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- 1. ric AAI, Inc. ~ i l m i n ~ ~~ o r t h ~ r o l i n ~ on, C Larry L. Augsburger David E. Nichols University of ~ a r y l a n d ~ u r d u e University Baltimore, ~ a r y l a n d West Lafayette, Indiana Douwe D. Breimer S t e ~ ~ e nS c h u l m ~ n G. Gorlaeus Laboratories University of Florida Leiden, The l ether lands Gainesvilie, Florida Trevor M. Jones Jerome P. Skelly The Association of the Alexandria, Virginia ~ritish Pharmaceutic~lIndustry London, United Kingdom Hans E. Junginger Felix Theeuwes Leiden/Amsterdam Center Aka Cor~oration for Drug Research Palo Alto, C~lifornia Leiden, The Netherlands Vincent H. L. Lee~niversity of Southern California University of Sheffield Los Angeles, California Royal Hallamshire H o s ~ i t ~ i Sheffield, United K i n ~ d o m Peter G. ~ e ~ ~ i n lnstitut de Recherche Jouvein~l Fresnes, France
- 2. tical Statistics: Practical a R. evised and Expanded, edjted by J o s e ~ h Robjnson and ~ n c e nH. Lee t rd A. Guarin~31 . Transdermal Controlled Systemic ~edications,edjted by Yje W. C ~ j e n32. elivery Devices: Fundamentals and Ap~lications, edjted by ravee en33. Pharmacokinetics; Regulatory lndustria~ erspect~ves, ed~ted by ~ e t e G. Welljn~ Francis L. S. Tse r and ~ l ~ n i cDrug Trials and Tribulations, edited by ~ l l e n Cat0 ai E. . ~ransdermalDrugDelivery: Deveiop~entalIssuesandResearchInitiatives, edjted by Jonathan a d ~ r a f t Ri~hard Guy ~ and H. DosageForms, edjtedby Coatingsfor P ~ a r ~ a c e u t i c a l ~ j l b S. ~ a n k eand Chrjsto~her7: ~hodes e r nufacturing and Pro~uctionTechnolo- W. ions, edjted by ~ a v j d Osborne and ~ n t o n H. A ~ a n n Stability: Principles and Practices, Jens 7. Carstensen istics: Practical and Clinical Applications, Second Edition, ed, Sanford ~ o l t o n radablePolymers as Drug De~ivery Systems, edjtedby ~ a asin in r ~46. ~ r e c l i n i c a ~ Disposition: A LaboratoryHandbook, ~rancjsf. S. 7seand Drug J a ~ e J. Jaffe s LC in the Pharmaceutical Industry, e ~ j t e d ~ o d ~W. n by j ~~n~ and Stanley r m a c e ~ t i ~ a l ~ i o e q u i v a l e n c e , by ~ e t e E. Well;n~, ~ranc;s edjted r f. and S ~ ~ j k a V.t ~ j n ~ h e n49. Pharmaceutica~ Dissolution Testing, ~ ~ e V. ~ a n a k a ~ s h50, Novel Drug ~elivery Systems:SecondEdition,Revisedand Expan~ed, Yie W. Chjen the Clinical Drug ~ e v e l o p m e n t Process, avid M. Cocchetto and ~ardj nufacturing Practices for ~harmaceuticals: A Plan dition, edited by Sjdney H. W;llj~ and J a ~ e s53. Prodrugs: Topical and Ocular Drug Delivery, edjted by ~ e n n54. P h ~ r m a c e u t i c a l i n h a ~ ~Aerosol Technolo~y, tion ed;ted by A~thony ~ ; c ~ e y J.
- 3. 55, Radiopharmaceuticals: ~ h ~ m i s t rand y Pharmacology, ~ ~ n n nd ~hrjster y s t r ~ m ~ Delivery, e ~ j t by~ j c h a eJ. ~athbone e ~ i in Pharmaceutical Development, e~ited fopment Increasing Process: Efficiency and Cost- Effectiveness, ~ d i t e dby Peter G, w1~lijng,Louis Las a, an^ U ~ e s hV. n~kar c r o ~ a ~ i c u l a tSystems for the Delivery of Proteins and Vaccines, e ~ d j ~ e y Srnadar ohe en and ~ o ~ a %ernstejn r d78. Good ~ a n u f a c t u r i n gPr ces for Pharmaceuticals: A Plan Control,FourthEdition,visedand Expanded, ~ j ~ n e y N, R. ~ t o k e r79. ~ q u e o ~Polymeric ~ o a t i n g s for s Pharmaceutical Dos Edition, Revised and Expanded, e ~ i t e d James w1 ~ c ~ j ~ j t y by Statistics:Practicaland ~ l i n ~ c Applicatjons,ThirdEdit~on, al81 . andb book ofPhar~aceutica~Granu~ationTechnoio~y, ~ ~ j t e d by 2. ~ n o l o ~ ~ntibiotics: Second ofy Edition, Revised and Ex~ande~, ~ j ~ e e by ~ j i i R. ~ t r~ h i j ~ o
- 4. rocess E n g ~ n e e r i n ~ ~ t ~ o nJ. n y ~epyrogenation, Second
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- 6. MARCELDEKKER
- 7. This book is printed on acid-free paper. ~~rters L Dekker, Inc. adison Avenue, New York, N Y 10016tel: 212-696-9000; fax: 212-685-4540Marcel Dekker ACHutgasse 4, Postfach 8 12, CH-4001 Basel, Switzerlandtel: 41-51-261-8482; fax: 41-51-261-8896http://www.dekker.comThe publisher offers discounts on this book when ordered in bulk quantities. For morei n f o ~ a t i o nwrite to Special Sales/Professional Marketing at the headquarters address above. ,Neither this book nor any part may be reproduced or transmitted in any form or by anymeans, electronic or mechanical, including photocopying, micro~lming, recording, or by andany infor~ationstorage and retrieval system, without permission in writing from thepublisher.Current printing (last digit):10987654321 CA
- 8. o my wife with gratitude for her~~nderstandi~g, support, and love
- 9. This Page Intentionally Left Blank
- 10. This book is an outgrowth of my notes for a graduate course given at the ~niversityof Wisconsin for several decades. It focuses on the ~ r i ~ c i ~ Z e s science of phar- of themaceutical sciences, not necessarily on details or particular examples, except whenthey are s~pportive material for the text. The solids area of the pharmaceutical sciences has been explored more often inthe last decade than in prior times. This, in particular, is due to the advent ofsophisticated instrumentation and computer access. However, such advantages canlead to a certain mental laziness, and much of what is written in today’s literature isdisregardful, in fact at times ignorant, of the principles on which the instruments andprograms are based, and much misinterpretation occurs. Parts of this book addressthis aspect. In so doing, the ref~rences often not new, but rather give credit to the arescientists of yore who really were the innovators. The book also presents some entirely new aspects, not pr viously published,concerning the proper basic consideration in the approach to certain areas of phar-maceutical solid science. The book is written for those who are interested in the actual pro~esses theonmicroscopic level, with particular emphasis on elucidating models for systems so thatthey can be of general use. The book should appeal to pharmaceutical scientists inindustry, as well as the more sophisticated segment ofpharmaceutical manufacturingpersonnel. It should appeal to scientists in government agenciproblem areas that might have bearing on, for example, New (NDAs). It should have appeal to attorneys in patent law as well as patent exam-iners, because it elucidates whether a given. type of solution to a problem is reallypatentable. Also, it should be appealing to graduate students and to advanced under- graduate students who desire a place in the pharmaceutical solid sciences area.
