THERMODYNAMICS OF
`
`PHARMACEUTICAL
`
`SYSTEMS
`
`An Introduction for
`Students of Pharmacy
`
`Kenneth A. Connors
`
`School of Pharmacy
`University of Wisconsin—Madison
`
`OXILEEIEEQENCE
`
`A JOHN WILEY & SONS, INC., PUBLICATION
`
`000001
`
`Exhibit 1 107
`
`ARGENTUM
`
`Exhibit 1107
`ARGENTUM
`IPR2017-01053
`
`IPR2017-01053
`
`000001
`
`

`

`Copyright © 2002 by John Wiley & Sons, Inc. All rights reserved.
`
`Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
`Published simultaneously in Canada.
`
`No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or
`by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as
`permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior
`written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to
`the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400,
`fax 978-750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission
`should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken,
`NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail: permcoordinator@wiley.com.
`
`For general information on our other products and services please contact our Customer Care Department
`within the US. at 877-762-2974, outside the US. at 317-572-3993 or fax 317-572-4002.
`
`Wiley also publishes its books in a variety of electronic formats. Some content that appears in print,
`however, may not be available in electronic format.
`
`Library of Congress Cataloging-in-Publication Data:
`
`Connors, Kenneth A. (Kenneth Antonio), 1932-
`Thermodynamics of pharmaceutical systems: an introduction
`for students of pharmacy / Kenneth A. Connors.
`p.
`cm.
`Includes bibliographical references and index.
`ISBN 0-471-20241-X (paper : alk. paper)
`I. Title.
`1. Pharmaceutical chemistry.
`2. Thermodymamics.
`[DNLM:
`1. Thermodymamics.
`2. Chemistry, Pharmaceutical.
`QC 311 C752t 2003]
`RS403.C665 2003
`615’.1941c21
`
`2002011151
`
`Printed in the United States of America.
`
`10987654321
`
`000002
`
`000002
`
`

`

`Thermodynamics of Pharmaceutical Systems: An Introduction for Students of Pharmacy.
`Kenneth A. Connors
`Copyright © 2002 John Wiley & Sons, Inc.
`ISBN: 0—471—20241—X
`
`
`This article is protected bycopyrighrand is provided byrhe Universny ofwisconsinr
`Madison under license from John Wiley &Sons. All riym reserved.
`
`
`
`
`10.1. SOLUBILITY AS AN EQUILIBRIUM CONSTANT
`
`The topic of solubility merits special attention because of its great importance in
`pharmaceutical systems. We can generally anticipate that a drug must be in solution
`if it is to exert its effect. Typically the type of system we encounter is a pure solid
`substance (the solute)
`in contact with a pure liquid (the solvent). We allow
`equilibrium to be achieved at fixed temperature and pressure, such that at equili-
`brium the system consists of (excess) pure solid phase and liquid solution of solute
`dissolved in solvent. According to Gibbs’ phase rule, P = 2 and C = 2, so
`F = C — P —|— 2 = 2 degrees of freedom. These are the temperature and pressure,
`which we have specified as fixed. Thus there remain n0 degrees of freedom;
`the system is invariant. This means that at fixed temperature and pressure, the con-
`centration of dissolved solute is fixed. We call this invariant dissolved concentration
`
`the equilibrium solubility of the solute at this pressure and temperature. (We say
`that the solution is saturated.) Our present concern is with how the equilibrium
`solubility depends on the temperature and on the chemical natures of the solute
`and the solvent.
`
`Expressed as a reaction, the dissolution process is
`
`Pure solute : solute in solution
`
`At equilibrium the chemical potentials of the solute in the two phases are equal, or,
`letting component 1 be the solvent and component 2 the solute
`
`1 1 6
`
`I12 (solid) : I12 (soln)
`
`000003
`
`000003
`
`

