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`Solvent Systems and Their
`Selection in Pharmaceutics
`and Biopharmaceutics
`
`Patrick Augustijns
`Catholic University of Leuven, Belgium
`
`Marcus E. Brewster
`Janssen Pharmaceutica N.V., Beerse, Belgium
`
`MYLAN EXHIBIT 1026
`
`iii
`
`

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`
`Patrick Augustijns
`Laboratory for Pharmacotechnology
`and Biopharmacy
`Catholic University of Leuven
`Belgium
`
`Marcus E. Brewster
`Janssen Pharmaceutica N.V.
`Beerse, Belgium
`
`Library of Congress Control Number: 2007924356
`
`ISBN-13: 978-0-387-69149-7
`
`e-ISBN-13: 978-0-387-69154-1
`
`Printed on acid-free paper.
`C(cid:2) 2007 American Association of Pharmaceutical Scientists.
`All rights reserved. This work may not be translated or copied in whole or in part without the written
`permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
`NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
`connection with any form of information storage and retrieval, electronic adaptation, computer
`software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
`The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
`not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject
`to proprietary rights.
`
`While the advice and information in this book are believed to be true and accuate at the date of going
`to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any
`errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect
`to the material contained herein.
`
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`springer.com
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`iv
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`1
`
`Principles of Solubility
`
`YUCHUAN GONG AND DAVID J.W. GRANT
`Department of Pharmaceutics, College of Pharmacy,
`University of Minnesota, Minneapolis, MN
`
`HARRY G. BRITTAIN
`Center for Pharmaceutical Physics, Milford, NJ
`
`Introduction
`Solubility is defined as the maximum quantity of a substance that can be com-
`pletely dissolved in a given amount of solvent, and represents a fundamental
`concept in fields of research such as chemistry, physics, food science, pharma-
`ceutical, and biological sciences. The solubility of a substance becomes especially
`important in the pharmaceutical field because it often represents a major factor
`that controls the bioavailability of a drug substance. Moreover, solubility and
`solubility-related properties can also provide important information regarding
`the structure of drug substances, and in their range of possible intermolecular
`interactions. For these reasons, a comprehensive knowledge of solubility phe-
`nomena permits pharmaceutical scientists to develop an optimal understanding
`of a drug substance, to determine the ultimate form of the drug substance, and
`to yield information essential to the development and processing of its dosage
`forms.
`the solubility phenomenon will be developed using
`In this chapter,
`fundamental theories. The basic thermodynamics of solubility reveals the re-
`lation between solubility, and the nature of the solute and the solvent, which
`facilitates an estimation of solubility using a limited amount of information.
`Solubility-related issues, such as the solubility of polymorphs, hydrates, solvates,
`and amorphous materials, are included in this chapter. In addition, dissolution
`rate phenomena will also be discussed, as these relate to the kinetics of solubility.
`A discussion of empirical methods for the measurement of solubility is outside
`the scope of this chapter, but is reviewed elsewhere (Grant and Higuchi, 1990;
`Grant and Brittain, 1995).
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`Chapter 1: Principles of Solubility
`
`Units for the Expression of Solubility
`A discussion of the thermodynamics and kinetics of solubility first requires a
`discussion of the method by which solubility is reported. The solubility of a sub-
`stance may be defined in many different types of units, each of which represents
`an expression of the quantity of solute dissolved in a solution at a given temper-
`ature. Solutions are said to be saturated if the solvent has dissolved the maximal
`amount of solute permissible at a particular temperature, and clearly an un-
`saturated solution is one for which the concentration is less than the saturated
`concentration. Under certain conditions, metastable solutions that are supersatu-
`rated can be prepared, where the concentration exceeds that of a saturated solu-
`tion. The most commonly encountered units in pharmaceutical applications are
`molarity, normality, molality, mole fraction, and weight or volume percentages.
