by single H-bonded chains (r s = 1) like alcohols, the main contribution to the solubility is the negative hydrophobic effect (F term). However, this contribution is partially counterbalanced by the positive O contribution associated to the formation of new H-bonds between the solute and the solvent. Solubility values lay therefore between those of the first and second classes; (iv)
in solvents associated by double H-bonded chains (r s = 2) like water, the solubility of testosterone propionate depends predominantly on the "hydrophobic effect" contribution. Its large negative value is at the origin of the very low aqueous solubility of testosterone propionate and of its hydrophobicity. In contrast to what is observed with the previous class (iii), the enthalpy gained by the newly-formed hydrogen bonds between water and the solute (O term) is now far from being sufficient to compensate the decrease of the mobile order entropy of water ( F term) when the solvent molecules are brought in a larger volume as a consequence of the addition of testosterone propionate. Finally, in the case of aqueous solubility, the small changes in the non-specific cohesion forces can
Table 7 P r e d i c t e d (Eq. 2) solubilities, cb°red, (in v o l u m e f r a c t i o n ) o f t e s t o s t e r o n e p r o p i o n a t e at 25°C a n d B, D, F, a n d O c o n t r i b u t i o n s to its solubility. T h e f l u i d i z a t i o n c o n s t a n t A = - 1.30 Solvent
th pred
B
D
F
0
Aliphatic hydrocarbons n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane n-Hexadecane Cyclohexane Methylcyclohexane t-butylcyclohexane Cyclooctane
0.00770 0.0111 0.0107 0.0120 0.0143 0.0140 0.0145 0.0150 0.0156 0.0239 0.0225 0.0271 0.0365
1.222 1.009 0.834 0.685 0.557 0.448 0.345 0.265 0.124 1.320 1.039 0.593 0.948
- 4.789 -4.209 -4.074 - 3.806 -3.503 -3.417 - 3.277 - 3.164 -2.982 -3.754 - 3.534 -2.902 -2.959
Aromatic hydrocarbons and polar non-associated solvents Benzene 0.582 Toluene 0.485 Tetraline 0.453 CCl 4 0.440 CHCI 3 0.611 1,2-Dichloroethane 0.589 trans-l,2-Dichloroethylene 0.615 c&-l,2-Dichloroethylene 0.627 Chlorobenzene 0.549 Nitrobenzene 0.546 CS z 0.707 Anisole 0.510
0.814 0.771 0.556 0.947 0.867 0.857 0.897 0.898 0.730 0.729 0.953 0.631
- 0.055 - 0.194 - 0.047 - 0.466 - 0.059 - 0.086 - 0.083 - 0.064 -0.029 - 0.034 - 0.000 -0.004
H y d r o g e n - b o n d e d s o l v e n t s ( r s = 1) Ethanol Propanol Butanol
0.233 0.157 0.133
2.240 1.849 1.485
- 0.537 - 0.915 - 1.046
-3.843 -3.301 -2.771
Hydrogen bonded solvents(rs= Water
0.659-10 -6
9.015
- 0.001
- 32.486
1.982 1.814 1.613
2) 10.540
P. Ruelle et aL / Pharmaceutica Acta Helvetiae 68 (1993) 49-60 0.8
success from the knowledge of its solubility in hexane and the results are analyzed according to the nature of the solvents.
0.7-
aromatic hydrocarbons and
t-"
.9
o.6-
polar non-associated solvents
#
0.5-
Acknowledgements
0.4~
The authors thank the Ciba-Geigy-Jubil~iums-Stiftung for financial support and are indebted to Professor P.L Huyskens (University of Leuven, Belgium) for his advices.
0.30.2-
~E~ 0
0.1-
aliphatic hydrocarbons alcohols •
~
mmm
References
0.0--0.1
59
....
I .... -3
I''''1''' -2 -1
'1 . . . . 0
I .... 1
2
B+D Fig. 3. Solubility (in volume fraction) of testosterone propionate in various solvents versus the sum of the contributions B and D of the solubility equation.
be neglected without further loss of accuracy in the prediction over against the huge hydrophobic effect.
3. Conclusion
The solubility equation derived from the thermodynamics of the mobile order is used to predict the solubility of proton-acceptor substances, i.e., five solid ketones and two esters, in a wide range of organic nonelectrolyte solvents including alcohols. With respect to apolar hydrocarbons previously studied, the present solutes are able to interact in solution with the alcohols by forming specific preferential contacts. The effect of these preferential contacts on the solubility is accounted for by the O term of the solubility equation, and its treatment is mainly based on stability constants, K o . Obtained from experimental solubilities, two different mean standard stability constants, i.e., 170 and 110 cm 3 mol-1, have been used to characterize respectively the ketone-alcohol and ester-alcohol hydrogen bonds. The addition of the O term in the predictions has a net positive influence on the solubility by increasing its value without modifying significantly the values of the other contributions involved in the solution process. Using the stability constants previously determined, the solubility of testosterone propionate in 28 solvents including alcohols and water has been predicted with
Abrams, D.S. and Prausnitz, J.M. (1975) Statistical thermodynamics of liquid mixtures. A new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 21, 116-128. Acree Jr., W.E. (1992) Enthalpy of fusion of some organic compounds. In Lide, D.R. (Ed.), Handbook of Chemistry and Physics, 72nd edition. CRC Press, Boca Raton, pp. 83-90. Acree Jr., W.E. and Bertrand, G.L. (1983) Thermochemical investigations of nearly ideal binary solvents. 6. Solubilities of iodine and benzil in systems of nonspecific interactions. J. Solution Chem. 12, 101-113. Acree Jr., W.E. and Procyck, A.D. (1988) Comment on "entropies of diphenylglyoxal solutions in non-polar solvents". Thermochim. Acta 130, 367-368. Acree Jr,, W.E. and Rytting, J.H. (198,2a) Solubilities in binary solvent systems. II. The importance of non-specific interactions. Int. J. Pharm. 71, 231-238. Acree Jr., W.E. and Rytting, J.H. (1982b) Solubility in binary solvent systems. I. Specific versus nonspecific interactions. J. Pharm. Sci. 71, 201-204. Bailey, A.V., Harris, J.A. and Skau, E.L. (1970) Solubilities of some normal saturated and unsaturated long-chain fatty acids methyl esters in acetone, n-hexane, toluene, and 1,2-dichloroethane. J. Chem. Eng. Data 15, 583-585. Davies, M. and Kybett, B. (1965) Sublimation heats of long-chain methyl esters. Trans. Farad. Soc. 61, 1893-1896. Garland, F.M., Hoerr, C.W., Pool, W.O. and Ralston, A.W. (1943) Solubilities of high molecular weight symmetrical normal aliphatic ketones. J. Org. Chem. 8, 344-357. Hasted, J.B. (1974) The dielectric properties of water. In Luck, W.A.P. (Ed), Structure of Water and Aqueous Solutions, Verlag, Chemie, Weinheim, pp. 377-389. Hildebrand, J.H. and Scott, R.L. (1962) Regular Solutions. Prentice Hall, Englewood-Cliffs, NJ. Hildebrand, J.H., Prausnitz, J.M. and Scott, R.L. (1970) Regular and Related Solutions: the Solubility of Gases, Liquids and Solids. Van Nostrand Reinhold, NY. Huyskens, P.L. (1989) Seminar of Pharmaceutical Preformulation: Basic Concepts and Applications. Champ6ry, Switzerland. Huyskens, P.L. (1990) Thermodynamic and spectroscopic entities. J. Mol Liquids 46, 285-296. Huyskens, P.L. and Siegel, G.G. (1988) Fundamental questions about entropy. III. A kind of mobile order in liquids: preferential contacts between molecular groups. Bull. Soc. Chim. Belg. 97, 821-824.