`

`THE IDEAL SOLUBILITY
`
`117
`
`Writing out the chemical potentials gives
`
`“‘2’ (solid) + RT ln a2 (solid) : u; (soln) + RT In 612 (soln)
`
`(10.1)
`
`where the standard state of the solid is the pure solid, and we will adopt as the
`standard state of the solute in solution the Henry’s law definition on the molar
`concentration scale. Rearranging Eq. (10.1) leads to
`
`Au° : —RT In
`
`a2 (soln)
`
`oz (solid)
`
`(10.2)
`
`where Aug 2 pg (soln) — u; (solid). But the solid is in its standard state,
`(12 (solid) : 1.0 by definition, and we obtain
`
`so
`
`A110 2 —RTlna2 (soln)
`
`(10.3)
`
`is the activity corresponding to
`is invariant—it
`We have seen that a2 (soln)
`the equilibrium solubility—so comparison of Eq. (10.3) with the fundamental
`thermodynamic result
`
`AG" 2 —RTan
`
`(10.4)
`
`leads to the conclusion that a2 (soln), the activity of the solute in a saturated
`solution, must have the character of an equilibrium constant. As a consequence,
`we can evaluate standard free energy, enthalpy, and entropy changes for the solution
`process in the usual manner (Chapter 4). These quantities are respectively called the
`free energy, heat, and entropy of solution.
`For nonelectrolyte solutes, particularly those of limited solubility, so that the
`saturated solution is fairly dilute, it will be acceptable to approximate the activity
`a2 (soln) by the equilibrium solubility concentration. This is usually in molar
`concentration units, and is often symbolized s.
`
`10.2. THE IDEAL SOLUBILITY
`
`A thermodynamic argument can predict the equilibrium solubility of a nonelectro-
`lyte, provided it dissolves to form an ideal solution. Ideal behavior does not mean
`that intermolecular interactions are absent. On the contrary, solids and liquids
`would not exist without the intermolecular forces of interaction. In the present
`context, ideal behavior means that the energy of interaction between two solvent
`molecules is identical to that between one solvent and one solute molecule, so
`that a solvent molecule may be replaced with a solute molecule without altering
`the intermolecular energies. (This requires that the solvent and solute molecules
`have the same size, shape, and chemical nature, a demanding set of limitations.)
`
`000004
`
`000004
`
`

`

`1 18
`
`SOLUBILITY
`
`Quantitatively, an ideal solution can be defined as one having the following proper-
`ties (Chapter 7):
`
`AHmix = 0
`
`AVmix = 0
`
`ASmix = —R(X11I1X1 +X21I1X2)
`
`(10.5)
`
`(10.6)
`
`(10.7)
`
`According to Eqs. (10.5) and (10.6), there is no heat or volume change on mixing
`the solute and solvent in an ideal solution, and the entropy change is given by
`Eq. (10.7). Since an +x2 = 1, the logarithmic terms are necessarily negative, so
`ASmix is positive, and this constitutes the “driving force” for dissolution, because
`of the relationship AG 2 AH — TAS.
`If the entropy of mixing is the driving force for dissolution, what is the “resis-
`tance”? It is the solute—solute interaction forces, which, for solids, lead to the
`“crystal
`lattice energy.” These must be overcome for the solute to dissolve.
`Now, the free-energy change for the dissolution process is the same no matter
`what reversible mechanism (path) is taken to pass from the initial state (pure solute)
`to the final state (saturated solution), so we can divide the process as follows (for a
`solid solute):
`
`Crystalline solute \— supercooled liquid solute
`
`Pure liquid solvent
`
`\— solvent containing cavity
`
`Supercooled liquid solute F saturated solution
`
`+ solvent-containing cavity
`
`Crystalline solute + pure liquid solvent ;‘ saturated solution
`
`Since in an ideal solution the solvent—solvent interactions match the solvent—solute
`
`interactions, the energy required to create molecule-sized cavities in the solvent is
`offset by the energy recovered when the solute molecules are inserted into these
`cavities. The energetic cost of the dissolution process then appears in the first
`step, the melting of the solid. An equivalent viewpoint (Grant and Higuchi 1990,
`p. 16) is that the enthalpy of solution is given by
`
`AI'Isoln : AIifusion + AI'Imix
`
`But AHmix : 0 for an ideal solution, so AHSOIn : AHfusion.
`The saturation solubility, we have seen, is an equilibrium constant, so the van’t
`Hoff equation [Eq. (4.29)] is applicable
`
`
`(11an _ AHf
`dT _ RT2
`
`(10.8)
`
`where the solubility is expressed as the mole fraction simply to maintain con-
`sistency with Eq. (10.7), and where AHf is the heat of fusion and T is the absolute
`
`000005
`
`000005
`
`