`The molarity (abbreviated by the symbol M) of a solution is defined as the
`number of moles of solute dissolved per liter of solution (often written as mol/L
`or mol/dm3), where the number of moles equals the number of grams divided
`by its molecular weight. A fixed volume of solutions having the same molarity
`will contain the same number of moles of solute molecules. The use of molarity
`bypasses issues associated with the molecular weight and size of the solute, and
`facilitates the comparison of different solutions. However, one must exercise
`caution when using molarity to describe the concentrations of ionic substances
`in solution, because the stoichiometry of the solute may cause the solution to
`contain more moles of ions relative to the number of moles of dissolved solute.
`For example, a 1.0 M solution of sodium sulfate (Na2SO4) would be 1.0 M in
`sulfate ions and 2.0 M in sodium ions.
`The normality (abbreviated by the symbol N) of a solution is defined as the
`number of equivalents of solute dissolved per liter of solution, and can be written
`as eq/L or eq/dm3. Normality has the advantage of describing the solubility of
`the ionic compounds since it takes into account the number of moles of each
`ion in the solution liberated upon dissolution of a given number of moles of
`solute. The number of equivalents will equal the number of grams divided by
`the equivalent weight. For ionic substances, the equivalent weight equals the
`molecular weight divided by the number of ions in the compound. Equivalent
`weight of an ion is the ratio of its molecular (atomic) weight and its charge.
`Therefore, a molar solution of Na2SO4 is 2 N with respect to both the sodium
`and the sulfate ion. Since the volume of solution is temperature dependent,
`molarity and normality can not be used when the properties of solution, such as
`solubility, is to be studied over a wide range of temperature.
`Molality is expressed as the number of moles of solute dissolved per kilogram
`of solvent, and is therefore independent of temperature since all of the quanti-
`ties are expressed on a temperature-independent weight basis. The molality of a
`solution is useful in describing solubility-related phenomena at various temper-
`atures, and as the concentration unit of colligative property studies. When the
`density of the solvent equals unity, or in the case of dilute aqueous solutions, the
`molarity and the molality of the solution would be equivalent.
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`Chapter 1: Principles of Solubility
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`3
`
`Expressing solution concentrations in terms of the mole fraction provides the
`ratio of the number of moles of the component of interest to the total number
`of moles of solute and solvent in the solution. In a solution consisting of a single
`solute and a single solvent, the mole fraction of solvent, XA, and solute, XB, is
`expressed as:
`
`(1)
`
`(2)
`
`(3)
`
`XA = nA
`nA + nB
`XB = nB
`nA + nB
`where nA and nB are the number of moles of solvent and solute, respectively.
`Obviously the sum of the mole fraction of the two components must equal one:
`XA + XB = 1
`Since mole fractions provide quantitative information of a mixture that can be
`readily translated down to the molecular level, this unit is most commonly used
`in thermodynamic studies of solubility behavior.
`Volume fraction is frequently used to define the composition of mixed solvent
`systems, or to express the solubility of one solvent in another. However, since the
`volumes of solutions exhibit a dependence on temperature, the expression of
`concentrations in terms of volume fraction requires a simultaneous specification
`of the temperature. In addition, since volume defects may occur during the mix-
`ing of the solvents, and since these will alter the final obtained volume, defining
`the solubility of a solution in terms of volume fraction can lead to inaccuracies
`that can be avoided through the use of other concentration parameters.
`The concept of percentage is widely used as a concentration parameter in
`pharmaceutical applications, and is expressed as the quantity of solute dissolved
`in 100 equivalent units of solution. The weight percentage (typically abbreviated as
`% w/w) is defined as the number of grams of solute dissolved in 100 grams of
`solution, while the volume percentage (typically abbreviated as % v/v) is defined as
`the number of milliliters of solute dissolved in 100 mL of solution. A frequently
`encountered unit, the weight-volume percentage (typically abbreviated as % w/v)
`expresses the number of grams of solute dissolved in 100 mL of solution. The
`choice of unit to be used depends strongly on the nature of solute and solvent,
`so the solubility of one liquid in another is most typically expressed in terms
`of the volume percentage. The use of weight or weight-volume percentages is
`certainly more appropriate to describe the concentration or solubility of a solid
`in its solution.