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P. Ruelle et aL / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
James, K.C. and Roberts, M. (1968) The solubilities of the lower testosterone esters. J. Pharm. Pharmacol. 20, 709-714. James, K.C., NG, C.T. and Noyce, P.R. (1976) Solubilities of testosterone propionate and related esters in organic solvents. J. Pharm. Sci. 65, 656-659. Jimenez, E. and Ordonez, D. (1986) Entropies of diphenylglyoxal solutions in non-polar solvents. Thermocbim. Acta 108, 221-228. Kohler, F. (1974) Ultrasonic relaxation due to hydrogen bond dissociation. In Luck, W.A.P. (Ed.), Structure of Water and Aqueous Solutions. Verlag. Chemic, Weinheim, pp. 491-503. Magnussen, T., Rasmussen, P. and Fredenslund, A. (1981) UNIFAC parameter table for prediction of liquid-liquid equilibria. Ind. Eng. Chem., Process Des. Dev. 20, 331-339. Mishra, D.S. and Yalkowsky, S.H. (1992) Ideal solubility of a solid solute: effect of heat capacity assumptions. Pharm. Res. 9, 958959. Prausnitz, J.M., Lichtenhale, R.N. and De Azevedo, G. (1986) Molecular Thermodynamics of Fluid Phase Equilibria, 2nd ed. Prentice Hall, Englewood-Cliffs, NJ. Ruelle, P., Rey-Mermet, C. Buchmann, M., H6 Nam-Tran, Kesselring, U.W. and Huyskens, P.L. (1991) A new predictive equation for the solubility of drugs based on the thermodynamics of mobile disorder. Pharm. Res. 8, 840-850. Ruelle, P., Buchmann, M., H6 Nam-Tran and Kesselring, U.W.
(1992a) The mobile order theory versus UNIFAC and regular solution theory - derived models for predicting the solubility ot solid substances. Pharm. Res. 9, 788-791. Ruelle, P., Buchmann, M., H6 Nam-Tran and Kesselring, U.W. (1992b) Comparison of the solubility of polycyclic aromatic hydrocarbons in non-associated and associated solvents: the hydrophobic effect. Int. J. Pharm. 87, 47-57. Ruelle, P., Buchmann, M. and Kesselring, U.W. (1993) The hydrophobic effect at the origin of the low solubility of inert substances in hydrogen-bonded solvents. J. Pharm. Sci. submitted. Scatchard, G. (1931) Equilibria in non-electrolyte solutions in relation to the vapor pressures and densities of the components. Chem. Rev. 8, 321-333. Scott, R.L. (1987) Models for phase equilibria in fluid mixtures. Acc. Chem. Res. 20, 97-107. Sedgwick, R.S., Hoerr, C.W. and Harwood, H.J. (1952) Solubilities of saturated fatty acid esters. J. Org. Chem. 17, 327-337. Siegel, G.G., Huyskens, P.L. and Vanderheyden, L. (1990) Competition between solute-solvent and solvent-solvent hydrogen bonds: pyridines in alcohols and in water. Ber. Bunsen-Ges. Phys. Chem. 94, 549-553. Simonelli, A.P. and Higuchi, T. (1962) Melting and freezing behavior of methyl stearate. J. Pharm. Sci. 51, 584-590.
Pharmaceutica Acta Helvetiae 68 (1993) 49-60
The effect of proton-acceptor sites of the solute on its solubility in proton-donor solvents Paul Ruelle
a . , E l i e S a r r a f a, L e e n V a n D e n B e r g e b, K a t a r i n e Ulrich W. Kesselring
a
S e g h e r s b, M i c h e l B u c h m a n n
a,
a
Institut d'Analyse Pharmaceutique, Ecole de Pharmacie, Universit~ de Lausanne, BEP, CH-1015 Lausanne, Switzerland b Department of Chemistry, University ofLeuven, Celestijnenlaan 200 F, B-3030 Heverlee, Belgium
Abstract
The solubilities of five solid ketones and two esters are predicted in common organic nonelectrolyte solvents using the solubility equation derived from the mobile order theory. In the framework of this theory, the formation of solute-solvent hydrogen bonds is treated on the basis of standard stability constants. Two different values characterizing the ketone-alcohol and the ester-alcohol hydrogen bonds, respectively 170 and 110 cm3/mol, have been determined. The formation of specific molecular interactions brings about a net increasing of the solubility without modifying the values of the other contributions relevant to the solution process. Using the predetermined values of the stability constants, the solubility equation is then successfully applied to predict the solubility of testosterone propionate in 28 solvents including alcohols and water from the sole knowledge of its solubility in hexane. Key words: Solubility; Mobile order theory; Hydrogen bond; Stability constant; Modified solubility parameter; Testosterone propionate
I. Introduction
The prediction of the solubility of a solid solute in a given solvent requires, from a thermodynamic point of view, the knowledge of its ideal solubility and activity coefficient. The ideal solubility of a solid solute is related to the energy needed to transform that solute from its solid state to its hypothetical liquid state at the considered experimental temperature. It only depends on the melting properties of the solute, i.e., the melting t e m p e r a t u r e (Tm) , the enthalpy (AmeltH) or the entropy (AmeltS) of fusion and the heat capacity difference (ACp) between the solid and the supercooled liquid forms of the solute. A m o n g all these parameters, the latter is obviously the most difficult to determine experimentally. Accordingly, two alternative approximations are commonly used, either ACp is assumed to be zero or is assumed to be equal to the entropy of fusion. These two assumptions have recently been
* Corresponding author.