- 11. This Page Intentionally Left Blank
- 12. Preface v ne ~omponent Systems 1 operties of Solids 13 3. Solubility 27 51 61 6. ~rystallization 89 7. Amorphates 107 117 erms with Crystalline Solids 133 e~o~si~era~ions 159 iagrarns and Eutectics 169 Particles and Surfaces 191 209 14.Solid State Sta~ility 223 olid-State Stability 267 lumes and Densities 28 1
- 13. viii17. Cohesion 299 30919. ~ o m m i n ~ t i o n 323 33521. et ~ r a ~ ~ ~ a t i o n 353 ard Shell Capsules 37523. Tablet Physics 387 rinciples of Tablets 407 sintegration and Dissolution 427 439 27. ~ o a t i n g Tablets of 455 28. Single Unit S ~ s t a i n e ~ Release Dosage 469 29. Sustained Release by ~icroenca~sulation 493 Index 51 1
- 14. 2 namic Functions 2 3 4 1.5. Methods 6 6 1.7. Introduction to Polymorphism 7 attice Energy, for Ionic Compounds 8 Symbols 11 eferences I1The purpose of pharmaceutical research is to explore the causes of properties ofdosage forms, in this case, solid dosage forms, The properties of the dosage foand a host of its qualities are a function of the neat drug. Characteri~ation the ofdosage form, therefore, requires characterization of the drug substance and what itsproperties are, so that the sources of derivative properties in the dosage form canbeadequately assigned. It is granted that such sourcing is never complete. Is the dis-solution rate of a drug in a dosage form, for instance, a function of the dissolutionrate of the drug substance, or is it influenced more by the excipients? Suchquestionscannot be answered a priori, but before an answer is attempted, the dissolution rateof the drug substance must first be known. Hence, this property (and many other)properties of the drug substance must be explored. Tools exist, nowadays, that allow sharp definition of a solid. Such character-ization of solid-state forms encompass microscopy, infrared (IR) spectroscopy, dif-ferential scanning calorimetry (DSC), t h e r ~ o g r a v i ~ e t r ianalysis (TGA), Karl cFischer titration, X-ray powder diffraction analysis, single-crystal X-raydiffraction,
- 15. and (at times) solution calorimetry (Ledwige, 1996). eference will be made to thesemethods in appropriate places in this book. n solid pharmaceutical-marketed products, both the drug substance and exci- are present. (The word “drug,” at times, also refers to the drug product, ction made in the present text now seems to be the accepted h research dealing with pharmaceutical products is directed ke them possible and also addresses the failures that mightor do occur. though many such failures stem from manufacturing and excipients, manyalso stem from the drug substance itself. It is, therefore, of importance to discuss the ropert ties and testing approaches of the neat drug (the “drug,” the “drug sub-stance”) to assess the properties and difficulties asso~iatedwith the final product(the “dosage form” or the “drug product”).There are three states of matter: (a) gases, (b) liquids, and (c) solids. Their definitionsare intuitive, but if defined in words, a gas needs a three-dimensional, closed con-tainer to contain it, a Z i q ~ needs simply an open three-dimensional container, and a i~s ~ Z simply needs a two-~imensiona~ i~ planar support. The definition, however, isnot specific in the terms of solids. As shall be seeninlater chapters, solids are either crystalline or amor~hous, amorphoussolids may and(above their glass temperature, ir’) be rubbery, and below this temperature, they areglassy. In the rubbery state they are to be likened to (or actually are) supercooledmelts or li~uids and, as such, are liquids. In the glassy state, however, a substancewill mimic many of the qualities of a crystalline solid; hence it may be considered ve a viscosity, it will,in this text, be conventional to even if it is amorphous, if its viscosity is higher thanwhat it is at the glass transition temperature. A viscosity at 2’of 10l2“often used (Lu and Zografi, 1997) and this willbe employedhere as the cutoffpoint for a solid. In this book, the followi~g terminology will be used for the four t h e ~ o d y n a ~ i c functions: E is free energy, F is Helmholz free energy, G is Cibbs’ energy, and N is enthalpy, and in differe~tialform they are related as follows, whereiris temperature, S is entropy, V is volume, P is pressure: (1.1)
- 16. The chemical energy terms are not included in the foregoing, but with these, it is inparticular Eq. (1.3) that is affected. dG ==: -SdT + VdP + (13where I;L is chemical potential and n is number of i-species transferred. It is particu-larly noted that dG = 0 during equilibrium, and that, for a voluntary process,dG < 0. G is a convenient function in that d T and d P are zero at constant tempera-ture and pressure, and that, under these conditions, (1*6) lpy change at constant pressure. An outcome of this is that (1*7)Often, in a chemical situation, at constant T? it is possible to indepen~entlydeter-mine AG and AH, and it is then possible to calculate A S fromAnother frequently employed relation isFor instance, for a chemical reaction with equilibrium constant (1.10)Inserting Eq. (1.9) into Eq. (1. 10)then gives d{AG/T}/d~ -~dln[K]/dT = -AH/T2 = (1.11)If ln[K] is known at several temperatures, A H for the reaction may be found, andA S may now be found from Eq. (1.8). ost often, in chemistry, systems are constant-~ressuresystems.h o ~ e v ~situations arise that call for constant-volume considerations a r,case, the chemical equilibrium criterion is that A F , not AG, be zero. It should finally bementioned that the entropy S, of a system isa measure of its oltzmanns law states that: (1.12) mber of ways in which a system can be made up. It is of interest to estimate the number of phases that can be present under one particular energetic condition. Suppose an ensemble in equilibrium consists of rz c o ~ p ~ n e n tand p phases. ecause there is equilibrium between phase 1 and 2, s, between phase 2 and 3, and so on the following holds. Note that Eq. (1.13) constitute p - 1 equations. There are - 1) equations for each of the n compounds, so the total number of equations is n(p - 1). temperature are variables and there are (n - 1) independent concentrations per
- 17. +phase, so that the number of variables is p(n - 1 ) 2. The number of degrees offreedom is the n u m ~ e r variables minus the number of equations, i.e., of d =p(n-1)+2-n(p--l)=n-p+2 f (1.14)This means that there are d variables that may be changed without the system f“losing” a phase. As an example, a beaker of water has one component, there are two phases(liquid and gas); hence, n = 1 and p = 2, so that by Eq. (1.14) there is 1 degree offreedom (i.e., one variable [either T or P may be changed). If the temperature is Iincreased a bit, no phase will be lost. However, it is not possible to change both Tand P at will, because a given T dictates a certain P and vice versa. The situation is different at the freezing point. Here, there are three phases, ice(solid), water (liquid), and vapor. Hence d = 0, and neither T nor P may be cham- fged without losing a phase. Increasing the t~mperature loses the solid phase (the icemelts) and lowerin it loses the liquid phase (the water freezes). Such a point is calleda t r ~ l point. e The use of Eq. (1.14) is often difficult and it is st~essedthat it applies only to anequili~riu~ sit~ation. When in doubt, it is prudent to actually do the derivation leading to Eq. (1.14) for the particular system and obtain [df - 21 as the difference between the number of equations and the number of unknowms. The term degree o ffree do^ in this contextis exactly the oppositef its statistical~ e a n i n ~ s o (where itis the n ~ ~ o ~po~nts i ~ the s f e r~ u number o equations~. f LLA lattice is a periodic array. Points in the (ideal) lattice are occupied by molecules orions, and these may arrange themselves in different fashions (Fig. 1.1). There are seven different crystal systems, as shown in Table 1.1 ositioning of atoms, molecules, or ions in the lattice may be visualized as aseries of layers. Depending on which direction the lattice is viewed,there are different“layers” in different directions. The distance between these layers is denoted d below,and the manner in which d is d e t e ~ i n e d as follows: is To get an idea, first of all, ofthe magnitude of d consider a solid compound ofmolecular weight 180 and a true density 1.5 The molar voll.”lle of such a corn- *pound would be 18011.5 = 120 cm3jmol. cause there are 6 x molecules in amole, each of these occupies 120/(6 x = 200 x cm3. If, for order of mag- b Example of crystal forms. The angles, u, v, and w are shown in the left figures,as are the possible distances, a, b, and e. These are referred to in Table 1.1.
- 18. Angle between Length of side Alternateaxes Examples name distances System a=b=c Regular Cubic NaCI a=b$c Tetragonal Pyramidal Rutil aZb#c 0rthorhombic AgN03 Rhombic a # W c Monoclinic ~-~minobenzoic acidu#v#w#9O0 a#b+ Triclinic K2Cr07u=t1=~#90" a=b=c Trigonal ~hombohedral NaN03u = w = 90" a =b =c # d Hexagonal Graphitew = 120"nitude calculations~the arrangement is assumed to be cubic, the side length of thecube encasing the molecule would be given by d3 = 200 x cm3 (1.15)or d = 5.85 x cm3 = 5.85 A (1.16) Awhere 1 is defined as 10" cm3.X-rays are of this order of magnitude and are usedfor measurement of atomic, molecular, and ionic distances within a lattice. This is performed according to rags law, which relies on the fact, that whentwo X-rays are in-phase, they will then reinforce one another, and the principle onwhich it is carried out is shown in Fig. 1.2.Two X-rays, 1 and 2, strike a surface at an angle of II . Ray 2 traverses a distanceABC (in bold in the figure) longer than ray 1; hence, for them to be in-phase, thisdistance must be a multiple of the wavelength h of the ray. The distance A.shown, by simple trigonometry, to be equal to 2d sin[q Le., for attenuation to bemonitored at the collector Q , this distance must be equal to nh; that is, 2d sin[U ] = nh (1.17) Schematic for Braggs law. The incident angle, U, equaling thereflected angle, U,isusually referred to as 8.