`For very dilute solutions, solubility is often expressed in units of parts per
`million (ppm), which is defined as the quantity of solute dissolved in 1,000,000
`equivalent units of solution. As long as the same unit is used for both solute and
`solvent, the concentration in parts per million is equivalent to the weight, vol-
`ume, or weight-volume percentages multiplied by 10,000. The descriptive terms
`of solubility that is expressed in units of parts of solvent required for each part of
`solute can be found in each edition of the United States Pharmacopeia (Table 1).
`
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`Chapter 1: Principles of Solubility
`
`Descriptive term
`
`Very soluble
`
`Freely soluble
`
`Soluble
`
`Sparingly soluble
`
`Slightly soluble
`
`Parts of solvent required
`for 1 part of solute
`
`Solubility < 1
`
`1 < Solubility < 10
`
`10 < Solubility < 30
`
`30 < Solubility < 100
`
`100 < Solubility < 1,000
`
`Very slightly soluble
`
`1,000 < Solubility < 10,000
`
`Practically insoluble, or Insoluble
`Table 1. Descriptive terms of solubility.
`Reproduced from:
`United States Pharmacopeia, 25th edition. United States Pharmacopeial Convention;
`Rockville, MD; 2002, p. 2363.
`
`Solubility > 10,000
`
`Thermodynamics of Solubility
`The equilibrium solubility of a substance is defined as the concentration of solute in
`its saturated solution, where the saturated solution exists in a state of equilibrium
`with pure solid solute. As solutes and solvents can be gaseous, liquid, or solid,
`there are nine possibilities for solutions, although liquid-gas, liquid-liquid, and
`liquid-solid are of particular interest for pharmaceutical applications. Among
`these, the most frequently encountered solubility behavior involves solid solutes
`dissolved in liquid solvent, so systems of this type will constitute the examples of
`the following discussions.
`For the particular system of a saturated solution, the dissolved solute in the
`solution and the undissolved solute of the solid phase are in a state of dynamic
`equilibrium. Under those conditions, the rate of dissolution must equal the
`rate of precipitation and hence the concentration of the solute in the solution
`remains constant (as long as the same temperature is maintained).
`For two phases in equilibrium, the chemical potential, μi, of the component
`in the two phases must be equal:
`
`μsolute = μsolid
`The chemical potential, also known as the molar free energy, can be represented
`by:
`
`(4)
`
`μ = μ◦ + RT ln a
`(5)
`◦ is the chemical potential of the solute molecule in its reference state,
`where μ
`and a is the activity of the solute in the solution. Since both the dissolved solute
`and the undissolved solid must refer back to the same standard state, it follows
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`Chapter 1: Principles of Solubility
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`5
`
`that the activities of the dissolved solute and that of the undissolved solid must
`be identical.
`The activity of a component in a solution is defined as the product of its
`activity coefficient, γ , and its mole fraction, X:
`a = γX
`For the solute B in a saturated solution:
`asolid = asolute = γB XB
`
`(7)
`
`(6)
`
`or
`
`XB = asolid
`γB
`
`(8)
`
`According to equation (8), the solubility of a substance would be propor-
`tional to the activity of the undissolved solid, and inversely proportional to its
`activity coefficient. Although the activity of a substance in its standard state is
`defined as unity, the activity of the undissolved solid must depend on reference
`state. A hypothetical, supercooled liquid state of solute at the temperature of
`interest is commonly taken as the standard state, making the activity coefficient
`a more complicated term. The activity coefficient will depend on the nature of
`both the solute and solvent, as well as on the temperature of the solution.
`
`Solubility in Ideal Solutions
`
`In order to understand the thermodynamics of solubility, it is appropriate to
`begin with a simplified model of solution, namely that of an ideal solution. An
`ideal solution is defined as one where the activity coefficient of all components
`in the solution equals one. Under these stipulations, the activity of the dissolved
`solute, the activity of the solid, and the molar solubility of the dissolved solute
`would be equal.
`
`(9)
`
`asolute = asolid = XB
`As discussed above, the absolute activity of the solid depends on the chosen
`reference or standard state, and the usual practice is to take the supercooled
`liquid state of the pure solute at the temperature of solution as the standard
`state of unit activity. At temperatures lower than the melting point, the liquid
`state of the solute is less stable than its solid state, making the activity of the
`corresponding solid less than one.