`

`THE IDEAL SOLUBILITY
`
`119
`
`temperature. We have seen above why the heat of fusion appears in a solubility
`expression. (Incidentally, a dissolved solid should be viewed as possessing some
`of the properties of the liquid state, consistent with the above view that fusion is
`the first step in the dissolution process.) Now suppose that AHf is independent
`of temperature, which is equivalent to writing, for the solute from Eq. (1.23):
`
`ACp = €1qu — c;°“d = 0
`
`(109)
`
`Then integrating Eq. (10.8) from Tm to T gives
`
`
`
`1an = —% (Tm _ T)
`
`R
`
`Tr,"
`
`(1010)
`
`where Tm is the melting temperature and T is the experimental
`Equation (10.10) allows us to calculate the ideal solubility.
`
`temperature.
`
`Example 10.1. The melting point of naphthalene is 802°C, and its heat of fusion at
`the melting point is 4.54 kcal mol‘l. What is the ideal solubility of naphthalene
`at 20°C?
`
`Logx2 2
`
`—4540calmol‘l
`
`(2.303)(1.987 cal mol’lK’l)
`= —0.577
`
`<
`
`60.2K
`
`353.35 K x 293.15 K
`
`>
`
`x2 = 0.265
`
`Deviations from ideality will be manifested by discrepancies from the ideal solubi-
`lity as calculated with Eq. (10.10). Table 10.1 lists equilibrium solubilities for
`
`Table 10.1. Naphthalene solubility at 20°C
`
`Solvent
`
`(Ideal)
`Chlorobenzene
`Benzene
`Toluene
`Carbon tetrachloride
`Hexane
`Aniline
`Nitrobenzene
`Acetone
`n-Butanol
`Methanol
`Acetic acid
`
`Water (25°C)
`
`x2
`
`0.265
`0.256
`0.241
`0.224
`0.205
`0.090
`0. 130
`0.243
`0. 183
`0.0495
`0.0180
`0.0456
`
`0.0000039
`
`000006
`
`000006
`
`

`

`120
`
`SOLUBILITY
`
`naphthalene in many solvents. Observe that those solvents most chemically like
`naphthalene, that is, aromatic and nonpolar solvents, show behavior most closely
`approximating ideal behavior.
`At the melting temperature Tm the solid and liquid forms of the solute are
`in equilibrium, so AG,« 2 0 and we get AHf = TmASf, giving Eq. (10.11) as an
`alternative form of Eq. (10.10):
`
`lIle = —
`
`mm, — T)
`RT
`
`1.11
`(0 )
`
`10.3. TEMPERATURE DEPENDENCE OF THE SOLUBILITY
`
`Since AHf is always a positive quantity, Eq. (10.10) predicts that the solubility of
`a solid will increase with temperature. Moreover, Eq. (10.10) shows that if two solid
`substances have the same heat of fusion, the one with the higher melting point will
`have the lower solubility. Conversely, if they have the same melting point, the one
`with the lower heat of fusion will have the higher solubility. All of these inferences
`from Eq. (10.10) refer to systems forming ideal solutions, so deviations from the
`predictions can occur for real systems. Nevertheless,
`the increase of solubility
`with temperature is very widely observed for solids. Even the relationship of solu-
`bility to melting point can be a useful guide, though confounding phenomena can
`introduce complications; for example, hydrogen-bonding or other polar interactions
`may raise both the melting point and the aqueous solubility. The comparison of the
`temperature dependence of solubility of solids and gases is instructive; see
`Table 10.2.
`
`Equation (10.10) can be rearranged to Eq. (10.12):
`
`lnxz = —
`
`AHf AHf
`RT +RTm
`
`10.12
`
`)
`
`(
`
`Table 10.2. The contrary effects of temperature on the solubilities of solids and gases
`
`01 : 1111 w:
`ShdAH‘l.
`.dAHv
`Amg
`q
`
`as
`
`Solids
`
`Gases
`
`Solution is the process of
`passing from solid to liquid
`(fusion, AHf)
`
`Solution is the process of
`passing from gas to
`liquid (condensation, AHC), which
`is the reverse of vaporization (AHU)
`
`AHf is positive, so x2 increases
`as Tincreases
`
`AH“ is positive, so AHc is negative; thus x2
`decreases as T increases
`
`000007
`
`000007
`
`