- 19. Ledwidge et al. (1996), for instance, reports an X-ray diffraction pattern. (using -X rays with h = 1.5418tf)of diclofenac ~-(2-hydroxyethyl)pyrrolidin.e andthe smallest 20-value where a peak occurs is 7.6". The d i s ~ ~ (i.e.,e the d-value) ~ccorresponding to this would be d = 1.54/(2 sin 3.8") = 11.6 A (1.18)The most common method is powder X-ray d ~ ~ ~ ~ In t this method, powder is c i u ~ .packed into a cell, and this is subjected to the type of detection shown in Fig. 1.2. lanes present themselves in sufficient abundance to allow determination of thecrystal lattice constants without determination of the position and direction ofatoms, molecules, or ions in the lattice ~ingle-crystalX-ray crystallography allows determination of the position anddirection of the ions, atoms, and molecules in the lattice. For instance, Turel et al.(1997) usedX-ray crystallography to determine the crystal structure of ciprofloxacinhexahydrate, and showed that it exists in zwitterionic form in the solid state. Thecarboxylic proton is present by the piperaz terminal nitrogen. Adjunctly, however, employed they , Raman spectroscopy, and thermalmethods to determine that the water in the hexahydrate was present in complicated anetwork governed by hydrogen bonding.Inorganic ionic compounds consist of fairly spherical entities, and their packing isrelated to the relative radii of the two components of the systems. Consider, for instance, the situation in Fig. 1.3, in which, a c o ~ p o u n d consistsof two ions, one smaller, with radius r, and one larger, with radius R. It is obviousfrom the figure at the right t~iangle, Ahypotenuse, CA = 2fz + 2r. Hence, (2Ry + (2R)2= (211 + 2r)2 (1.19)or (1.20) Schematic for derivation of the radius ratio rule.
- 20. which has the positive root: r = R(1 - V2) = 0.141R (I 21)Similar relations can be obtained for other arrangements (crystal systems), and therules in Table 1.2 emerge.Whereas inorganic compounds often (if not most often) crystallize in one particularcrystal system, organic compounds have the capability of c~stallizing several indifferent ~ Q Z forms ( ~ o r ~ ~ s )this p~enomenon denoted ~ o Z ~ ~ o r ~ ~ ~ ) and , is ecause there are seven crystal systems, it might be tempting to think that therecould, at most, beseven different po~ymorphsof one compound; however, thenumber is not limited to that. The molecules may be in different lattices, because their orientation is differentin thetwo different polymorphs (of the same crystal system). The lattice constants, a,6, and e, then might or would be different. olymorphs will be su et to a special chapter (see Chapter 8) but at this pointthe following will be noted. two polymorphs, one (form I) will be (con~guration-ally) more stable than the other (e.g., form 11) for the following reasons. 1. There areno rules for the nomenclature I, 11, and so on. numbers simply signify the chronological order in which they were pro- duced. 2. The less stable form, at a given temperature, will have a higher vapor pressure. 3. The less stable form, at a given temperature, will have a higher ap~arent solubility. This concentration of drug in the solvent is reproduci~le, but the solution is not thermodynamically stable. Eventually precipitation of a more (the more) stable form will occur, and the concentration will level off at the the~odynamic e~uilibrium solubility. 4. It is not possible, in a practical sense, to talk about the “most stable” polymorph, for a more stable polymorph may be discovered at a later time. From a fictional point of view, this is the subject of the book ~ ~ t Examples of the Radius Ratio Rule ApplicationsRatio = r/R coordinatio~ number Lattice Example 0-0.155 2 Carbon dioxide0.155-0.225 3 Hexagonal Boron nitride0.225-0.414 4 Tetrahedral blende Zinc0.414-4.733 6 Octahedral NaCX0.733-1 8 Body-centered cubic csc121 12 Face-centered cubic and also hexagonal
- 21. r l ~ r ~ ~ l Kurt ~ o n n e g u t . ere a more stable, higher-melting form of by e , water (Ice Nine) eventually causes the world’s oceans to freeze over. 5. The molecules in solutions created by either a less or more stable poly- morph are the same.The section to follow has been developed quite rigidly for inorganic ions.extensions to organic crystals are possible. In the development, the term ~ ~ r t j c Z e willoften be used to signify “ion” or in some cases “molecule,” hen bonding occurs between two molecules, a minimum will occur in thepo~ential energy curve that exists betweenthem. This distance is known as the latticeconstant Ro (Fig. 1.4). nergy curves, as a function of atomic or molecular distance, are rationalized aron and Prutton, 1965) by the existence of two opposing forces between the ms or molecules: an attractive force and a repulsive force. The attractive force is, theoretically, inversely proportionalto the seventhpower of the interatomic or intermolecular distance. The repulsive term dependson distance by some (the nth) power of the separation, The potential energy u‘ ofthe interaction between two neighboring ions, therefore, may be written as -~ ~ d =( ~ 1 (By) ) (I .22) The value o f n is, ordinarily, from 10 to 13. ach interparticular distance (rY)is expressedas a number (pii) multiplied with ration (R)between two particles. Examplewise, in the situation shown in Fig.1.5, thejth and the ith molecule wouldseparated by 2 “units” so that rij would equal In general this may be written as rij = pijR (123) ) isnowsummedover all interaction possibilities, which then gives theenergy, ( , one ~ ~ r t i c ~ e . pfor ”I Potential energy curve.
- 22. <"_"""" "" """"> r10,12 Schematic of interacting atoms or molecules. (1.24)where summationis over all i # j . The following termsare introduced for thesake ofconve~ience: (1.25)and (1.26)For a given crystal a and b are constants, so that Eq. (1.24) becomes: 4 = ( A u / R 7 )- ( ~ b / R n ) (1.27) The distance at equilibrium, Ro, is obtained by obtaining the first ~erivativeand equating it to zero (and at the same time ensuring that thesecond derivative will ~ 4 / = (-7Au/R8) ~ R +( ~ ~ b / R n " ) (12 8 ) occur this must be zero, soFor equilibriu~ to + (-7AaIR~) ( ~ ~ b / R ~ " ) (1.29)or 7AaIR; = n ~ ~ / R ~ ~ ~ (1.30)or (7/n)(Aa)/R: = ~ b / R ~ (1.31)This is now inserted into Eq. ( 1 2 7 ) to give 40 [Aa/u:l - [ ( 7 / ~ ) ( A a ) / RAla / R ; J I { l - ( 7 1 ~ ) ) [: = (1.32) of The energy per mole U can now be obtained by multi~lication 4 0 , ~ i t h theAvogadro number N,so that
- 23. u =N ~ A ~ / - (7/n)}~ l R ~ ~ (1.33) onv vent ion ally? U is equated with the enthalpy of sublimation, because solidsare considered constant-volume (rather than constant-pressure) situations. Theactual value of a is obtained geometrically. For ions, the terms alternate in sign(each secondion being negative,thus exerting attraction, each second beingoppositein sign and giving rise to attraction). For molecules (many organic molecules) theforce terms are all positive.The large negative value ofn, the exponent in the secondterm in Eq. (1.27), is most often acceptable to use only one or two terms m a ~ i n g itthe s~mmation fairly simple. Ro values may be obtained by X-ray analysis, leaving but two unknowns, Aand y1 in Eq. (1.33). The value of n may be obtained by lo~-temperature compres-sibility measurements. The definition for compressibility k, is k -(l/V)dV/dP (1.34) t low temperature, the 7“ term in the definition for U vanishes so that dU =: -PdV, ‘35) (1so that 1/k = Vd2U/dV2 (1.36) olar volume with the nomenclature used here is v = NR3 (1.37)so that dU/dV = ( d U / d ~ ) ( d R / d ~ ) (1.38) he second derivative, hence, is + d2 U / d V 2 = (dU/d~)(d2R/dV2) (d2U/dR2)(dR/d~)2 (1.39) t equilibrium^ dU/dR = 0, so that the first term vanishes, reducing Eq. (1.39) toU/d d2 V 2 = (d2U/dR2)(dR/d V)2 (1 .40) rom Eq. (1.36) we have (dR/dV)2 = (3NR2)2== 9N4R4 (1.41) q. (1.40) becomes d2U/d V2 = (d2U/dR2)(9N4R4) (1.42)This is combined with Eq. (1.36) to give ~)(9N4R~)(d2 U/dR2) = 9N3Ri(d2 / d R 2 ) U (1.43)Equation (1.43) when differentiated twice gives d2U / d R 2 = 56(NAa)[1 - (7/n)](R~) (1.44)which inserted in I l k = 504N4(~a)[1 (7/n)](R~2) - (1.45)which allows calculation o f n.