`An ideal solution requires that the scope of solute-solute, solvent-solvent,
`and solute-solvent intermolecular forces be all the same. Thus, the net energy
`change associated with breaking bonds between two solute molecules and two
`solvent molecules, and then forming new bonds between solute and solvent
`molecules must be zero. Moreover, the mixing process is ideal as well, so that the
`total volume of the solute/solvent system does not change during the mixing
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`6 p
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`Chapter 1: Principles of Solubility
`
`rocess.
`
` Umix = 0
` Hmix = 0
` Vmix = 0
`
`(10)
`
`(11)
`
`(12)
`
`where Umix is the energy of mixing, Hmix is the enthalpy of mixing, and Vmix
`is the volume change of mixing. The ideal entropy of mixing, Smix, can be
`derived from pure statistical substitution
` Smix = −R(nA ln XA + nB ln XB)
`
`(13)
`
`where nA and nB are the number of moles of the solvent (A) and the solute (B),
`respectively. Because the mole fractions of the solvent and the solute, XA and XB,
`are less than unity, it follows that Smix is always positive. From this analysis, one
`can conclude that the mixing processes associated with an ideal solution would
`be thermodynamically favored.
`The dissolution of a solid in a solvent can be considered as consisting of two
`steps. The first step would be, in effect, a melting of the solid at the absolute
`temperature (T) of the solution, and the second step would entail mixing of the
`liquidized solute with the solvent. The enthalpy of solution ( Hs) is therefore
`equal to the sum of the enthalpy of fusion ( HT
`f ) and the enthalpy of mixing
`( Hmix). However, since the enthalpy of mixing must equal zero for an ideal
`solution, it follows that the enthalpy of solution must equal the enthalpy of
`fusion of the solid at the given temperature, T:
` Hs = HT
`
`(14)
`
`f
`
`f
`
`f
`
`(15)
`
`For those situations where the temperature of study is not the same as the melt-
`(cid:3)= Hm
`f , where now Hmf
`ing point, then HT
`is the enthalpy of fusion at the
`f
`
`melting point( Tm). If one makes the approximation that the enthalpy of fusion
`is constant over the temperature range in the vicinity of the melting point, then:
` Hs = HT
`≈ Hm
`(cid:3)
`(cid:2)
`
`Applying the Clausius-Clapeyron equation to the solubility calculation yields:
`= Hs
`RT2
`
`P
`
`(16)
`
`∂ ln a
`∂T
`
`Integration of equation (16) provides the relationship known as the van’t Hoff
`equation, which expresses the temperature dependence of the solubility of a
`solid solute (identified as species B) in an ideal solution:
`− 1
`ln XB = ln aB = − Hs
`R
`Tm
`
`(cid:3)
`
`1 T
`
`(cid:2)
`
`(17)
`
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`Chapter 1: Principles of Solubility
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`7
`
`(18)
`
`(19)
`
`(cid:3)
`
`1 T
`
`(cid:2)
`
`f
`
` G m
`f
`
`By combining equations (15) and (17), one finds that the molar solubility of the
`solute in an ideal solution (expressed in natural logarithmic form) is given by:
`ln XB = ln aB = − Hm
`− 1
`R
`Tm
`Since the solid solute and its corresponding molten solid must be in a state of
`equilibrium at the melting point, it follows that:
`= 0
`= Hm
`− Tm Sm
`f
`f
`where the enthalpy of fusion ( Hm
`
`f ) is equal to Tm Smf , where Smf
`is the entropy
`
`of fusion at the melting temperature. Under these circumstances, equation (18)
`(cid:2)
`(cid:3)
`may also be written as:
`
`f
`
`− 1
`
`(20)
`
`ln XB = ln aB = − Sm
`Tm
`T
`R
`The enthalpy and entropy of fusion, and the melting temperature may all be mea-
`sured through the use of differential scanning calorimetry (DSC), and therefore
`equations (18) and (20) provide a simple way to predict the solubility of a solute
`in an ideal solution.