`

`TEMPERATURE DEPENDENCE OF THE SOLUBILITY
`
`121
`
`Inx2
`
`1/T
`
`Figure 10.1. Hypothetical solubility van’t Hoff plots for polymorphs.
`
`If AHf is essentially constant over the experimental temperature range, Eq. (10.12)
`predicts that a plot of In x2 against
`1 / T will be linear with a slope equal to
`—AHf/R. The line should terminate at the melting point, where 1 / T = 1/ Tm. Often
`such lines are straight, probably because the usual range of temperatures is small.
`The slope gives AHf in principle, but in actuality the quantity evaluated from the
`slope is not precisely AHf because the solution is seldom ideal, and instead the
`quantity found in this way is termed the heat of solution.
`Throughout this discussion we have been assuming that the solid phase consists
`of the pure solid and not a solid solution. Another possible complication arises if
`the solid substance can exist in two crystalline forms (polymorphs; Chapter 6),
`which interconvert at transition temperature T[. The van’t Hoff plot can resemble
`Fig. 10.1a or Fig. 10.1b depending primarily on the kinetics of the transformation.
`In Fig. 10.1a, the two forms are sufficiently stable that their solubilities can be
`separately measured at the same temperatures, which are below the transition
`temperature. Nevertheless, the crystal form having the higher solubility (at a given
`temperature) is thermodynamically unstable (it is said to be metastable, since its
`kinetics of transformation permit it to exist for some period during which it acts
`as if it were stable), and will ultimately be converted to the stable form. Extrapola-
`tion of the lines to the transition temperature may be possible. Sulfathiazole in 95%
`ethanol shows the Fig. 10.1a behavior (Milosovich 1964; Carstensen 1977, p. 7).
`In Fig. 10.1b, one form exists in one temperature range, the other form in a
`temperature range on the other side of TI. The melting point observed will be that
`of the higher-melting polymorph. Carbon tetrabromide exemplifies this behavior
`(Hildebrand et a1. 1970, p. 23).
`
`000008
`
`000008
`
`

`

`122
`
`SOLUBILITY
`
`Let us return to the assumption that the change in heat capacities, AC , is zero,
`for all the subsequent discussion was based on this assumption. If AHf in fact is a
`function of temperature, then ACI, is not zero. Suppose we make the more reason-
`able assumption that ACI, is a nonzero constant, and write AHf as
`
`AHf 2 AH?“ — AC,,(Tm — T)
`
`(1013)
`
`where AH;n is the heat of fusion at Tm. Equation (10.13) is inserted into Eq. (10.8),
`which can be rearranged and integrated to give Eq. (10.14):
`
`
`. —AHm Tm — T
`AC Tm — T
`111x; =
`f
`"
`+
`R
`TTm
`R
`T
`
`
`Tm
`AC
`”111—
`R
`T
`
`—
`
`(10.14)
`
`This equation is useful for assessing the error that may be introduced by making the
`
`simple assumption ACI, = 0. Suppose, for example, that the experimental tempera-
`ture is 25°C and the melting point is 100°C. Then the last two terms in Eq. (10.14)
`become equal
`to 0.25ACp/R — 0.22 ACp/R : 0.03ACp/R. Thus considerable
`compensation can take place, making the approximation ACI, : 0 more acceptable
`than it might have seemed.
`
`Example 10.2. These are solubility data for nitrofurantoin in water (Chen et al.
`1976). Analyze the data to obtain the heat of solution.
`
`I (0C)
`
`24
`30
`37
`45
`
`105;;2
`
`6.01
`8.57
`13.16
`18.99
`
`The data are manipulated as required to make the van’t Hoff plot according to
`Eq. (10.12):
`
`T (103K)
`
`3.37
`3.30
`3.23
`3.14
`
`log x2
`
`—5.22
`—5.06
`—4.88
`—4.72
`
`The plot is shown in Fig. 10.2. It is possible that the points describe a curve, but this
`is uncertain with the data as given, for conceivably the scatter is a consequence of
`experimental random error. A straight line has therefore been drawn. Its slope is
`
`000009
`
`000009
`
`

`

`SOLUBILITY OF SLIGHTLY SOLUBLE SALTS
`
`1 23
`
`logX2
`
`3.1
`
`3.2
`
`3.3
`
`3.4
`
`Figure 10.2. van’t Hoff plot for nitrofurantoin solubility.
`
`103/ T
`
`—2300 K, so we calculate
`
`AHsoln = (2300 K)(1.987 cal mol’l K”)
`= 4570 cal mol‘l
`
`= 4.57 kcalmol‘l
`
`= 19.1 kJ mol’1
`
`Note that the enthalpy change is labeled AHsoln to indicate explicitly that this is a
`heat of solution.
`
`10.4. SOLUBILITY OF SLIGHTLY SOLUBLE SALTS
`
`Many salts exhibit very low solubilities in water. Silver chloride is an example;
`if aqueous solutions of silver nitrate and sodium chloride are mixed, solid silver
`chloride precipitates. It is conventional to describe this process as the reverse of
`the precipitation reaction, namely, as the dissolution of the salt. Let us begin
`with the simplest case of a 1 2 1 sparingly soluble salt MX. The solid crystalline
`form is ionic. When it dissolves in water the ions dissociate, and no ion pairs are
`detectable. We therefore write the equilibrium as
`
`MX(s) : M+ + x—
`
`(10.15)
`
`Proceeding as we have done for several earlier processes, we equate the chemical
`potentials of the solid and the dissolved solute at equilibrium:
`
`u<s> = u<soln>
`
`000010
`
`000010
`
`