- 24. A constant in the energy versus distance equation = adelung constant adelung constant B = constant in the energy versus distance equation d = distance between molecular layers E = energy H = enthalpy anns constant; (b) compressibility N = Avogadros number n = (a) number of particles, ions, molecules; (b) integer in (c) exponent in potential energy versus distance equation. P = pressure pij = factor forexpressing the distance between the ith andjth ion in units of = distance between the ith and jth ion = (a) distance between particles, ions, molecules; (b) ionic radius of larger ion r = ionic radius of a smaller ion Ro = ~quilibrium distance between particles, ions, molecules S = entropy 1 = absolute temperature ( " U = crystal energy u = potential energy between two ions Y = volume W = number of ways of building up a system , = chemical potential u 8 = incident angle of an X-ray h = wavelength enettlnetti 6, Giordano F, Fronza 6 , Italia A, Pellegata R, Villa M, Ventura P (1990). J Pharm Sci 79:470.Carstensen JT (198 1). Solid Phar~aceutics,MechanicalProcesses andRate "Phenomena. Academic Press, New York, pp 6-7.Kittel (1962). Introduction to Solid State Physics. pp 70"79. T, Draper SM, Wilcock DJ, Corrlgan 0 1 (1996). J Pharm Sci. 85:16.Lu Q, Zografi G (1997). J Pharm Sci 86: 1374.Maron SM, Prutton CF (1965). ""Principles o f PhysicalChemistry,4th ed. London, pp 728-729. (1997). Int J Pharm 15259.
- 25. This Page Intentionally Left Blank
- 26. 2.1. 13 2.2. 16 2.3. Classic Heat Capacity 18 2.4. The Einstein Equation 19 2.5. 22 2.6. 2.7. References 25The text, as mentioned earlier, will deal first with properties of solids that are not,primarily, a function of their subdivision. In essence they may be considered theproperties of an infinitely large slab of the solid, Later chapters will deal with proper-ties that are a function of the subdivision of the solid (e.g., particle size).Crystals are never perfect. As they grow (a point that will be discussed later) planesmay grow over one another on the surface (Fig. 2.1), shunting out areas of voids.They may also grow as a screw (a so-called screwdislocation), and in this case thereis a sort of pore that penetrates the crystal as the axis of the screw. Several types ofdefects are depicted in Figs. 2.2 and 2.3. From a statistical-mechanical point of view, defects are to be expected. Thedevelopment of this concept in the following is based on the Schottky defect, but itwould also apply to vacancies of other natures. Suppose (Fig. 2.3), that a crystalcontains nine molecules. There is but one wayof arranging them. If one of theinternal molecules is moved to the surface, there will be { lo1}= 10 ways of doing
- 27. A E 3 c D A nucleus or crystal (A) grows on the surface, and two sites are shown. Furthergrowth and a site in a second layer are shown in (C) and in (D) the growth in “higher” sitesgrow over the lower sites creating a “hole.” Schottky Frenkel Vacancy Frenkel, Schottky, and screw defects. 0 000 0 0 000 9 Posit’ions 10 Positions 11 Posltlons One Way 10 Ways 55 ways Situation where one and two Schottky defects are created in a crystal with (origin- ally) nine lattice sites.
- 28. MW 2 x 74.5Cl K Cl K C1 KK 6 K C1 K C1 1C K C1 K C1 K lK C1 K C1 K C1K C1 C1 K C1 KC1 K 6C1 Ca 1K C1 K C1 K C1C1 K C1 K Cl K MW 75.5 + 35.5 Calcium replacing IC. as a means of creating a vacancy.this. Itf two molecules were moved the surface, then the number of ways would be to{112}= 11 x 10/2 = 55 ways. olt~mann states that the entropy of a system S is proportional to the lawlogarithm of the number of ways in which it can be made up: S = kln(ways the system can be made up) (2.1) oltzmann constant. For a system of N + n positions with n vacancies?the entropy would be S = k ln[(N + n)!/{N!n!} (.) 22Use is now made of Sterlings formula lnN! = N l n N - NLe., as applied to this system ln[(N + n)!/{N!n!} N l n N - ( N - n ) l ~ ( N n) - nln(n) = -so that S = k{NlnN - ( N - n)ln(N - n) - nln(n)} (2.5) olid systems are usual1 onsidered constant volume systems, so that in equilibriumconsiderations?it is the lmholtz free energy (rather than the Gibbs energy) that isapplied. IC = nEs - TSwhere Es is the energy associated with one vacancy. This isnow differentiatedrelative to n to give the equilibrium condition: dF/dn = EL* kT ln([N - n]/n)= 0 - (2.3where the argum~nt Eq. (2.5) differentiated relative to n. This rearranges to: is E:s= -kT ln(n/[N - n]) (2.8)or n == NemEsk 1 / "Normal range of vacancies is ofthe order of 0.001%.
- 29. efects are often creating by doping (i.e., introducing a foreign molecule into ce of the compound in question). For instance, with KCl, the potassium ion ) may be replaced with relative ease bycalcium ion (MW 40), becausetheir asizes are appro~imatelyequal. If one considers a crystal with N ions of KCl, theneach time a calcium ion (MW 40) is introduced, a hole with one missing K 39) ~~~is created. The loss in weight,therefore, is 38 per calcium ion. If there are n calciumions in a crystal with N positive ion sites, then the density is calculated as follows(Figs. 2.4 and 2.5) without vacancies, should be (~~ of C1 being 35.5) N“ = 2(N + n)74.5 (2.10)The weight W+ with vacancies would be W+ = 2N74.5 + n75.5 + n35.5 (2.11)The difference between these two numbers is A W = -38n (2.12)The volume of the crystal is N x V , where V is the molecular volume, so that thedifference in density would be(2.13)A ~ / =N-38n/NV ~ttenrauch (1983), H~ttenrauch and einer(1979 a,b), Longuemard et al. ersey and Krycer (1981), Moriata etal. (1984), Grant and York (1986), and nd Grant (1987) have called attention to the fact that processing of solidscauses lattice defects,givingrise to an increase in disorder. ancock and Zografi(1997) claim that this would give the particle a certain viscoelasticity. Hiestand (1997) states that “the ever present, plastic deformation profiles anexplanation why lot-to-lot problems are c o ~ m o n . ” yield value ofthe particles is Thedependent on defects in the crystals, and changes may occur in nearly all processing.Usual production sp~cifications not include criteria for mechanical properties. doThere are several different definitions of ~ e n ~ i tThe ideal density ofa crystal can be y.calculated from knowledge of its lattice parameters and the molecular weight. n/N x 10000 Change in density of KC1 doped with calcium ions. The lower line is the theoreticalline, the upper line the experimental line. (Data from Pick and Weber, 1950.)