tested and critically discussed by Mishra and Yalkowsky (1992). From this discussion, it appears that none of these approximations can be considered to be superior or yielding better results. Once the solute is dissolved, the activity coefficient indicates the degree of deviation of the solute-solvent system formed with respect to the ideal behaviour. An activity coefficient which differs from 1 is an expression of excess (non-ideal) free-enthalpy, and may furthermore be decomposed into a partly enthalpy-governed residual activity coefficient and an excess-entropygoverned combinatorial activity coefficient. Various models dealing with the estimation of the activity coefficients can be found in the literature (Prausnitz et al., 1986; Scott, 1987). Depending on the origin attributed to the deviation, some of these models consider exclusively entropic effects while others consider entirely enthalpic effects. The latter include the methods based on the Hildebrand regular solution theory (Scatchard, 1931; Hildebrand and Scott, 1962; Hildebrand et al., 1970) and the universal group activity coefficient (UNIFAC) method (Magnussen et al., 1981). U N I Q U A C
0031-6865/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved SSDI: 0031-6865(93)E0004-Z
50
P. Ruelle et al. / Pharmaceutica Acta Heh'etiae 68 (1993) 49-60
(Abrams and Prausnitz, 1975) is one of the models that include both the enthalpic and entropic contributions to the activity coefficients. However, no single approach has by now been completely satisfactory. The ability to predict thermodynamic activity coefficients of substances in any given environment would however be of great value for understanding solution behaviour in general, and for predicting solubility and distribution properties of the molecules in particular. As the magnitudes of the activity coefficients are largely determined by the extent and the nature of molecular interactions between the dissolved molecules and their surrounding solvent molecules, the ability to predict solubility directly depends on how well one can quantify the various solute-solute, solvent-solvent and solute-solvent interactions involved in the solution process. An important step towards the predictions of the solubilities was made around 1930 when Scatchard published an equation, based on the previous work of Hildebrand, which allows to obtain the heat of mixing of two liquids on the basis of their molar heats of vaporization and molar volumes, or more precisely on the basis of their solubility parameters. This equation still constitutes the background of practically all theoretical approaches of solubility prediction found in the literature. It is a kind of modern formulation of the empirical principle of the alchemists "Similia similibus solvuntur". As a matter of fact, according to that equation, the best solvent would be that with the same solubility parameter as the solute. The ScatchardHildebrand equation is based on two fundamental assumptions: (i) the hypothesis of the geometric mean (the cohesive energy per unit volume between the solute and the solvent is the geometric mean of the values of the pure components); and (ii) the interactions between molecules in the solution occur "at random" and there are no preferential contacts. If these two assumptions can be regarded as crude approximations for the dispersion and dipole-dipole cohesion forces, they however contradict the very essence of hydrogen bonds. The principle of the alchemists is entirely erroneous when H-bonding interactions are involved because hydrogen bonds do not follow the rule of the geometric mean and do not lead to random contacts. One of the most important characteristics of Hbonding in the liquid phase is the establishment of "highly preferential" contacts between the protondonor and proton-acceptor sites of the various molecules. These particular contacts confer to Hbonded liquids a higher degree of "order" and a kind of structure. The structuration produced by H-bonding
is however intermittent or labile: contrary to valence bonds, H-bonds in liquids are regularly broken, their life times lying between 10 -5 and 10 -I1 s (Hasted, 1974; Kohler, 1974). The fundamental difference with the crystalline state lies in the "mobility" of the Hbonds in the liquid phase: the disorder or the order in a liquid is a dynamic one. Both the order introduced in the liquid by the formation of the hydrogen bonds and the perpetually moving character of these bonds constitute the basic foundations of the "mobile order" theory initiated by Huyskens and Siegel (Huyskens and Siegel, 1988; Huyskens, 1990; Siegel et al., 1990). The mobile order theory is at the basis of a new thermodynamic treatment of the liquid state, the quantitative development of which led to equations describing the effect of solvent-solvent, solute-solute and solute-solvent interactions on the chemical potential of the solute. A universal predictive equation for solubility (in volume fraction) of solid and liquid substances has therefore been derived (Ruelle et al., 1991) and has already been applied with success to predict the solubility of inert systems in non-polar, polar and hydrogen-bonded solvents (Ruelle et al., 1992a,b, 1993). The aim of this paper is to further apply this new theoretical approach to predict the solubility of crystalline proton-acceptor substances able to interact by hydrogen bond with proton-donor solvents like alcohols, and to determine the stability constants characterizing these hydrogen bonds. For that purpose, five solid ketones (10-nonadecanone, 11-heneicosanone, 12-tricosanone, p-benzoquinone and benzil) and two solid esters (hexadecanoic methyl ester and octadecanoic methyl ester) have been chosen. Their solubilities are predicted in a series of common organic nonelectrolyte solvents of widely differing polarities.
2. Results and d i s c u s s i o n
The predictive solubility equation takes into account the various contributions to the free-enthalpy change accompanying the solubilization process. As far as only chemically inert solid solutes are concerned, their solubilities are predicted, whatever the nature of the solvent, by the sum of four contributions (Eq. 1): the fluidization of the solute (A-term), the placing (exchange) entropy correction resulting from the difference in the molar volumes of solvent and solute (Bterm), the changes in the non-specific cohesion forces upon mixing (D-term), and the effect of the self-association of the solvent (F-term), i.e., the hydrophobic effect. On the other hand, when proton-acceptor so-
P. RueUe et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
lutes are considered, one has to add to the previous terms an additional contribution (O-term) to the solubility, in order to account for the effect of the H-bond formation between the proton-acceptor site of the solute and the proton-donor group of the solvent. For such substances, the solubility results therefore from the sum of five terms (Eq. 2). solubility of apolar substances In ~bB = A + B + D + F ,
(1)
solubility of proton-acceptor substances In c k B = A + B + D + F + O .
(2)
Characterizing single physical phenomena related to the solubilization process, each term of the above equations corresponds to well-defined contribution favourable or unfavourable to the solubility. A = -AmeltH(1/T-
1/Tm)/R
B = O.5cks(VB/Vs - 1) + 0.5 ln(~bB + cksVB/Vs)
D = - 62Vn(¢5~ - 3~)2/(RT) F = -rsqbsVR/V s
O = ln(1.0 + K o ( c k s / V s - ckB/VB)) In these expressions, AmeltH and Tm represent the molar heat and the temperature of fusion respectively; VB, 6'B, ckB, Vs, 6~, 4~s stand for the molar volume, the modified non-specific cohesion parameter and the volume fraction respectively of the solute B and of the solvent S; r s represents the "structuration or mobile order factor" of the solvent amounting approximately to 1 for solvents associated by single H-bond chains like alcohols, 2 for diols and water, and 0 for non-associated solvents; K o is the stability constant which
Table 1 Physical properties of the solutes Solute
10-Nonadecanone ll-Heneicosanone 12-Tricosanone p-Benzoquinone Benzil Hexadecanoic methyl ester Octadecanoic methyl ester
VB (cm3/ mol)
t~ (jl/2 cm - 3/2)
(kJ/mol)
Tm (°C)
338.0 371.0 404.0 79.6 180.0 309.0 342.0
17.30 17.20 16.93 24.92 21.72 17.63 17.11
67.264 76.200 78.034 18.531 19.762 55.647 64.434
57.00 63.50 69.00 115.00 95.20 30.60 39.05
a b b c c c c
AmeltH
a b b ~ d f f
a b b e d f g
a this work; b Huyskens (1989); c determined from the experimental solubility of the solute in hexane or in heptane; d Acree Jr. (1992); Acree Jr. and Ry ning (1982a); f Davies and Kybett (1965); g Simonelli and Higuchi (1962).