`

`124
`
`SOLUBILITY
`
`Expanding these gives
`
`u°(s) + RTlna(s) = u: +RTlna+
`+ u: +RTlna_
`
`and collecting terms (and noting that a(s) = 1 by our standard state definition),
`
`A}? 2 —RT lna+a,
`
`(10.16)
`
`the
`where Au" : 111+ 11‘: — u°(s). Evidently then [compare with Eq. (10.4)],
`
`product a+a_ is an equilibrium constant. By Eq. (8.23a) we see that a+ a_ : c122,
`
`
`
`
`where a: is the mean ionic activity, and since 61?, = 721:1, Eq. (10.16) can b
`written
`
`
`
`Aw = —RT1nyEc?
`
`(10.17)
`
`If no extraneous ions are present, so that the ionic strength is due solely to the ions
`from the sparingly soluble salt (and hence is very low), the activity coefficient term
`is essentially unity. Moreover, the molar concentrations of the cation M+ and the
`anion X‘ are equal, and each is numerically equal
`to the equilibrium molar
`solubility of the salt, which is commonly denoted s. Thus Eq. (10.17) becomes
`
`Aw = —RTln 52
`
`(10.18)
`
`Equation (10.17) is exact; Eq. (10.18) is usually a reasonable approximation, and
`both implicitly define the equilibrium constant for Eq. (10.15). This constant is
`symbolized Ksp and is called the solubility product. Since solubility products are
`very small numbers, it is common to state them as szp, where szp = —log Ksp.
`Table 10.3 lists some szp values.
`
`Table 10.3. Solubility products for slightly soluble salts“
`
`Salt
`
`BaSO4
`CaCO3
`Ca(OH);
`Ca3(PO4)2
`CuI
`AuCl
`AuCl3
`Fe(OH)2
`Fe(OH)3
`
`pK,1,
`
`9.96
`8.54
`5 .26
`28.7
`1 1.96
`12.7
`24.5
`15.1
`37.4
`
`“In the temperature range 18725°C; water is the solvent.
`
`Salt
`
`PbCO3
`PbS
`MgCO3
`HgZS
`HgS (red)
`HgS (black)
`AgBr
`AgCl
`Agl
`
`pKg1,
`
`13.13
`27.9
`7.46
`47.0
`52.4
`51.8
`12.30
`9.75
`16.08
`
`000011
`
`000011
`
`

`

`SOLUBILITY OF SLIGHTLY SOLUBLE SALTS
`
`1 25
`
`Example 10.3. What is the solubility of silver chloride in water? From Table 10.3,
`szp = 9.75 for AgCl, so K5,, = 1.78 x 10*". From Eq. (10.18), K3,, = s2, so
`5 = K5,, = 1.33 x 10-5 M.
`
`In the general case of the salt whose formula is Mqu the solubility product is
`defined, in accordance with the usual formulation of equilibrium constants:
`
`KSp = 65:46;
`
`(10.19)
`
`The quantity that we label s then depends on the stoichiometry.
`
`Example 10.4. What is the molar solubility of ferrous hydroxide?
`From Table 10.3, szp = 15.1, or Ksp = 7.9 X 10716. The dissolution reaction is
`
`Fe(OH)2 : Fe2+ + 20H—
`
`so Ksp = cFecéH. (The charges on the subscripts are omitted for clarity.) Since each
`molecule of Fe (OH)2 that dissolves yields one Fe2+ ion, we define the solubility as
`the concentration of ferrous ion, or CFe = s. The stoichiometry yields 601.1 2 2 cFe,
`so the result is1
`
`Ksp = s X (25)2 = 453
`
`Therefore s = 5.8 X 10’6 M.
`
`Example 10.5. What is the solubility of silver chloride in 0.02MKCl? Assume
`activity coefficients are unity.
`is defined
`the solubility. The solubility product
`Again we set
`cAg = s,
`Ksp = cAgcCl; however,
`the chloride concentration has been augmented by the
`addition of potassium chloride, so we write cc] = 0.02 + s; that is, the chloride con-
`centration is the sum from two sources, the KCl and the AgCl. We therefore have
`Ksp : s(0.02 —|— s), which is a quadratic equation that can be solved for 5. Before
`doing that, however, it is worth trying the approximation CC] = 0.02, which involves
`neglecting the relatively small contribution from dissolution of the AgCl. Thus
`
`K,,, = 0.02s = 1.78 x 10—10
`s = 8.9 x 10‘9M
`
`First note that the approximation seems well justified. More interestingly, observe
`that the solubility of silver chloride has been reduced from about l x 10’5 M in
`water (Example 10.3) to about l x 10‘8 M in 0.02 M KCl. This is an example of
`the common ion efi‘ect. The solubility of any slightly soluble salt can be reduced by
`adding an excess of one of its constituent ions.
`
`The accuracy of such calculations can be improved by making use of the Debye—
`Hiickel equation to estimate the values of mean ionic activity coefficients.
`
`000012
`
`000012
`
`