`To achieve a better prediction of the solubility of a solute, one must consider
`the temperature dependence of the enthalpy of fusion, which is described by
`the Kirchoff equation
`
`(cid:2)
`
`(cid:3)
`
`= C p
`
`(21)
`
`(22)
`
`∂ Hf
`∂T
`P
`where Cp is the difference between the heat capacities of the supercooled liquid
`and that of the corresponding solid. Therefore:
`= Hm
`− C p (Tm − T)
` HT
`f
`f
`With the assumption that Cp is independent of temperature, integration of
`equation (16) and the replacement of Hs by HT
`f , yields the Hildebrand equa-
`(cid:2)
`(cid:3)
`tion
`Tm − T
`− C p
`− 1
`ln XB = ln aB = − Hm
`+ C p
`Tm
`T
`R
`T
`R
`Tm
`R
`Equation (23) provides a better prediction of the solubility of a solute in an ideal
`solution.
`Prediction of solubility in an ideal solution can also be performed using the
`entropy approach developed by Hildebrand and Scott (Hildebrand and Scott,
`1962). Assuming that Hs ≈ T Sm
`≈ T C p, they found that:
`ln XB = ln aB = − Sm
`R
`m
`Equation (24) is similar to equation (20), except that ln(XB) is correlated to
`ln(T) instead of 1/T. The solubility prediction using equation (24) was found
`
`ln
`
`(23)
`
`1 T
`
`f
`
`(24)
`
`T T
`
`f
`
`ln
`
`f
`
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`Chapter 1: Principles of Solubility
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`8 t
`
`o have a better tolerance for the non-ideality of the solution than that obtained
`using equation (20).
`Several approaches have been used to predict the entropy of fusion required
`for the prediction of solubility. According to Walden’s rule, the entropy of fusion
`f ) is approximately equal to 13 cal/K·mol for most organic compounds
`( Sm
`(Walden, 1908). Use of this approximation reduces equation (20) to:
`
`ln XB = ln aB = − θm − 25
`
`298.15
`
`(25)
`
`where θm is the melting point of the solute in degrees centigrade.
`Yalkowsky proposed that the entropy of fusion of an organic compound is
`the sum of translational, rotational, and internal entropy changes when it is
`released from the crystal lattice (Yalkowsky, 1979):
` Sf = Strans + Srot + Sint
`
`(26)
`
`while, the translational entropy change consists of the components associated
`with the expansion and change of position as the solid melts.
` Strans = Sexp + Spos
`
`(27)
`
`Yalkowsky also proposed empirical values and limits for these components. Both
`the Walden and Yalkowsky models provide ways by which one can predict the
`entropy of fusion, and therefore predict the solubility of the solute in an ideal
`solution.
`Over a small temperature range, the enthalpy of solution of a solid can be
`assumed to be independent of temperature. The van’t Hoff equation shows that
`ln(XB) increases with temperature, until the solid melts at T = Tm. At this con-
`dition, the solid forms a liquid in the absence of solvent, and since XB = 1, the
`slope of the van’t Hoff plot is equal to ( HS/R). The degree of ideality associ-
`ated with a given solution may therefore be tested by evaluating the degree of
`linear correlation between ln(XB) and 1/T. Figure 1 shows the ideal behavior
`of naphthalene dissolved in benzene and xylene, which is due to the similar
`nature of the molecules involved, and the strength of intermolecular interac-
`tions such as polarity, polarizability, molecular volume, and hydrogen-bonding
`characteristics (Grant and Higuchi, 1990). On the other hand, the molecular
`properties of ethanol are very different from those of naphthalene. Thus one
`finds that for solutions of naphthalene in ethanol, ln(XB) does not exhibit a
`linear dependence on 1/T, which is taken as an indication of the non-ideal
`character of the solution.
`Typically, one finds that the solubility that would be predicted assuming the
`model of an ideal solution is normally much higher than the solubility that is
`actually measured for a non-ideal solution.
`
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`Chapter 1: Principles of Solubility
`
`9
`
`Figure 1. Van’t Hoff plot of the molar solubility of naphthalene in benzene, xylene,
`and ethanol as a function of the reciprocal of the absolute temperature. The solid
`line corresponds to equation (17) for the ideal solubility of solid. Reproduced from
`DJW Grant, and T Higuchi, Solubility Behavior of Organic Compounds, John Wiley &
`Sons, New York, NY, 1990, p. 17.