51
governs the H-bonding formation between the protonacceptor solute and the proton-donor solvent. All these physico-chemical properties, except K o, needed for the solubility predictions are reported in Table 1 for the solid solutes (five ketones and two esters), and in Table 2 for the solvents in which solubilities are predicted. Remember that the molar volumes of the solutes in solution are calculated by adding group contributions (Ruelle et al., 1991) for the various groups constituting the molecule. To study the effect of active proton-acceptor sites of the solute on their solubility, all the solubilities have been first calculated using Eq. 1 by considering the ketone and ester solutes as apolar substances unable to form hydrogen bonds with proton-donor solvents. The results expressed in molar fraction (Table 3) are compared to the experimental values found in the literature or determined by ourselves at the appropriate temperature by using the visual thermoturbidimetric technique. The quality of the predictions is demonstrated (Figs. 1 and 2) by the solubility ratios, x p r e d / x exp, between the predicted and experimental values. In the case of the non-associated so!vents, the ratios vary from 0.62 to 1.39 (except for 12-tricosanone in acetonitrile and nitroethane, and for octadecanoic methyl ester in acetone and acetonitrile). However, in the case of alcohols, one observes that all the predicted solubilities are substantially lower than the experimental results, with similar solubility ratios amounting to about 0.35 for ethanol to octanol, and 0.05 for methanol. Such results clearly demonstrate the power and the validity of Eq. 1 to predict the solubility of polar proton-acceptor substances in a wide range of non-associated nonelectrolyte solvents for which the whole solubility process does not differ from that of apolar substances, but also show that Eq. 1 is not sufficient when proton-donor solvents are considered. As a matter of fact, the formation of solute-solvent hydrogen bonds in solution increases the solute solubility by decreasing the free-energy of the system. This additional contribution to the solubility is treated in the frame of the mobile order theory by the O term (Ruelle et al., 1991) included in Eq. 2. Contrary to the currently used approaches based on Hildebrand (or Hansen) solubility parameters which do not take preferential contacts and entropy effects originating from H-bonding into account, the present treatment is based on stability constants, K o , and on the concentration of the active sites of the solvent, 4~s/Vs, which allows to include the important entropy effects accompanying the formation of specific molecular complexes. To make solubility predictions by means of Eq. 2,
P. Ruelle et al. / Pharmaceutica Acta Heh,etiae 68 (1993) 49-60
52
one therefore needs to know the order of magnitude of the stability constants, K o , characterizing the hydrogen bonds formed between the O H groups of the alcohols and the ketone or ester groups of the solutes. However, the stability constants, K o, appearing in Eq. 2 are influenced by the self-association of the solvent and hence do not correspond to those which one should find (by IR spectroscopy for instance) when diluting the associated solvent in an inert one so as to suppress the self-association bonds. Moreover, as these stability constants depend on the solute and on the solvent, different values should be used for each particular
system, which would make the predictions impossible. Fortunately, it appears that, for a given class of solutes, the values of K o, deduced from experimental solubilities by means of the solubility equation, do not vary very much in a given family of solvents (Table 4). Without impairing the predictive character of the model, one can therefore use mean standard values of stability constants, K o , which characterize a given type of hydrogen bond. In this work, two different values, i.e., 170 and 110 cm 3 mo1-1 as mean values of the particular stability constants reported in Table 4 (excluding the results related to methanol), have been
Table 2 Molar volume, Vs, and modified non-specific cohesion parameter, 6~, of various solvents at 25°C Solvent n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane n-Tetradecane n-Hexadecane Decaline Squalane Cyclohexane Methylcyclohexane t-Butylcyclohexane Cyclooctane Isooctane Benzene Toluene Ethylbenzene p-Cymene m-Xylene p-Xylene Mesitylene Tetraline Carbon tetrachloride Chloroform Dichloromethane 1,1-Dichloroethane
Vs
6~
(cm3/mol)
(j1/2 cm-3/2)
116.1 131.6 147.5 163.5 179.6 195.9 213.3 228.6 260.3 294.1 156.9 525.0 108.8 128.3 173.9 134.9 166.1 89.4 106.9 123.1 156.0 123.2 123.9 139.6 137.1 97.1 80.7 64.5 84.8 1,2-Dichloroethane 78.8 1,1-Dibromoethane 92.9 1,2-Dibromoethane 97.0 n-Chlorobutane 105.0 1,4-Dichlorobutane 112.1 Chlorobenzene 102.1 o-Dichlorobenzene 113.1 Bromobenzene 105.3 trans-l,2-Dichloroethylene 78.0 cis-l,2-Dichloroethylene 76.0 Nitrobenzene 102.7
14.18 14.56 14.66 14.85 15.07 15.14 15.25 15.34 15.49 15.61 18.90 16.25 14.82 15.00 15.50 15.40 14.30 18.95 18.10 18.02 20.20 17.20 17.30 17.00 19.43 17.04 18.77 20.53 18.51 20.99 18.77 20.75 17.12 19.78 19.48 18.77 21.22 18.41 18.61 21.77