`

`126
`
`SOLUBILITY
`
`10.5. SOLUBILITIES OF NONELECTROLYTES: FURTHER ISSUES
`
`Salt Effects. In Example 10.5 we encountered one type of salt effect. There is
`another type of salt effect that is observed when the solubility of a nonelectrolyte
`is studied as a function of ionic strength (or of the concentration of an added
`electrolyte). Compare the nonelectrolyte solubility in the absence and presence
`of added salt. Since the solid solute is present in both cases
`
`u(solid) 2 ”(CS 2 0) = “(09
`
`where cs is the concentration of added salt. Therefore a(cS : 0) : a(cs), or
`
`so'yo = .97
`
`(10.20)
`
`where so and s are the solubilities in the two cases. Thus 7/70 = 50/5; and since
`70 = l is a reasonable assumption, 7 = so/s, and we have a method for measuring
`nonelectrolyte activity coefficients. Moreover, it is found experimentally that the
`quantity log (so/s) often varies linearly with cs, or
`
`Logs—0 = kscs
`S
`
`(10.21)
`
`If so/s > 1, then kS is positive, and the nonelectrolyte is said to be “salted out”; if
`so/s < 1, then kS is negative, and the solute is “salted in.” These are called the
`“salting-out and salting-in effects,” and the constant kS is known as the Setschenow
`constant.
`
`Regular Solution Theory. We have seen that an ideal solution has thermody-
`namic mixing quantities AHmix = 0 and ASmix = —R(x1 In x1 + x2 ln x2). A regular
`solution is defined to be one having an ideal entropy of mixing but a nonideal
`enthalpy of mixing. Recall also that the ideal solubility of a nonelectrolyte (i.e.,
`the solubility when a nonelectrolyte forms an ideal solution) is given by
`
`
`lnxz = _AHf T'“ _ T
`R
`TTm
`
`(1022)
`
`where ACI, is assumed to be zero or negligible. The molecular interpretation of an
`ideal solution is that the energy of interaction of a solute molecule with a solvent
`molecule is identical with the energy of interaction of two solvent molecules.
`in
`The molecular interpretation of regular solution theory is quite different;
`regular solution theory the energy of 1—2 interactions (where l is the solvent, 2
`is the solute) is approximated as the geometric mean of l—l and 2—2 interaction
`energies, or2
`
`U12 = (U11U22)1/2
`
`(10.23)
`
`000013
`
`000013
`
`

`

`SOLUBILITIES OF NONELECTROLYTES: FURTHER ISSUES
`
`127
`
`This approximation results in regular solution theory being applicable mainly to
`fairly nonpolar systems, that is, nonpolar nonelectrolytes dissolved in nonpolar
`solvents. For our present interest, the essential result (Hildebrand and Scott 1964,
`p. 271) of regular solution theory is embodied in Eq. (10.24), which may be
`compared with Eq. (10.22):
`
`lIle = ——
`R
`
`AHf (Tm — T)
`
`TTIn
`
`
`—
`
`V20?
`
`(231 — 52)2
`
`(10.24)
`
`where V2 is the molar volume of solute and (p1 is the volume fraction concentration
`of solvent in the solution. The quantities 51 and 52 are the solubility parameters of the
`solvent and solute. These are physical properties with the following significance.
`
`The term AHvap, the molar heat of vaporization, is the enthalpy required to effect
`the transformation of one mole of liquid to its vapor state. During this process all
`the solvent—solvent interactions (which are responsible for the existence of the
`liquid phase) are overcome. A quantity called the cohesive energy density (CED)
`is defined
`
`AHvap — RT
`
`CED =
`
`(10.25)
`
`where V is the molar volume of the liquid. We anticipate, and we find, that liquids
`with strong intermolecular interactions (especially polar “associated” liquids
`having the potential for strong dipole—dipole and hydrogen-bonding interactions)
`have larger ced values than do nonpolar liquids. Table 10.4 lists some CED values.
`Because of the manner in which CED appears in regular solution theory equa-
`tions, Hildebrand (Hildebrand et a1. 1970; Hildebrand and Scott 1964, p. 271)
`defined the solubility parameter 5 by Eq. (10.26). Table 10.4 also gives 5 values.
`
`5 = (CED)‘/2
`
`(1026)
`
`Referring now to Eq. (10.24), note that if 81 = 52, we recover Eq. (10.22) for the
`ideal solution; in other words, the condition 51 = 52 is equivalent to the condition
`AHmix = 0. The greater the difference 51 — 52 (or of 52 — 51, because the differ-
`ence is squared), the greater the deviation from ideality, and, as Eq. (10.24) shows,
`the lower the solubility that is predicted. This provides a guide for experimental
`design; to achieve maximal solubility according to regular solution theory, strive
`to equate the solubility parameters of solvent and solute. Since the solute identity
`is usually established by the nature of the problem, the experimental variable is the
`solvent identity. Sometimes mixed solvent systems function better than do pure sol-
`vents for this reason. For example, a mixture of ether (5 = 7.4) and ethanol
`(5 = 12.7) dissolves nitrocellulose (5 = 11.2), although neither pure liquid serves
`as a good solvent for this solute.3
`Although the cohesive energy density, and therefore the solubility parameter, is a
`well-defined physical property for any solvent, regular solution theory is limited
`
`000014
`
`000014
`
`