`
`Solubility in Regular Solutions
`
`One rarely encounters ideal solutions in practice, and practically all solutions of
`pharmaceutical interest are non-ideal in character. For such non-ideal solutions,
`the activity coefficient (γ B) of the solute does not equal one because the range
`of solute-solute, solvent-solvent, and solute-solvent interactions are significant.
`Therefore, one must consider the effect of the activity coefficient in order to
`predict the properties of non-ideal solutions:
`XB = aB
`γB
`ln XB = ln aB − ln γB
`
`(28)
`
`(29)
`
`In equations (28) and (29), aB is the activity of the dissolved solute and the
`undissolved solid, which may be evaluated using the hypothetical supercooled
`liquid as the standard state of unit activity. ln(aB) may be expressed by equation
`
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`10
`
`Chapter 1: Principles of Solubility
`
`1 T
`
`(cid:2)
`
`(17), as was the case for ideal solutions. Therefore:
`ln XB = − Hs
`− 1
`− ln γB
`R
`Tm
`
`(cid:3)
`
`(30)
`
`The value of the activity coefficient depends on many factors, and for non-ideal
`solutions the activity coefficient may be predicted from knowledge of the nature
`of the solute and the solvent.
`For the sake of simplicity, the prediction of activity coefficients in regular so-
`lutions, the simplest non-ideal solution, will be discussed. For a regular solution,
`the energy of mixing and the enthalpy of mixing are not negligible because the
`intermolecular solute-solute, solvent-solvent, and solute-solvent interactions are
`different. However, the total volume is still assumed to be unchanged during
`mixing.
`The activity coefficient in a regular solution can be estimated by considering
`the changes in intermolecular interaction energies that accompany the mixing
`of solute and solvent. For this purpose, the solution process may be divided
`into the three steps illustrated in Figure 2. The first step would consist of the
`removal of a solute molecule from its pure solute phase into the vapor phase,
`the second step would be the creation of a hole in the solvent for incorporation
`of the solute molecule, and the third step is the process where the free solute
`molecule fills the hole created in the solvent (Higuchi, 1949; Hildebrand and
`Scott, 1950; Martin, 1993).
`To begin the analysis, the potential energy of solute-solute, solvent-solvent,
`and solute-solvent pairs is identified as w BB, w AA, and w AB. In the first step, an
`energy equal to 2w BB must be absorbed to break the solute-solute interaction
`between two adjacent solute molecules in the solid. After the solute molecule
`
`Figure 2. Hypothetical steps in solution process.
`
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`Chapter 1: Principles of Solubility
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`11
`
`is removed to the vapor phase, the hole created in the solute closes, which
`releases an energy equal to w BB, making the net energy change associated with
`liberation of a solute molecule equal to w BB. In the second step, energy equal to
`w AA is absorbed to separate a pair of solvent molecules, and to produce a hole in
`the solvent which the solute molecule may occupy. Finally, the solute molecule
`liberated from its solid phase is inserted in the hole in the solvent, forming two
`solute-solvent interactions and releasing an energy equal to 2w AB. The overall
`potential energy change, u, is therefore:
` u = w AA + w BB − 2w AB
`Using this simplified model, Hildebrand and Wood (1933) proposed
`ln γB = (w AA + w BB − 2w AB)
`VB 2
`A
`RT
`where VB is the molar volume of the solute in the supercooled state, A is the
`volume fraction of the solvent in solution, R is the gas constant, and T is the
`absolute temperature of the solution.
`The attractive interactions between pairs of solute and solvent molecules are
`assumed to be derived from van der Waals forces, so the solute-solvent interaction
`energy (w AB) may be represented by the geometric mean of the solute-solute
`(w BB) and the solvent-solvent (w AA) interaction energies:
`
`(31)
`
`(32)
`
`w AB = √
`
`w AA w BB
`
`(33)
`
`Therefore, equation (32) becomes:
`ln γB = ((w AA) 12 − (w BB) 1
`
`2 )2 VB 2
`RT
`The square root of the interaction energy is defined as the solubility parameter,
`δ, and so equation (34) can be rewritten as:
`ln γB = (δA − δB)2 VB 2
`RT
`where δA and δB are the solubility parameters of the solvent and solute, respec-
`tively. In the case of a mixed solvent system, the total solubility parameter of the
`solvent mixture is given by:
`
`A
`
`A
`
`(34)
`
`(35)
`
`δA = φ1δ1 + φ2δ2 + ···
`where δ1 and δ2 refer to the respective solvent parameters of pure solvents 1 and
`2, and φ1 and φ2 are the respective volume fractions in the solvent mixture.