Solvent Nitroethane Pyridine N,N-Dimet hylformamide Acetonitrile Carbon Disulfide Thiophene Acetone Methyl ethyl ketone Diethyl ketone Methyl isobutyl ketone Dipropyl ketone Anisole Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Dipropyl ether Dibutyl ether Dipentyl ether Dioxane Methyl formate Methyl acetate Ethyl acetate Butyl acetate Hexyl acetate Ethyl propionate Diethyl adipate Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol Isobutanol t-Butanol 1-Pentanol 1-Octanol Cyclohexanol Furfuryl alcohol Water
Vs
~
(cm3/mol)
(j1/2 cm-3/2)
71.8 80.9 77.0 52.9 60.0 79.6 74.0 90.2 106.4 125.8 140.4 119.1 104.8 120.4 124.3 141.8 170.3 204.0 85.8 62.1 79.8 98.5 132.5 165.8 115.5 202.2 40.7 58.7 75.1 76.9 92.0 92.4 92.8 94.3 108.6 158.3 106.0 86.5 18.1
22.44 20.94 22.15 23.62 20.50 18.70 21.91 20.90 20.13 19.95 19.49 20.20 18.78 15.70 16.12 17.96 17.45 17.30 20.89 22.96 21.67 20.79 19.66 19.15 20.05 18.17 19.25 17.81 17.29 16.80 17.16 16.60 16.14 15.78 16.85 16.38 17.88 18.99 20.50
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
53
Table 3
Experimental, X exp, and predicted (Eq. 1), X pred, solubilities of the ketone and ester solutes and A, B, D, F contributions to their solubility Solute/Solvent
X exp
10.Nonadecanone (25°C): A = - 2.630 CCI 4 0.110 Benzene 0.109 Cyclohexane 0.0635 Hexyl acetate 0.0532 Butyl acetate 0.0448 Hexadecane 0.0431 Tetradecane 0.0429 Diethyl ketone 0.0467 Methyl isobutyl ketone 0.0448 Decane 0.0445 Hexane 0.0474 Methyl ethyl ketone 0.0333 Ethyl acetate 0.0317 Acetone 0.0152 Nitroethane 0.00704 Acetonitrile 0.00221 Octanol 0.0217 Butanol 0.0109 Isopropanol 0.00766 Propanol 0.00726 Methanol 0.00207 l l - H e n e i c o s a n o n e (25°C): A = - 3.515 GEl 4 0.0739 p-Xylene 0.0606 Benzene 0.0575 Butyl acetate 0.0135 Methylisobutyl ketone 0.0172 Ethyl acetate 0.0107 Methyl acetate 0.00416 Butanol 0.00403 Propanol 0.00269 Ethanol 0.00109 12-Tricosanone (30°C): A = - 3.529 CC14 0.0505 Benzene 0.0447 Cyclohexane 0.0306 Hexane 0.0204 Butyl acetate 0.0125 Methyl ethyl ketone 0.0719 Ethyl acetate 0.00569 Acetone 0.00188 Nitroethane 0.00195 Acetonitrile 0.000485 Butanol 0.00262 Isopropanol 0.00142 Methanol 0.000473 Benzil(25°C):A = - 1.519 CH2CI 2 0.277 1,2-Dichloroethane 0.260 CHC13 0.297 Benzene 0.179 Toluene 0.149 CCI 4 0.0790 Cyclooctane 0.0145 Cyclohexane 0.0107 Octane 0.00726 Heptane 0.00657 Hexane 0.00573 Isooctane 0.00587
X pred
B
D
0.104 0.0934 0.0581 0.0570 0.0516 0.0513 0.0495 0.0483 1.01 0.0453 0.0410 0.0327 0.0314 0.0144 0.00691 0.00203 0.00956 0.00376 0.00231 0.00224 0.000143
1.391 1.550 1.393 0.790 1.111 0.136 0.263 1.465 1.205 0.592 1.147 1.837 1.674 2.406 2.557 3.582 0.930 1.964 2.417 2.481 4.706
-0.005 -0.192 -0.591 -0.370 -0.586 -0.345 - 0.392 - 0.810 - 0.753 -0.544 -0.831 -1.392 - 1.345 - 2.546 - 3.377 - 5.307 - 0.111 - 0.003 - 0.033 -0.000 -0.517
0.0521 0.0431 0.0424 0.0188 0.92 0.00981 0.00474 0.00121 0.000693 0.000290
1.767 1.386 1.979 1.352 1.456 1.983 2.545 2.204 2.761 3.576
-0.003 -0.001 -0.327 -0.816 - 1.031 -1.793 -2.862 -0.000 - 0.001 -0.055
0.0555 0.0407 0.0301 0.0204 0.0152 0.0704 0.00690 0.00236 0.00100 0.000228 0.000949 0.000566 0.0000167
1.902 2.164 1.833 1.513 1.520 2.424 2.203 3.045 3.162 4.328 2.427 2.949 5.610
-0.001 - 0.461 -0.572 -0.794 -1.091 -2.376
0.244 0.238 0.185 0.183 0.124 0.0708 0.0136 0.00810 0.00750 0.00623 0.00573 0.00415
0.802 0.655 0.707 0.615 0.496 0.653 0.306 0.572 0.098 0.208 0.338 0.082
-0.029 -0.013 -0.280 -0.266 -0.621 - 1.224 -2.801 -3.370 - 3.376 -3.570 -3.670 -3.967
- 3.878 - 4.815 - 7.153 - 0.009 -0.003 -0.864
F
Ref.
-2.092 -3.624 -4.351 -4.456 -8.295
Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) this work Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Garland et al. (1943) Ruelle et al. (1991) Ruelle et al. (1991) Garland et al. (1943) Ruelle et al. (1991) Garland et al. (1943)
-4.013 -4.923 -6.309
Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens (1989) Huyskens(1989) Huyskens (1989) Huyskens(1989) Huyskens (1989) Huyskens(1989)
-4.373 -5.238 -9.925
Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland Garland
et et et et et et et et et et et et et
al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943) al. (1943)
Jimenez and Ordonez (1986) Jimenez and Ordonez (1986) Jimenez and Ordonez (1986) Acree and Procyck (1988) Acree and Procyck (1988) Acree and Procyck (1988) Acree and Rytting (1982b) Acree and Bertrand (1983) Acree and Rytting (1982b) Acree and Procyck (1988) Acree and Procyck (1988) Acree and Rytting (1982b)
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
54 Table 3 (continued) Solute/Solvent
X exp
X pred
(25°C): A = - 1.733 0.0344 0.0264 0.00872 0.0116 0.00800 0.0105 0.00749 0.00771 0.00652 0.00704 0.00664 0.00664 0.00665 0.00534 Hexadecanoic m e t h y l e s t e r (20°C): A = - 0.797 CHCI 3 0.561 0.479 CCI 4 0.544 0.476 Benzene 0.499 0.475 Toluene 0.500 0.474 1,2-Dichloroethane 0.500 0.452 Cyclohexane 0.438 0.434 Butyl acetate 0.387 0.441 Acetone 0.349 0.435 Ethyl acetate 0.412 0.430 Hexane 0.394 0.394 Octane 0.396 0.377 Butanol 0.117 0.0362 Propanol 0.0961 0.0218 Methanol 0.0330 0.00171
B
D
F
Ref.