`

`128
`
`SOLUBILITY
`
`Table 10.4. Cohesive energy densities and solubility parameters
`
`Solvent
`n—Pentane
`
`Cyclohexane
`1,4-Dioxane
`Benzene
`
`Diethyl ether
`Ethyl acetate
`Acetic acid
`
`n—Butyl alcohol
`n Propyl alcohol
`Acetone
`Ethanol
`Methanol
`Acetonitrile
`
`Dimethylformamide
`Ethylene glycol
`Glycerol
`Dimethylsulfoxide
`Water
`
`CED (cal cm’3)
`50.2
`
`«3(ca11/2 cm‘3/2)
`7.0
`
`67.2
`96
`84.6
`
`59.9
`83.0
`102
`
`130.0
`141.6
`95
`168
`212
`141.6
`
`146.4
`212
`272
`144
`547.6
`
`8.2
`10.0
`9.2
`
`7.4
`9.1
`10.1
`
`11.4
`11.9
`9.9
`12.7
`14.5
`1 1.9
`
`12. 1
`14.6
`16.5
`12.0
`23.4
`
`(e.g., by the geometric mean approximation) to solutions of nonpolar substances. It
`should therefore not be expected to apply quantitatively to polar systems such as
`aqueous solutions.
`
`Example 10.6. Predict the solubility of naphthalene in n-hexane at 20°C. The solu-
`bility parameters are 51 = 7.3 and 52 = 9.9 (both in call/2 cm_3/2), and the molar
`volumes are V1 = 132 cm3 mol’1 and V2 = 123 cm3 mol’l. See Example 10.1 for
`additional data.
`
`We use Eq. (10.24), which in Example 10.1 was expressed in terms of log x2. In
`that form the first term on the right had the value —0.577, which we need not recal-
`culate. Now we consider the second term. We lack only the quantity (p1, the volume
`fraction of solvent. This appears to be a dilemma, because we cannot estimate (p1
`until we know x2, which is what we want to calculate.
`If we anticipate that the solute has a low solubility, it may be acceptable to make
`the approximation (p1 = 1. An alternative is to take the result for an ideal solution
`(Example 10.1, which gave x2 = 0.265) as a basis for estimating (p1. We will do the
`problem in both ways.
`(a) Let (p1 = 1. Then from eq. (10.24),
`
`Log x2 = —0.577 —
`
`(123 cm3)(1)2(7.3 — 9.9 call/2 mr3/2)2
`(2.303)(1.987 cal mol’l K-1)(293.15 K)
`= —0.577 — 0.620 = —1.197
`
`x2 = 0.064
`
`000015
`
`000015
`
`