- 30. If, for instance, the lattice parameters of a orthorhombic crystal are 5, 7 , and8A, and its molecular weight is 240, then the mass of one molecule is180/(6 x =3 x g. The volume it occupies is 5x 7x 8x= 2.8 x em3, so that the crystallographic density wouldbe 3/28 = 3/1.92 =1.07 g/cm3.Nowadays, crystallographic densities are reported routinely in studies ofthe crystallographic details of a particular form of the compound. As an example,Ceolin (1997) has reported the volume of the triclinic unit cell of carba~azepine tobe 2389 A3. ecause of lattice defects and vacancies, the actual density would be less, Theactual particle density is determined by either wet pycnometry or by helium pycno-metry (Fig. 2.6). In wet pycnometry, a liquid in which the solid is insoluble, is selected (e.g.,water for a poorly water-soluble compound). The pycnometer has a given volume V em3, and the weight of the contents W is determined. The pycnometer is filledto amark giving the density p l , of the solvent: Now M grams of solidare added, having the (unknown) density o f p2. Thesegrams occupy M/p2 em3, so that the liquid now occupies{ V - ( M / p 2 ) } em3. The netweight ( M 2 )of the ensemble is obtained ex~erimentally(Fig. 2.7), and is given byThe only unknown is p2, which is the quantity sought. isadvantages are (a) that the solid may be somewhat soluble in the pycn-ometer liquid, and (b) air entrapment. Theformer is marginal at best if the solvent isselectedwith care. At high dilution, ideal solutions are approached, so that thevolume contraction or expansion considerations are negligible. None of theseproblems exist inthe use of the helium pycnometer, which workson the same principle, except the “liquid” is helium. Therefore, it is not to be expectedthat theparticle density isthe “true” density.This could be derived only by knowing the lattice para~eters, ~ozwiakowski ai. (1996) reported on the solubility behavior of lamivudine etand in this process report on the lattice constants of the compound.for the compound is C8N I 1016S. The following program in BAmolecular weight em~loying precise atomic weights the Ground Glass Stopper With Capillary Inserting Glass Stopper Allows Excess Liquld t o Escape, Yielding A n Exact Volume Liquid pycnometer,
- 31. y - 0.70668 + - 5.0131e-3x RA2 0.591 l.G r 120 130 140 150 160 170 Molecular Weight .7 Densities as afunction.ofsubstituent for aseriesofmonoclinic 4“ubstitutedbenzoic acids. (Data from Musa, 1972.) 1 = (8 * 12.01115) X3 = 11 * 1.00797 X4 = 3 * 15.9994 2 + x 3 + X4 + x 5 ”; X6 U1 = 10.427 U4 = Ul*~2*Y3/20 ecVol in AA3 ”; U4 = INT “Mol Vol = ”;21 INT “Vol/grarn = ”; 2 2 st. Density = ”; 2 3 eat capacity plays a part in several pharrnaceutical considerations on a theoreticalplane. If a process goes from stage A to stage A+ (2.16)is accompanied by an enthalpy. The process could be, for instance, solubility, andthe heat associated with this would be the heat of solution. This is often considered aconstant. The heat capacity Cp of the solution is given by Cp = dAH/dT (2.17)
- 32. and assuming that it is constant then implies that AH is temperature-indepen~ent.There is but little difference between Cp and Cvfor solids, and they may be inter-changed freely. he considerations to be outlined in the following are mostly based on work etals, but they translate to organic molecules as well. The heat capacity is assumed to be associated with the energy E of the mole-cules in the lattice, and these are assumed to be harmonic oscillators. In classictheory, the average energy of a system is kT per degree of freedom, where k is the oltzmann constant and 7 is absolute temperature. For an ensemble of N harmonic oscillators, with three degrees of freedom (themolecule may oscillate in three directions), the average energy is: Eavg =3 N ~ ~ (2.18)or, for a mole Eavg 3RT = (2.29) (~.20)So that, for a solid, the heat capacity should be Cv = 6 cal/deg-mol (2.22)Table 2.1 shows examples of this. It will be shown later that indium is used as a calibrator for diff~rential scan-ning calorimetry ( equation. The equation is called the ~ e t i t - ~ ~ Z o ~ ~The foregoing holds in a classic sense, but the problem with it is that it predictsconstancy. The data in the table fairly well substantiates the Petit-but at lower temperatures, the heat capacities begin to drop. with systems for which one assumes constant enthalpies in atemperature range, it is possible to ~ o m m i terrors, and it becomes important toobtain an idea of at which te~perature(a,, the so-called Einstein te~perature),deviations may start to occur. Heat Capacities at 25°C Heat capacity Molecular Heat capacityCompound cal/"-g weight cal/"rnolCa 0.156 40.08 6.25CU 0.092 63.54 5.85In 0.056 114.82 6.43Mg 0.243 24.31 5.90CO 0.109 58.93 6.42
- 33. In this type of development, the quantum mechanical concept that the energytakes on values only as integers of one another, is used. The energy, for a harmonicoscillator is given by E = nhu = n(h/211.)(~211.) n h u = (2.22) cks constant (6.624 erg-s); v is frequency; h (i.e., h/211.) the is ks constant (1.054 and u is the angular frequency; y1 is ntum number, and is an integer. In an ensemble of N molecules, there will be various energy levels, El (withy1 = l), E2 (with n = 2), and so on. The fraction Vn) the molecules in energy of state n iven by the ~ o l t ~ m a n n distribution, Le., total number of molecules is givenby (2.24)The energy of all the molecules is given by (2.25) y introducing Eq. (2.20), the average energy may now be calculated as (2.26)~ntroducingEq. (2,19) - ~ ~ / R= -hu/RT = X T (2.27)we may write Eq. (2.24) as (2.28)If we use the notation Y = (1 + ex + e2dV,+ - a (2.29)then dY/dx = (e" + 2e" + - .) (2.30)so that, in Eq. (2.28) Eavg nhud In Y/dx = (2.3 1) .29)] is a geometric series with factor ex, so that the sum is Y = 1/(1 - ex) (2.32) ence, - Eavg hu/{(exp(hu/~T) 1) = (2.33)This should be applicable at all temperatures, but at high temperatures k + ( e x ~ ( ~ ~-/ 1 x~ 1) (ho/rcT) + . . - 1 = (hu/kT) (2.34)
- 34. so that Eavg ho/(ho/kT) == ==I kT (2.35)that is, at temperatures higher than a given temperature c3pE (the so-called Einsteintemperature), the energy equals the classic energy. The Einstein model gives profiles in reasonable agreement with experimentaldata, provide^ a suitable choice is made of the fundamental oscillator frequency.Both the terns h and kT are energy terms, and it is more convenient to talk about otemperatures than about f~equencies, so it is conventional to tie this in with theEinstein tem~erature by: .ho z== k@E (2.3~)~ i t this terminology, Eq. (2.31) becomes h Eavg = kc3p,/{eXP(@E/T) - 1) (2.37)so that the heat capacity becomes - LdE/dT = Lk(c3p,lT)2{exp(c3p,/T))/{exp(c3p~/T) (2.38)where L is Avogadro’s number. ith experimental data, it is possible nowto find (byiteration) a value of c3pE that makes the data fit the best. Figure 2.8 is an example ofthis. The severe assumption in the Einstein model is that there is only one funda-mental frequency. (There should at least be three, one for each degree of freedom.)Debye later refined the model to include many frequencies and obtained an evenbetter fit. The important lesson to draw from this is that heat ca~acities ( ~ e ~ eon ~ i are nthe c o m ~ o u ~ d ) times s u f ~ c i e n t ltewlperature-dependent and thatthis ~ h o ~ be at ~ ldtaken into account. The most common appro~imation that is hH=Q+gT (2.39)For instance, ~ n t e ~ n a t i o n a l~ r i t i cTables uses this, and higher polynomial ~lap~roximations,when tabulating heat capacities and enthalpies as a function ofte~perature. 0.0 0.2 0.4 0.6 0.8 1.0 TI0 iarnond heat capacities compared with the Einstein. model with a 4of 1320 >,
- 35. 0th liquids and solids have vapor pressures. Vapor pressures ofsolids may be quite ow, although some pharmaceutical substances (nitroglycerin7ibuprofen) have sig-nificant vapor pressures at room temperature. Vapor pressure of a solid is measured by means of a so-calledwhich measures the escaping tendency of the gas through a capillary. For less precise, but more easily attainable vapor pressures, thermal gravi-metric analysis (TGA) is employed. A covered pan with a pinhole is a l l o ~ e d tostay at a given temperature in the TGA, and the loss rate (dWx/d~) measured. isThis loss rate is proportional to the vapor pressure in the pan. A substance withknown vapor pressure PC ( e g , benzoic acid) is run in parallel, and the vapor pres-sure of the solid (Py) is The development to follow holds for any condensed phase of a one-componentsystem. It will be assumed that the equilibrium is between a solid and its vapor. ibbs, phase rule states that cf = C - - P + 2 i (2.41)where G is number of components, P is number of phases, and df is the degrees of hases. For a one-component system this becomes df= --E"+ (2.42) n e~uilibrium two phases have the same chemical potential, that is, the Pg = Ps (2.43)where the subscript g denotes gas and s denotes solid. The phase rule states that inthe described situation there will be two degrees of freedom (e.g., temperature andpressure) that may be changed, so that at equilibrium, the following must hold:It is recalled that dG -SdT + Vdp (2.45)and that lu, is the G function per mole, SO that { d P ~ / d T= ~ } "8s (2.46)andwhere s and v are molar entropy and volume. A similar set of equations for the solid ence, Eq. (3.2) may be written (2.48) (2.49)or
- 36. (2.50) (~.51)where A H is the heat absorbed at constant temperature and pressure when X mol ofsubstance passes from the solid to the gaseous state (Le.? it is the molar heat of egarding the volumes, the molar volume in the s d stateis negligi~le that in the gas phase, and if this is considered (VI - v,) 25 VI =RT/P (2,52)~ntroductionof Eqs. (2.51) and (2.52) into Eq. (2.50) gives: (2.53) (2.54)This integrates to lnb] = - ~ ~ / /( ~ ~ ) 3 + (2.55)whereis an integra~ion constant. An example of this is the vapor pressure of benzoic acid, The direct data are plotted in Fig. 2.9 and the logarithmic transformation isplotted in Fig. 2.10. It is noted that the heat of vaporization is AH == 1.99 x ’7.685 = 15.4 ~cal/mol It may also be noted that it is assumed that the enthalpy of vaporization is nottemperature-depend~nt, (from the source) it is not so in the temperature interval andshown. If a substance is at a temperature suf~cientlyhigh for it to be in a melted ~ondition, its vapor pressure curve will follow the Clausius Clapeyron equation, except that now the slope is AHvap (i.e., the heat of vaporization). Vapor Pressure of Benzoic Acid as a Function o f Tempe~ature P = vapor pressure Temp (“C) lOOO/T K_-l 1nVl 60 0.1065 3.002 -2.244 70 0.2085 2.914 -1.568 80 0.3928 2.832 -0.934 90 0.7147 2.754 -0.336 100 1.2592 2.680 0.230 110 2.1539 2.610 0.767 Source: West and Selby (1967).