p-Benzoquinone
CC14 Dodecane Cyclooctane Octane Cyclohexane Heptane Isooctane
Octadecanoic
-
0.185 0.848 0.465 0.614 0.289 0.536 0.626
-
1.910 2.926 2.876 3.234 3.244 3.359 3.606
Acree Acree Acree Acree Acree Acree Acree
and and and and and and and
Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a) Rytting (1982a)
0.555 0.503 0.532 0.474 0.607 0.518 0.427 0.657 0.558 0.480 0.381 1.612 2.101 4.262
-
0.008 0.003 0.013 0.002 0.080 0.099 0.075 0.130 0.110 0.187 0.213 0.022 0.012 0.325
- 2.982 - 3.769 - 7.495
Sedgwick et al. (1952) Sedgwick et al. (1952) Sedgwick et al. (1952) Bailey et al. (1970) Bailey et al. (1970) Sedgwick et al. (1952) Sedgwick et al. (1952) Sedgwick et al. (1952) Sedgwick et al. (1952) this work this work this work this work this work
0.981 1.126 1.085 1.009 0.905 0.690 0.900 1.349 2.060 3.586 1.947 2.474 3.250 4.754
-
0.000 0.071 0.105 0.239 0.415 0.380
-
0.900 1.916 5.650 0.000 0.004 0.067 0.639
-3.571 - 4.426 - 5.736 - 8.379
Sedgwick et Sedgwick et Sedgwick et Sedgwick et this work this work Sedgwick et Sedgwick et Sedgwick et Sedgwick et this work this work this work Sedgwick et
methyl ester (20°C): A = - 1.613
CCI 4 CHCI 3 Benzene Cyclohexane Hexane Octane Butyl acetate Ethyl acetate Acetone Acetonitrile Butanol Propanol Ethanol Methanol
0.271 0.324 0.237 0.167 0.156 0.144 0.140 0.110 0.0373 0.00151 0.0248 0.0144 0.00640 0.000643
0.244 0.240 0.228 0.194 0.156 0.151 0.156 0.115 0.0607 0.00400 0.0109 0.00633 0.00270 0.000334
used to characterize respectively the ketone-alcohol a n d e s t e r - a l c o h o l H - b o n d e q u i l i b r i u m in s o l u t i o n .
al. al. al. al.
(1952) (1952) (1952) (1952)
al. al. al. al.
(1952) (1952) (1952) (1952)
al. (1952)
s o l v e n t s r e s u l t s t h e r e f o r e in a p o s i t i v e e f f e c t b y i n c r e a s ing t h e solubility.
On the basis of the above given values, the solubili-
To gain deeper understanding of the driving forces
t i e s o f t h e five s o l i d k e t o n e s a n d o f t h e t w o e s t e r s a r e
o f t h e s o l u b i l i t y , it is i n t e r e s t i n g t o c o m p a r e , f o r a g i v e n solute, the relative contribution of each term appearing
r e c a l c u l a t e d in a l c o h o l s u s i n g E q . 2, a n d t h e r e s u l t s , c o n v e r t e d i n t o m o l a r f r a c t i o n , a r e r e p o r t e d i n T a b l e 5. Compared
i n E q s . 1 o r 2 t o t h e s o l u b i l i t y , i.e., t h e f l u i d i z a t i o n
t o t h e v a l u e s g i v e n in T a b l e 3, t h e a c t u a l
constant (A), the hydrophobic effect (F), the correc-
s o l u b i l i t i e s a r e e n h a n c e d a n d a r e in b e t t e r a g r e e m e n t with the experimental results; the agreement between t h e o b s e r v e d a n d p r e d i c t e d solubilities, i n d i c a t e d by
t i o n f a c t o r f o r t h e p l a c i n g e n t r o p y ( B ) , t h e c h a n g e in the non-specific cohesion forces upon mixing (D), and
the
ratio
xpred/x
o f p r e d i c t e d vs e x p e r i m e n t a l s o l u b i l i t y , exp, is n o w o f t h e s a m e o r d e r o f m a g n i t u d e
than that observed previously for the non-associated solvents. The presence of proton-acceptor sites on the s o l u t e s a b l e t o f o r m h y d r o g e n b o n d s in p r o t o n - d o n o r
the effect of hydrogen-bond formation (O). The values o f t h e s e c o n t r i b u t i o n s a r e r e p o r t e d in T a b l e 3 w h e n n o s o l u t e - s o l v e n t h y d r o g e n b o n d is c o n s i d e r e d , a n d in T a b l e 5 w h e n h y d r o g e n b o n d i n g is t a k e n i n t o a c c o u n t , e.g., f o r t h e s o l u b i l i t y p r e d i c t i o n s in a l c o h o l s . D e s c r i b ing u n f a v o u r a b l e p r o c e s s e s to t h e solubility, t h e contri-
P. Ruelle et aL / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
55
and the solvent respectively becomes large. In associated solvents like alcohols, the solubility value is mainly governed by the hydrophobic effect (term F) which varies according to the following relationship: the smaller the size of the solvent, the larger the hydrophobic effect (the decrease of the mobile order entropy) and the lower the resulting solubility. Finally, it can be observed that, when a hydrogen bond is formed between a solute and a solvent, the corresponding contribution (term O) is just added to the previous effects, without modifying significantly their values (cf. the values of the contributions B, D and F to the solubilities in alcohols reported in Tables 3 and 5). To investigate the transferability of the stability constants between molecules forming the same type of hydrogen bond with the solvent, we propose to predict the solubility of testosterone propionate, a pharmaceutical substance for which numerous experimental solubilities are given in the literature. Testosterone propionate displays two active sites, the ester and the ketone
Diethyl Methyl isobutyl Methyl ethyl Hexyl Butyl Ethyl
l~No~de~n~e
0.0
0.2
0.4
0.6
0.8
1.0
1.2
solubility ratio X"d/X °~
Fig. 1. Predicted versus experimental solubilityratio (xPred/x exp) of 10-nonadecanoneat 25°C.
butions A, D and F appear with negative signs, the relative magnitudes of which influence the order of magnitude of the solubility. For instance, in the same solvents, the solubility of 12-tricosanone remains always lower than that of octadecanoic methyl ester because of its lower fluidization constant (-3.529 versus -1.613), while they have similar values of their modified non-specific cohesion parameters. In contrast, the contributions B and O are positive and favour solubilization, except when the molar volume of the solvent is greater than that of the solute in which case the term B becomes negative. This is in particular the case with p-benzoquinone for which, in non-associated solvents, all the contributions tend to decrease its solubility. From a general point of view, the solubility of a given solute in non-associated solvents will depend on the balance between the relative opposite effects stemming from B and D, which are as great as the differences of the molar volumes and of the modified non-specific cohesion parameters between the solute
Eth~ 1.2-Dichl¢
Hexadecanoic methyl ester 0.0
0.2
0.4
0.6
0.8
1.0
1.2
solubility ratio XP"~/X °~ (xpred/x exp) of hecadecanoicmethylester at 20°C.