`Introducing equation (35) into equation (30) yields the Hildebrand solubil-
`(cid:3)
`(cid:2)
`ity equation describing regular solution behavior:
`− (δA − δB)2 VB 2
`ln XB = − Hs
`− 1
`R
`Tm
`RT
`According to equation (37), if the difference between δA and δB is very small,
`then the second term approaches zero. The implication of this is that a regular
`
`(36)
`
`A
`
`(37)
`
`1 T
`
`

`

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`
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`
`12
`
`Chapter 1: Principles of Solubility
`
`2
`
`(38)
`
`and/or hydrogen-bonding interactions exist in the solution, w AB (cid:3)= √
`
`solution would behave in an ideal manner when the solute and solvent have
`similar chemical properties. It may be seen that the Hildebrand solubility equa-
`tion enables the prediction of solubility in regular solutions, as long as one has
`knowledge of the solubility parameters of both components in the solution.
`Following the introduction of the Hildebrand model, the topic of solu-
`bility parameters has been extensively discussed (Hildebrand and Scott, 1962;
`Hildebrand et al., 1970; Kumar and Prausnitz, 1975; Barton, 1983), and values
`of δ can be found in these reference works. As a general rule, compounds hav-
`ing stronger London forces will be characterized by larger solubility parameters
`values.
`Hildebrand and Scott (1950) proposed that the solubility parameters of sim-
`ilar molecules could be calculated using the enthalpy of vaporization ( Hv) and
`(cid:2)
`(cid:3) 1
`the molar volume of the liquid component (Vl) at the temperature of interest:
` Hv − RT
`δ =
`Vl
`Predictions of the solubility of non-polar solutes in non-polar solvents have
`been successfully achieved using the Hildebrand solubility equation (Davis et al.,
`1972). These solutions may be classified as regular solutions since the primary
`intermolecular interactions are London dispersion forces. However, the equa-
`tion does not provide a good prediction of solubility for solutions involving
`polar components. When dipole-dipole, dipole-induced-dipole, charge transfer,
`w AAw BB,
`and with the presence of hydrogen bonding the entropy of mixing is no longer
`ideal. In addition, Vmix will not equal zero if the dimensions of the solute and
`solvent molecules are very different.
`Modifications to the Hildebrand solubility parameter model have been ad-
`vanced in attempts to achieve better degrees of solubility prediction (Taft et al.,
`1969; Rohrschneider, 1973). Among these, the three-dimensional solubility pa-
`rameter introduced by Hansen and Beerbower (1971) showed the most practical
`application. These workers calculated the total solubility parameter (δtotal) using
`three partial parameters, δD, δP, and δH:
`= δ2
`δ2
`D
`P
`H
`total
`where the parameters δD, δP, and δH account for dispersion, polar, and hydrogen-
`bonding interactions, respectively. Some of the values deduced for δD, δP, δH,
`and δtotal are listed in Table 2. Another modification of Hildebrand solubility
`parameter considered the effects of polar interaction and hydrogen bonding,
`and was found to yield good solubility predictions in many cases (Kumar and
`Prausnitz, 1975). However, the modified Hildebrand solubility equation can only
`be used empirically in predicting solubility in polar solvents, since the original
`assumptions associated with regular solutions do not apply in polar solvents
`(Grant and Higuchi, 1990).