- 37. 50 60 70 80 9 0 100 110 120 Temp ("C) Vapor pressure of benzoic acid as a function of temperature. y - 20.826 - 7 . 6 8 5 1 ~R"2 - 1.000 1 0 -12.8 2.7 "2.6 2.9 3.0 3.1 1000/T Data in Table 2.2 (see Fig. 2.9) treated according to Eq. (2.55). The heat of fusion AHmelt is the difference between the two, that is, Vapor pressure curves (Fig. 2.11) and melting points will assume a special signifi- cance when further discussion on polymorphism is presented. olymor~hism the phenomenon of a chemical entity being able to exist in two is different crystal forms. It will be discussed ingreater detail elsewhere in this text, but a few points and examples are appropriate to mention at this point. Ceolin et ai. (1997), have reported on p , T diagrams of carbamazepine. Car~amazepine (USP) is monoclinic, but other polymorphic forms exist. ~ u b l i ~ ~ t gives a triclinic polymorph, but single crystals are ~ i f ~ c u l tproduce ion to in this manner. The authors produced a crystal of dimensions 10 x 70 x 430 p m that they used for single-crystal characterization of the polymorph. They show the following topological p , T diagram (Fig. 2.12).
- 38. 51 Melting Point v40 GO 80 100 120 140 160 Temp ("C) Vapor pressure diagram of benzoic acid ( ~ e l t i n g point 122°C). E Vapor Phase !32 190 Temperature ( " C ) The literature caption (the reference Fig. 4) should read: B is the triple pointbetween triclinic, monoclinic and vapor; D is the triple point between triclinic, liquid, andvapor. (Data from Ceolin et al., 1977.) They found the transition point by using a tube heated at the position of thesolid, and by monitor in^ the deposit and the temperature alongthe tube; they foundthat 132 was the triple point. , Toscanini S , Gardette M.-F, Agafonov VN, Dzyabchen~oAV, Bachet B (1997). J Pharm Sci 86:1062.Einstein A (1907). Ann Physik 22: 180.Grant DJW, York P (1986). Int J P h a m 30:161.Hersey JA, Krycer I (1981). Int J Pharm Techno1 Prod Manuf 2(2):55.Hiestand E (1997). J P h a m Sci 86:987.Huttenrauch R (1983). P h a m Ind 45(4):435.Huttenrauch R, Keiner I (1979a). Int J P h a m 259.Hutten~auchR, Keiner I (1979b). Powder Techno1 22289.
- 39. Jozwiakowsk~ MJ,Nguyen NT, Sisco JJ, Spankcak CW (1996). J Pharrn Sci 87:193.Longuemard P, Jbilou My Guyot-Herrnann A- , Guyot J-C (1998). Int J P h a m 17051.Moriata M, Nakai Y, Kukuoka E, Nakajima SI (1984). Chern Pharrn (19’72). PhD dissertation, University of Wisconsin, Madiso try and Physics, 48th ed. The Ch~micai ubber Co., Cleveland, OH, p. D 141. 143.
- 40. 3.1. E ~ u i l i b r i u ~ Solubility 28 3.2. eat of Solution 28 3.3. etermination: Effect of Temperature 32 3.4. trolytes on Solubility 37 3.5. ixed Solvent Systems 37 3.6. lectric Constant on Solubility Parameters 38 3.7. ultiple Solubility Peaks 39 3.8. ~ o ~ ~ ~ e ~ a t i o ~ 41 3.9. ~ yclodextri~s 423.10. Solub~lityand p 423.11. rediction Equations for Solubility in 443.12. 443.13. 453.14. ffect of Surfactants 463.15. 46 46 47Solubility of c o ~ ~ o u nis of great importance in pharmaceutics, and the subject has dsbeen subdivide^ into the foregoing subtopics.
- 41. henever the tern solubility is employed, it is tacitly assumed that it is equilibriumsolubility. In other words, it assumes that a (stable) solid (the solute) is laced incontact with a li~uid (the o l ~ eand~the system is all owe^ to be agitate^ for a long s ~ ~ , while, or by other ~ e a n allowed to reach a state o e ~ u i l i b r i ucharacteri~ed the s f ~, byfact t ~ a the co~cent~ation solute has reached a co~stant t of level. This de~nition by no means easy to establish in practice. Such things as small is temperature ~uctuations,and that solubility maybe a function of particle size,makes the experimental establishment of solubili~yof acompound dif~cult to achieve. Add to that the fact that solids of higher energetics(metastable poly~orphs or amorphates) have higher apparent solubilities also confounds the issue. At times (e.g., in the case of benzodiazepam), the drugsubstance, as first produced (in clinical trials), turned out to be a metastable polymorph. Apparent equilibrium solubilities esta~lishedand were t h o u ~ h t be true equilibrium solubilities because the to figures were reproducible, until one day the more stable form happened to be pro- duced, and this had a lower solubility. Because it is never reallycertain that any drug substance produced is a c ~ ~ a lthe stable polymorph, the term e ~ u i l i b r i u ~ ly solubility is clouded to some degree with uncertainty. Inthis chapter to follow it is going to be assumed that solubility is exactly what the foregoing italicized de~nition purports it to be. For the purposes of this book, there are four types of equilibri~msituations ill, 1933) that may be considered: e solid phase is a pure compound, and there is one liquid phase. e solid phase is a pure compound, and there is more than one liquid phase. 3. The two components form a solid solution in such a way that there is unlimited solubility in the solid phase. 4. There aretwo solid solutions forming (Le., there is limitedsolubility in the solid phase). f these, case 1 is ove~helmingly most cosituation. the Case 2 is at portant in differential scanning calorimetry work. If the melts ofcompoundAand compound Bare immiscible, then the DSC thermogram willshowtwo peaks, one at each compound’s melting point, otherwiseone broadpeak will occur. This will be discussed in a later chapter. When a substance (the solute) dissolves in a solvent there are certain changes that ome solutions are ideal solutions, and in such solutions the volumes, for property thatis of importance in the following isthe heat associated with the solution of a solid drug substance in a solvent (most often water), and it will become a ~ ~ a r e that the effect of temperature on solubility is associated with an aspect of nt this thermal phenomenon. There is a fair amount of misinterpretation of the “heats of solution” in lit- erature, and in this aspectitis fruitful to quote a very old, but comprehensive