Fig. 2. Predicted versus experimental solubility ratio
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
56 Table 4
CH 3 0
Stability constants, K o , of s o l u t e - a l c o h o l H-bonds in solution, derived from the experimental solubilities, ~b~xp (volume fraction)
K o (cm3/
Solute
Solvent
6~xp
10-Nonadecanone (25°C)
Methanol Propanol Isopropanol Butanol Octanol Ethanol Propanol Butanol Methanol Isopropanol Butanol
0.01697 0.03186 0.03280 0.03898 0.04522 0.00680 0.01310 0.01600 0.004674 0.007403 0.01140
Methanol Propanol Butanol
0.2065 0.3036 0.3069
310.31 152.11 141.47
Methanol Ethanol Propanol Butanol
0.005376 0.03615 0.06240 0.08640
36.49 72.50 84.62 103.83
mol)
11-Heneicosanone (25°C)
12-Tricosanone (30°C)
Hexadecanoic methyl ester (20°C)
Octadecanoic methyl ester (20°C)
504.44 158.24 166.97 165.65 196.13 157.00 209.07 207.17 1081.08 113.78 158.56
Of
V
V
Scheme 1. Testosterone propionate.
groups (Scheme 1). While the solubility predictions in non-associated solvents require only the knowledge of the melting properties and of the modified non-specific cohesion p a r a m e t e r of testosterone propionate, the predictions in alcohols or in water require the knowledge of the stability constants, K o i , characteristic for the ester. • • alcohol, ketone • • • alcohol, ester. • • water and k e t o n e . . , water associations. Moreover, as the two active sites of testosterone propionate are positioned far away from each other and are not connected via delocalized electrons, we assume independency of those groups and additivity of their interactions with the solvent. Accordingly, the O term of the solubility equation has then to be written as: O = ln(1.O + K o l ( q b s / V s - 4~B/VB))
Table 5 Experimental, X ex°, and predicted (Eq. 2), X pred, solubilities of the ketone and ester solutes in alcohols and A, B, D, F, 0 contributions to their solubility S o l u t e / S o l v e n t X exp
X pred
B
D
F
O
- 2.046 - 3.528 -4.249 -4.350 - 8.253
0.697 1.013 1.138 1.154 1.639
1 0 - N o n a d e c a n o n e (25°C): A = - 2.630
Octanol 0.0217 Butanol 0.0109 Isopropanol 0.00766 Propanol 0.00726 Methanol 0.00207 l l - H e n e i c o s a n o n e (25°C): Butanol 0.00403 Propanol 0.00269 Eth anol 0.00109 1 2 - T r i c o s a n o n e (30°C): A Butanol 0.00262 Isopropanol 0.00142 Methanol 0.000473
0.0201 0.912 - 0 . 1 0 6 0.0111 1.920 - 0.002 0.00776 2.369 - 0 . 0 3 2 0.00765 2.431 - 0.000 0.000760 4.685 - 0 . 5 1 2 A = - 3.515 0.00351 2.187 - 0 . 0 0 0 0.00230 2.742 - 0.001 0.00115 3.560 - 0 . 0 5 5 = - 3.529 0.00274 2.411 - 0.008 0.00185 2.932 - 0 . 0 0 3 0.0000866 5.607 - 0 . 8 6 3 H e x a d e c a n o i c m e t h y l e s t e r (20°C): A = - 0.797 Butanol 0.117 0.0964 1.372 - 0 . 0 1 5 Propanol 0.0961 0.0704 1.796 - 0 . 0 0 8 Methanol 0.0330 0.00759 4.104 - 0 . 2 9 8 O c t a d e c a n o i c m e t h y l e s t e r (20°C): A = - 1.613 Butanol 0.0248 0.0257 1.860 - 0.000 Propanol 0.0144 0.0170 2.375 - 0.004 Ethanol 0.00640 0.00849 3.160 - 0 . 0 6 2 Methanol 0.000643 0.00129 4.721 - 0 . 6 2 9
- 3 . 9 7 6 1.035 - 4 . 8 8 4 1.173 - 6 2 7 5 1.354 - 4 . 3 3 9 1.037 - 5 . 2 0 3 1.159 - 9.918 1.643 - 2.473 0.580 - 3.138 0.709 - 7 . 1 7 6 1.263 - 3.385 -4.221 - 5.549 - 8.313
0.723 0.848 1.019 1.300
+ ln(1.0 + Ko2(dPs/Vs - d~B/VB) ) where Kol and Ko2 stand respectively for the interaction of the ester and the ketone groups. For the predictions in alcohols, we take the standard values of the stability constants Kketone... alcohol = 170 cm 3 m o l - 1 and Ke~ter...alcoho1 = 110 cm 3 mo1-1 determined previously. For predictions in water, we use for both Kketone...wate r and Kester...wate r the value of 3500 cm 3 m o l - t which corresponds to the mean association constant determined between water and the ester group (Ruelle et al., 1991). The melting point of testosterone propionate is 393.0 K. Its molar heat of fusion, determined by differential scanning calorimetry (James and Roberts, 1968), is 22100 J m o l - 1. These p a r a m e t e r s used together with a AC o value of 169.58 J K -1, leads to a fluidization constant A of - 1 . 3 0 at 25°C. Combined with the molar volume of 294 cm 3 mol-1, reported by James et al. (1976), one deduces the modified non-specific cohesion p a r a m e t e r of testosterone propionate, i.e., ~ = 2 0 . 5 8 4 j 1 / 2 cm-3/2, from its experimental solubility in hexane. From this unique experimental solubility value and using the stability constants established on a basis that
P. Ruelle et al. / Pharmaceutica Acta Helvetiae 68 (1993) 49-60
has nothing to do with testosterone propionate itself, its solubility is predicted in 28 solvents of known modified non-specific cohesion parameters, 6~, including alcohols and water (Table 6). It must be pointed out that the reported results are really predictions. They are not obtained by fitting experimental values with adjustable parameters. A remarkable agreement is observed between experimental and predicted solubilities. The worst predictions yield solubilities which are only two times greater or lower than the experimental results. For any solvent, the solubility equation yields a correct prediction of the order of magnitude of the solute solubility spanning over seven orders of magnitude. Like for any other solute, the solubility of testosterone propionate results from the balance of elementary processes favourable or unfavourable to the solution process. The analysis of the relative contributions to solute solubility (Table 7) and of their variations through the solvents brings therefore useful information concerning the origin of solubility. Based on the main contribution and neglecting the fluidization con-
57
tribution (which is constant and independent of the solvents), four categories of solvents may be distinguished (Fig. 3): (i) in aliphatic hydrocarbons, the sum of the contributions D and B is always negative and the absolute value of the ratio B/D is lower than one. In these solvents, the testosterone propionate solubility is mainly governed by nonspecific cohesion forces (D term) owing to the large difference of the modified non-specific cohesion parameters between the solvents (14.0 < 6~ < 16.0) and the solute (8~ = 20.58). The D term playing against solubilization, it can easily be predicted that the solute solubilities in aliphatic hydrocarbons will be lower than in solvents of the second class; (ii) in the aromatic hydrocarbons and in the polar non-associated solvents, the sum of the contributions B and D is always positive and the absolute value of the ratio B/D is greater than one. In these solvents, the solubility of testosterone propionate depends essentially on the correction term for the placing entropy (B term) which favours dissolution. In this class of solvents, solute solubilities are 10 to 100 times greateI than in class (i). However, looking at the absolute
Table 6
Experimental, X exp, and predicted (Eq. 2), X pred, solubilities (in molar fraction) of testosterone propionate at 25°C Solvent
X exp
X ored
X pred//X exp
Ref.