`In solvent systems where polar interactions exert a major role, the molecular
`and group-surface-area (MGSA) approach provides a better quality solubility
`prediction (Yalkowsky et al., 1972, 1976; Amidon et al., 1974, 1975). Instead of
`
`+ δ2
`
`+ δ2
`
`(39)
`
`

`

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`
`16:46
`
`Solvents
`
`n-Butane
`
`n-Hexane
`
`n-Octane
`
`Diethyl ether
`
`Cyclohexane
`
`n-Butyl acetate
`
`Carbon tetrachloride
`
`Toluene
`
`Ethyl acetate
`
`Benzene
`
`Chloroform
`
`Acetone
`
`Acetaldehyde
`
`Solubility parameter (cal/cm3)1/2
`
`δD
`
`6.9
`
`7.3
`
`7.6
`
`7.1
`
`8.2
`
`7.7
`
`8.7
`
`8.8
`
`7.7
`
`9.0
`
`8.7
`
`7.6
`
`7.2
`
`δP
`
`0
`
`0
`
`0
`
`1.4
`
`0
`
`1.8
`
`0
`
`0.7
`
`2.6
`
`0
`
`1.5
`
`5.1
`
`3.9
`
`δH
`
`0
`
`0
`
`0
`
`2.5
`
`0.1
`
`3.1
`
`0.3
`
`1.0
`
`3.5
`
`1.0
`
`2.8
`
`3.4
`
`5.5
`
`δtotal
`
`6.9
`
`7.3
`
`7.6
`
`7.7
`
`8.2
`
`8.5
`
`8.7
`
`8.9
`
`8.9
`
`9.1
`
`9.3
`
`9.8
`
`9.9
`
`Carbon disulfide
`
`10.0
`
`Dioxane
`
`1-Octanol
`
`Nitrobenzene
`
`1-Butanol
`
`1-Propanol
`
`Dimethylformamide
`
`Ethanol
`
`Dimethyl sulfoxide
`
`Methanol
`
`Propylene glycol
`
`Ethylene glycol
`
`Glycerin
`
`Formamide
`
`9.3
`
`8.3
`
`9.8
`
`7.8
`
`7.8
`
`8.5
`
`7.7
`
`9.0
`
`7.4
`
`8.2
`
`8.3
`
`8.5
`
`8.4
`
`0
`
`0.9
`
`1.6
`
`4.2
`
`2.8
`
`3.3
`
`6.7
`
`4.3
`
`8.0
`
`6.0
`
`4.6
`
`5.4
`
`5.9
`
`12.8
`
`0.3
`
`3.6
`
`5.8
`
`2.0
`
`7.7
`
`8.5
`
`5.5
`
`9.5
`
`5.0
`
`10.9
`
`11.4
`
`12.7
`
`14.3
`
`9.3
`
`10.0
`
`10.0
`
`10.3
`
`10.9
`
`11.3
`
`12.0
`
`12.1
`
`13.0
`
`13.0
`
`14.5
`
`14.8
`
`16.1
`
`17.7
`
`17.9
`
`20.7
`7.8
`7.6
`Water
`Table 2. Solubility parameters for some common solvents.
`Reproduced from:
`Hansen C, and Beerbower A. Solubility Parameters. In: Standen A. Kirk-Othmer
`Encyclopedia of Chemical Technology, 2nd ed. Supplement Volume. New York, NY: Wiley;
`1971. 889–910.
`
`23.4
`
`13
`
`

`

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`
`March 15, 2007
`
`16:46
`
`14
`
`Chapter 1: Principles of Solubility
`
`the potential energy term that was used in equation (32), a free energy model
`was used in the MGSA approach to represent the change of the interactions at
`mixing. The power of this approach is that changes in enthalpy and entropy are
`included:
`
`(40)
`
`ln γB = (WAA + WBB − 2WAB)
`VBφ2
`A
`RT
`In equation (40), W is reversible work which represents the internal free en-
`ergy. Yalkowsky et al. (1976) used the molar surface area (A) and the surface
`tension (σ ) to replace the molar volume (V ) and reversible work. Under those
`circumstances, equation (40) becomes:
`ln γB = σAB AB
`kT
`where σ A and σ B are the surface energies of the pure liquids A and B, while
`σ AB is the interfacial energy between the two liquids. The interfacial tension can
`be experimentally measured for substances of different polarity, and therefore
`equation (41) better predicts solubility in polar solvents.
`
`(41)
`
`Intermolecular Interactions in Non-Ideal Solutions
`
`Pr