`

`SOLUBILITIES OF NONELECTROLYTES: FURTHER ISSUES
`
`129
`
`(b) The volume fraction is defined as follows:
`
`(P1
`
`MW
`2—
`n1V1 + 11sz
`
`10.27
`
`)
`
`(
`
`Suppose n1 +n2 = 1; from Example 10.1, x2 = 0.265, or n2 2 0.265 and n1 2
`0.735. Using these numbers in Eq. (10.27) gives (p1 = 0.748. (Note how close
`(p1 is to am, because V1 and V2 are similar.) Repeating the calculation gives
`
`Log x2 = —0.577 — 0.335 = —0.912
`
`x2 = 0.122
`
`We therefore predict that x2 is between 0.064 and 0.122, and we might take the
`average as our best estimate. The experimental result (Table 10.1) is x2 = 0.090.
`
`Prediction Of Aqueous SOIubiIities. Water is the preferred solvent for liquid
`dosage forms because of its biological compatibility, but unfortunately many drugs
`are poorly soluble in water. To be able to predict the aqueous solubility of com-
`pounds, even if only approximately, is a valuable capability because it can guide
`or reduce experimental effort. Water is a highly polar and structured medium in
`which nonideal behavior is commonly observed, so we must abandon hope that
`the ideal solubility prediction of Eq. (10.10) will be useful, and even the regular
`solution theory [Eq.
`(10.24)] is ineffectual
`in solving this problem. Effective
`approaches may be guided by thermodynamic concepts, but they incorporate
`much empirical (i.e., experimental) content.
`Although the ideal solubility equation will not suffice to predict nonelectrolyte
`solubility in water,
`the solute—solute interactions responsible for maintaining
`the crystal lattice must nevertheless be overcome, so Eq. (10.10) will still be applic-
`able as a means of estimating the solute—solute interaction. What must be done in
`addition is to take account of the solvent—solvent and solvent—solute interactions,
`for these will in general not offset each other. In a paper that includes a valuable
`collection of solubility data, Yalkowsky and Valvani (1980) have developed a very
`useful method based on this approach. They start with Eq. (10.10), which they
`transform to Eq. (10.11), repeated here:
`
`lIlX2=—
`
`Asfm, — T)
`RT
`
`1.2
`(0 8)
`
`They then carry out an analysis of experimental entropies of fusion, reaching these
`conclusions:
`
`For spherical (or nearly so) molecules: ASf = 3.5 cal mol’1 K—1
`
`For rigid molecules: ASf = 13.5 cal mol’1 K’1
`
`000016
`
`000016
`
`

`

`130
`
`SOLUBILITY
`
`having n > 5
`For molecules
`(n — 5) calmol’l K—1
`
`flexible
`
`chain
`
`atoms: ASf = 13.5 + 2.5
`
`In the following we will use only the result for rigid molecules.
`Yalkowsky and Valvani then take the log P value of the solute (where P is the
`1-octanol/water partition coefficient) as an empirical measure of the solution phase
`nonidealities. They combine this with Eq. (10.28), convert to molar concentration,
`and apply a small statistical adjustment, finally getting Eq. (10.29) for the calcula-
`tion of rigid nonelectrolyte molar solubility in water at 25°C:
`
`Log (:2 = —o.o11(:m — 25) — log P + 0.54
`
`(10.29)
`
`where tm is the solute melting point in centigrade degrees. For liquid nonelectro-
`lytes tIn is set to 25, so the first term vanishes. Log P may be available from
`experimental studies, but it may have to be estimated by methods cited in Chapter 7.
`Yalkowsky and Valvani applied Eq. (10.29) to solubility data on 167 compounds
`whose solubilities ranged over nine orders of magnitude, finding that the estimated
`solubilities agreed with the observed solubilities to within 0.5 log unit for all but
`eight compounds, and in no case was the error greater than a factor of 10. Equation
`(10.29) is a very practical solution to the problem of predicting aqueous solubilities.
`Amidon and Williams (1982) refined the approach of Yalkowsky and Valvani,
`achieving better accuracy but at the cost of increased complexity in the equation.
`Grant and Higuchi (1990) describe alternative methods of calculation that are based
`on different pathways from the initial to the final state.
`Equation (10.10) and equations derived from it, such as Eqs. (10.28) and (10.29),
`contain the difference (Tm — T), showing that a higher melting temperature is
`reflecting stronger solute—solute interactions in the solid state. As a general but
`not precise rule, we may anticipate that very polar molecules (or functional
`groups) will conduce to strong intermolecular interactions by means of electrostatic
`forces, which for certain groups may include hydrogen bonding. Thus high
`molecular polarity tends to be associated with high melting temperature, and higher
`melting temperatures lead to lower solubilities, at least as they are described by
`Eq. (10.10).
`Now consider the special case of water as a solvent. Water is a very polar solvent
`and is capable of functioning as a hydrogen bond donor and acceptor. Very polar
`solute molecules will tend to interact strongly with the solvent water; these are the
`solvent—solute or solvation interactions that increase solubility. But we have seen
`that highly polar substances tend to have high melting temperatures, so we are
`led to the tentative conclusion that melting temperature may be an approximate
`indicator of the extent of solvent—solute interaction. It follows (still arguing in
`this approximate mode) that the opposing factors of solute—solute (crystal lattice)
`and solvent—solute (solvation)
`interactions are both measured by, or a