n-Pentane n-Hexane n -Heptane n-Octane n-Nonane n -Decane n-Undecane n-Dodecane n-Hexadecane Cyclohexane Methylcyclohexane t-butylcyclohexane Cyclooctane Benzene Toluene Tetraline CCI 4 CHCI 3 1,2-Dichloroethane
0.00300 0.00500 0.00580 0.00810 0.00740 0.00950 0.00850 0.0112 0.0100 0.0120 0.0145 0.0153 0.0181 0.260 0.200 0.200 0.160 0.354 0.294 0.320 0.300 0.270 0.220 0.180 0.251 0.0279 0.0401 0.0489 0.774.10- 7
0.00305 0.00500 0.00538 0.00672 0.00880 0.00936 0.0106 0.0117 0.0139 0.00898 0.00994 0.0162 0.0171 0.297 0.255 0.279 0.206 0.302 0.292 0.297 0.303 0.297 0.296 0.330 0.297 0.0571 0.0453 0.0457 0.406.10- 7
1.02 1.00 0.93 0.83 1.19 0.98 1.25 1.04 1.39 0.75 0.69 1.06 0.94 1.14 1.28 1.39 1.29 0.85 0.99 0.93 1.0l 1.10 1.34 1.83 1.18 2.05 1.13 0.94 0.52
James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James and Roberts James and Roberts James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James et al. (1976) James and Roberts this work this work James and Roberts
tram-l,2-Dichloroethylene cis- 1,2-Dichloroethylene Chlorobenzene Nitrobenzene CS 2 Anisole Ethanol Propanol Butanol Water
(1968) (1968)
(1968)
(1968)
P. Ruelle et al. /Pharmaceutica Acta Heluetiae 68 (1993) 49-60
58
values of the B and D contributions, it appears that the very reason of the greater solubilities in the present class is not due to an increase of the B term (with respect to the first class), but to the important decrease of the D contributions which is related to the smaller differences between the modified non-specific cohesion parameters of the solvents (17.0 < 3~ < 22.0) and the solute ( ~ = 20.58); (iii) in solvents associated by single H-bonded chains (r s = 1) like alcohols, the main contribution to the solubility is the negative hydrophobic effect (F term). However, this contribution is partially counterbalanced by the positive O contribution associated to the formation of new H-bonds between the solute and the solvent. Solubility values lay therefore between those of the first and second classes; (iv)
in solvents associated by double H-bonded chains (r s = 2) like water, the solubility of testosterone propionate depends predominantly on the "hydrophobic effect" contribution. Its large negative value is at the origin of the very low aqueous solubility of testosterone propionate and of its hydrophobicity. In contrast to what is observed with the previous class (iii), the enthalpy gained by the newly-formed hydrogen bonds between water and the solute (O term) is now far from being sufficient to compensate the decrease of the mobile order entropy of water ( F term) when the solvent molecules are brought in a larger volume as a consequence of the addition of testosterone propionate. Finally, in the case of aqueous solubility, the small changes in the non-specific cohesion forces can
Table 7 P r e d i c t e d (Eq. 2) solubilities, cb°red, (in v o l u m e f r a c t i o n ) o f t e s t o s t e r o n e p r o p i o n a t e at 25°C a n d B, D, F, a n d O c o n t r i b u t i o n s to its solubility. T h e f l u i d i z a t i o n c o n s t a n t A = - 1.30 Solvent
th pred
B
D
F
0
Aliphatic hydrocarbons n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane n-Hexadecane Cyclohexane Methylcyclohexane t-butylcyclohexane Cyclooctane
0.00770 0.0111 0.0107 0.0120 0.0143 0.0140 0.0145 0.0150 0.0156 0.0239 0.0225 0.0271 0.0365
1.222 1.009 0.834 0.685 0.557 0.448 0.345 0.265 0.124 1.320 1.039 0.593 0.948
- 4.789 -4.209 -4.074 - 3.806 -3.503 -3.417 - 3.277 - 3.164 -2.982 -3.754 - 3.534 -2.902 -2.959
Aromatic hydrocarbons and polar non-associated solvents Benzene 0.582 Toluene 0.485 Tetraline 0.453 CCl 4 0.440 CHCI 3 0.611 1,2-Dichloroethane 0.589 trans-l,2-Dichloroethylene 0.615 c&-l,2-Dichloroethylene 0.627 Chlorobenzene 0.549 Nitrobenzene 0.546 CS z 0.707 Anisole 0.510
0.814 0.771 0.556 0.947 0.867 0.857 0.897 0.898 0.730 0.729 0.953 0.631
- 0.055 - 0.194 - 0.047 - 0.466 - 0.059 - 0.086 - 0.083 - 0.064 -0.029 - 0.034 - 0.000 -0.004
H y d r o g e n - b o n d e d s o l v e n t s ( r s = 1) Ethanol Propanol Butanol
0.233 0.157 0.133
2.240 1.849 1.485
- 0.537 - 0.915 - 1.046
-3.843 -3.301 -2.771
Hydrogen bonded solvents(rs= Water
0.659-10 -6
9.015
- 0.001
- 32.486
1.982 1.814 1.613
2) 10.540
P. Ruelle et aL / Pharmaceutica Acta Helvetiae 68 (1993) 49-60 0.8
success from the knowledge of its solubility in hexane and the results are analyzed according to the nature of the solvents.
0.7-
aromatic hydrocarbons and
t-"
.9
o.6-
polar non-associated solvents
#
0.5-
Acknowledgements
0.4~
The authors thank the Ciba-Geigy-Jubil~iums-Stiftung for financial support and are indebted to Professor P.L Huyskens (University of Leuven, Belgium) for his advices.
0.30.2-
~E~ 0
0.1-
aliphatic hydrocarbons alcohols •
~
mmm
References
0.0--0.1
59
....
I .... -3
I''''1''' -2 -1
'1 . . . . 0
I .... 1
2
B+D Fig. 3. Solubility (in volume fraction) of testosterone propionate in various solvents versus the sum of the contributions B and D of the solubility equation.
be neglected without further loss of accuracy in the prediction over against the huge hydrophobic effect.
3. Conclusion
The solubility equation derived from the thermodynamics of the mobile order is used to predict the solubility of proton-acceptor substances, i.e., five solid ketones and two esters, in a wide range of organic nonelectrolyte solvents including alcohols. With respect to apolar hydrocarbons previously studied, the present solutes are able to interact in solution with the alcohols by forming specific preferential contacts. The effect of these preferential contacts on the solubility is accounted for by the O term of the solubility equation, and its treatment is mainly based on stability constants, K o . Obtained from experimental solubilities, two different mean standard stability constants, i.e., 170 and 110 cm 3 mol-1, have been used to characterize respectively the ketone-alcohol and ester-alcohol hydrogen bonds. The addition of the O term in the predictions has a net positive influence on the solubility by increasing its value without modifying significantly the values of the other contributions involved in the solution process. Using the stability constants previously determined, the solubility of testosterone propionate in 28 solvents including alcohols and water has been predicted with
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