- 42. reference (Taylor, 193 1). If solubility ofa compound in a solvent isplotted versus thetemperature then, in its simplest form, the curve will either rise or fall. If heat isevolved when the solid is dissolved in an ( a l ~ o s st ~ t u r a t e d s o l ~ tthen, the solubi- ~ io~lity of the compound will decrease with increasingtemperature, and the opposite, inthe simple case, is also true. owever, to quote Taylor (1931): nowledge that when water is poured upon solid p o ~ ~ s s i uhydroxide, m much heat is evolved; if one deduced therefrom that the solubility of the compound decreased with the temperature, the error would be flagrant. The initial heat of solution is positive; it may be that the total heat of solution is positive, but the final heat of solution, representing the dissolving of the last increment entering the solution at the saturation point, is negative,and hence a rise of temperature will result in the dissolving of another increment. The relations between partial molarand inte~ralheats of solutions are ransted (1943a),in the followingwords,directly translated (theword t ~ ~ ~ ~ o d uy ~ t ia n~ i ~ the tern A in the translatio~): f n n o used for To visualize the connection one may utilize a graphic presentation in which one most advantageously utilizes the x-concentration scale and in place of A which applies to n1 + n2 molecules of mixture [utilizes] Ai, the integral mixing [ ~ h e ~ o d y ~ a m i c ] func~ion for one mole of mixture. The equation corresponding to this may be derived in direct analogy with the [previously cited equations] containing nl + n2 moles but may also be obtained by introducing: A = (nl + n2)Al and One, hence, obtains the following equation, valid at constant temperature and pressure: as well as the relation between the differential [thermodynamic functions] The connection between AI, Az, and Ai is shown in Fig. 2 [re~onstructed as 3.1 in Fig. this text]. At a further point in the text ansted (1943b) states that A the t h e ~ o d y -namic function “can be the S, V, E, 6, or H functions.” The foregoing text talks to the difference between differential heats of solutionand integral heats of solution; examplewise the heat evolved per mole of sulfuric acidadded to 1 mol of water. The heat of solution (IT)of a mixture of n1 moles of acompound A in n2 moles of a solvent
- 43. Enthalpy N M 0 x=o x= 1 ti20 Solute “The abscissa is x (Br~nsted,1933~): The thermodynamic function in the figure is alpy H . The distance DJ is equal to the slope at L, mu~tipliedbyJD = (1 - x ) a ~ / a ~which, according to the Brransted Eq. (20) equals At - A,. ),value at D corresponds to H , = aHr/ax. The distance QP equals x times the slope at L, (Le.,x a H / ~ ~which according to the Brransted Eq. (20) equals A, - A 2 . Hence, the ordinate value ),at P equals A2 = a ~ ~ / a ~ .whereand ted the partial molar quantities of compone (subscri~t and compo- 1) ubscript 2). Table 3.1 lists the heats of sol of the sulfuric acid watersystem. that the first column is the third column divide the second column. se data are plotted, then a graph, such as shown ig. 3.2 ensues. If thetangent is drawn at any point of the curve, then the interce- th the left axis givesHI = 3H/dnl and the intercept with the right axis gives p 2 = 3H/dn2 where n1 andn2 are the molesof water and acidin the particular amount of solution i.e., +x = ~ / ( n l na). his is shownfor a composition of 0.55 mol ofsulfuric acid added ater (i.e., a mole fraction of 0. ). It is seen that the partial molar lution of water (A) and sulfuric acid ( are 6200 and 510 cal, respectively. is dissolved in water, there is a limiting value for x; namely, thatcorrespond in^ to the solubility. Hence onlypart of graph would apply, as shown in
- 44. Water and Sulfuric Acid. Heat of Solution as a Function of Compos~tion ole fraction 1000 x heat evolved mole per A H per mole ofacid ( X ) acid of solution0 0 00.1 15.6 -1.560.2 12.94 -2.590.3 10.71 -3.210.35 9.65 -3.380.4 8.63 -3.4520.45 7.68 -3.4560.5 6.73 -3.370.55 5.81 -3.200.6 4.87 -2.930.65 4-06 -2.640.7 3.2 -2.300.75 2.6 - 1.950.8 1.97 -1.580.85 1.42 -1.210.9 0.93 -0.840.95 0.45 -0.431.oo 0 0Source: Data from Brmsted, 1909; Marshall, 1933. 0.0 0.2 0.4 0.G 0.8 1.0 x eats of solution of the sulfuric acid system: The abscissa is the mole fraction ofsulfuric acid.
- 45. Fig. 3.3. It is noted that in Fig. 3.3 the heats of solution are terminated by thesolubility X . statements may (incorrectly) imply that there is either anincrease or a decrease in solubility of a compound with temperature. There aremany exceptions. ~ m m o n i u mnitrate solubility in water, for instance, exhibitsbreaks at 32"C, 83"C, and 126°C Maxima and minimain solubility/temperaturecurves also occur, and some of the situations of this will be discussed later in thischapter.The subject of eutectic diagrams willbe taken up in a later chapter, but a shortoutline will be given at this point. The simple solution situationreferred to in Sec. 3.1 exhibits a eutectic diagramsuch as shown in ig.3.4a. The so-called liquidous line in the right part of theeutectic, QU, is a mpositional line where, at a given temperature, T, there is anequilibrium between solid solute I and a solution of in water of composition x. 3This, in essence, is a solubility curve, and if the axes are ~ i p p e as ,shown in Fig. ~3Ab, then a conventional repr~sentation solubility versus temperature results. of Solubility of solids are determined by placing an excess of solid in contact withthe solvent in a hermetic containers (ampoule or closed testtube) and agitating it a inconstant te~perature bath. It is conventional to use 7 2 h for e~~ilibration. If less time is used, then the solubility may be obtained by iterative extrapola-tion, as demonstrated Table 3.2. Samples are taken after certain in times (here multi-ples of 12 h), and the supernatant is assayed. The concentrations are then plotted asa function of time, as shown in Fig. 3.5. It is seen that the data"seem" to level off at59, so the solubilities are subtracted from 59 (see column 3 inlogarithm takenof these numbers. These are plotted in Fig. 3.6. 5: X Q 0 Q c( - I V 0.0 0.2 0.4 0.G 0.8 1.0 x Heats of s o l u t i ~ ~ s depicted in Fig. 3.1 but t e r ~ i n a t e ~ the solubility X , repre- bysenting the highest concentration.
- 46. ilitMeltingPomt,T Mole Fraction x 1 L 1 L L U 1 $0 Solid B+ Water Q a Ice + Solid 8 4 1 Mole Fraction, X Temperature, T H20 Solute, B (a) Eutectic diagram of water and a solute, B. (b) The right side of the eutecticdiagram from Fig. 3.4a plotted with reversed axes (i.e., solubility as a functi~n oftemperature). 60M00.-.L M 40 30 20 10 0 0 20 40 GO 80 Time (hours) Data from Table 3.2. Example of ~olubilityDetermination by Iterative ~ x t r a p o l a ~ i o ~Time (h) Solub~lity(g/lOOO 59 g) -s h[59 - 21 0 4.078 0 5912 3.367 30 2924 2.639 45 1436 52.5 6.5 1.87248 56 3 1.09960 0.18 57.8 1.2
- 47. y = . 4 . 1 3 8 2 - 6.4403e-2x R*2 0.998 5 4 3 2 1 0 0 20 40 60 T m e (Hours) Data from Fig. 3.4 treated by iteration.repeated with a figure different from 59. The value of the iterant thatgives the best fit(the least sum of residual squares) is then assigned as the solubility. lubility is best expressed as molality or as weightof solute per gram of .e., not per cubic centimeter of solution). The conventional t r e a t ~ e n t ofsolubility as a function of temperature is to note that the chemical potential of acompound in solution, at a concentration level correspo~ding to an activity of a, isgiven by p1 = po + RTlna (3.4) ere, po is a reference state, and obviouslyis the chemical potential when the is unity (i.e., when a = 1 molal). hen there is eq~ilibrium (of between a solid and a saturated solution activity the chemical potential of the solid p,, equals that of the compound insolution, given by Eq. (3.4), that is, (3.5) ividing through by T and di~erentiatingrelative to T now gives = T + { a ( ~ . ~ / T ) / a T } ~ d{a(po/aT}~dT Rd(ln a,) (3.6)It is recalled thatThis when inserted in Eq. (3.6) then gives (after rearrangement) --{(h, - ho~/T2}dT ~ d ( l n ~ a , ~ ) =h" - h = -(hs - ho)is the enthalpyassociated with tran~ferri~g , 1 mol of solid into a quantity of saturated solution and h" - h, is commonly simply denoted h nes h, as "the partial molar enthalpy of the component in the erefore, at a given tem~erature,be the partial (3.9) (3.10)
- 48. where , is an integration constant; a, is the activity of the solute at saturation andis &given by a, = Y,S (3. 11)where S is the saturation concentration (in molality) and y, is the activity coat saturation. If this is assumed to be unity, then Eq. (3.10) becomesthe welland often used equation ln[q = AH/(^^)} +BThis is referred to as a Van’ ff plot (although this latter, properly, iswith equilibrium constants9 solubilities). More correctly Eq. (3.12)written:If y, is temperature-independent9then the logarithm of the saturation on cent rationis linear in reciprocal absolute temperatur a plotting mode that is often useexample of this is shown in Linearity of the Van’t ependent, and (b) activity coefficient (y,) is temperature-independent. If they are no off plot will not be linear. An example of this is shown in Fig. 3 rant et al. (1984) hypothesizedthat if, rather than “the partial molof solution of the solute, is inde~endent temperature, we assume that it is a oflinear function oftempe , as follows: AH; = a + bT.’9They interpret that “ amaybeconsidered to be the ~ y ~ ~ t ~valueof c AH; at the absolute zero of e t ~ a ~tem~erature b is the change in the apparent partial molar heat capacity of t,he andsolute at constant pressure, AC;2, whichisitselfassumed to be independent oftemperature. There isevidence that the introduction of terns containing hipowers of T, e.g., cT2’ etc., is unnecessary.” If the curvature in Fig. 3.by the heat of solutio^ not being temperature-independent, i.e., (3.15) d ln S/dT = A H / ( ~ ~ 2 ) (3.16)~ombining with Eq. (3.15) then gives this Td In S/dT = { A / ( (3.1’7) §olubility of ~ r t h ~ r h o m ~u~fanilamide Ethanol bi~ inTemp (“6) Solubility (g/ 1000 g) 1n[sl47.4 28.22 3.34 31 1240.3 23.34 3.15 3.1929.6 16.78 2.82 3.3024.1 14.15 2.65 3.36Source: Data from ~ i l o s o ~ i c 1964. h,
- 49. y = 12.380 - 2 . 8 9 6 1 ~R*2 - 1.000 1000/T able 3.3 plotted according to Eq. (3.12).Equation (3.17) integrates to Ins= (3.18) This may be fittedby nonlinear programs, butfor these to work, one must havea good estimate of A and B. To obtain good estimates, most graphing programswillcalculate (d In Cs)which may then be multiplied by I“ and plotted by way of Eq.(3.17) versus 1/T. This should produce a straight line with intercept B / R and slope A and 13 may now be estimated from the slope and intercept of this line, andmay be used a s . ~ r s t a p p r o x i ~ a ~ iin n s~ o n l i program, This approach has been oa ~ e ~ ~employed byseveralrecent investigators (~udipeddi,1998; ~ozwiakows~i ai., et1~96). It should, again, be emphasized that the enthalpy term in both(3.18) corresponds to “the partial molar enthalpy o f the c o ~ p o n e n tin the . . .solution . , . ii.e.1 the heat absorbed, at constant temperature and pressure, when 1mole of the component dissolves in the . . . solution.” ( ~ e n b i g h1961). T h i s ~ a c tin , ,itself, ~ a ~ itequite unde~standablewhy the Van’t ~o~ can not be expected to be slinear. Consider the diagram in Fig. 3.1. Suppos the depicted compound at a tern-perature of TI had a solubility corresponding to and at a higher tem~erature a had 1.9 1.8 1.7 1.6 1.5 1.4 3.1 3.4 3.2 3.3 3.5 1000/T Solubility of dl -p[pseudoephedrine]. (Data from ~ u ~ i p e d d1996.) i,
- 50. y 331.78 - 9 3 . 3 6 2 ~R-2 = 0.992 4 0 F"25 10 3.15 I I I 1000/T Derivative curve (d I S ) of data in Fig. 3.8 versus n T". solubility corresponding to L, then the ~ ~ f e r e n t i a l e n t h ~ ~ i e s ~ a of solution^ w o u be ~ f ~ ~ c t i of nt e ~ ~ e r a t ~ r e ; it is not unexpected that the Vant Hoff plot is not hence, linear, but it is rather to be expected. To assign the change in heat capacity as an explanation to the nonlinearity is rational only in the sense that the composition changes with temperature; hence, the change in heat capacity also changes. There are many examples of this; for instance, Longuemard et al. (1998) have reported on the solubility of aspirin in 38% alcohol; they failed to obtain linearity according to the Vant Hoff, although in this case the curvature may be because the ordinate is in grams per liter (g/L), rather than in grams per 1000 g (g/lOOO g) of solvent * The solvent has a great i ~ ~ u e n on solubility and should always be speci~ed. ce aqueous solutions, the concentration of electrolytes may greatly affect the sol~bility of a compound. (It will be seen later, that this is particularly true for a compoun~ that is, itself, an electrolyte). Figure 3.10 shows the effect of sodium chloride con- centration on the solubility of a bisnaphthalimide derivative. The use of mixedsolvent systems isoften necessary in pharmaceuti~s when a drug is poorly soluble. Cosolvents used are Ethanol Propylene glycol Glycerin olyoxyethylene glycols Ternary diagrams are used to visualize wheremaximum solubility occurs when more than one solvent is used (Fig. 3.1 1). The length of PA is the percentage of water, the length WB is the amount of ethanol and, here, EC is the percentage of glycerol. The lines in this presentation mode are parallel to the sides in the triangle. In a different presentation mode they
- 51. 0 -1 -2 -3 -4 -5 0.00 0.05 0.10 INaCl]. M Effect of salt concentration on the solubility of a bisnaphthalimide. (Data from et al., 1996.)are cast perpendicularly to the axes. A point inside the triangle, such asone given composition. If solubilities are determined for many solvent compositions, then the solubilitywould be the same (10 mglg, 20 mglg, etc.) for given compositions of the solvent, andsuch points can be connected to form isotherms and diagrams, such aswould result. The figure to the left in Fig. 3.12implies a maximum solubility,whereas in the other diagram, the more of one cosolvent that is added, the largerthe solubility is. requently the solubility is a fun~tionof the dielectric constant of the medium. ften, the relation is that of the Jaffe equation: + In[SJ = ( A / & ) l3 . (3.19)where A and B are constants and E is the dielectric constant of the solvent. Anexample of this is shown in Table 3.4, in which the solubility of a c o m ~ o u n ~ istabulated as a function of the dielectric constant of the medium (glycerin/water Water Glycerol Ternary diagram.
- 52. H 20 Hz0Glyerol Ethanol Glycerol Ethanol Ternary diagrams of the two types of solubility. ost often, with hydrophobic drugsthe solubility decreases withincreas- constant. The opposite happens at times, and an example of this isshown in Table 3.4 and Fig. 3.13. It is particularly useful, from a practical point of view, to carry out solubilitiesin solvent pairs of different ratios tovary the dielectric constant. Graphswill often belinear when plotted asin Fig. 3.13, but they will often show maximumsolubility at agiven dielectricconstant (Fig. 3.14), and the practical part of this is that once this isestablished, almost any other solvent pair willshow maximum stability atthat her than using dielectric constant as a measure, the Hildebrand solubility~arameter is often employed. Shino~a 6 (1978) defines this as (3 20) s the heat of vaporization of the solvent, V its molar volume, and 3.15 gives an example both of plotting the solubility of a com~ound(caffeine) in solvent rnistures with different solubility parameters, the plotting as afunction of their dielectric constant. olubility profiles vis-a-vis the solubility parameter of the solvent at times showsmult~plepeaks. This is the so-called cha~eleonic effect (Sunwoo and Eisem, 1971; Effect of Dielectric Constant on Solubility of ~isnaphthalimideDielectric constant Solubility ( S , mg/mL) 1nES3 1O O / ~ ~78.5 2.49 0.912 1.27474.9 2.30 0.833 1.33565.9 2.00 0.693 1.51752.6 1.42 0.35 1 1.90045.45 1.09 0.086 2.20042.5 0.9% -0.02 4.333
- 53. , y = 2.0006 - 8 6 . 4 7 6 ~ R^2 = 0.999 -0.2 0.0 12 1 0.0 17 I 0.022 l/(l)ielectric Constant) ffect of dielectric constant on the solubility of bisnaphtha.limide. (Data from aghavan et al., 1996.) 0 10 20 30 40 50 Dielectric Constant Effect of dielectric constant onthe solubility of phenobarbital in four systems: A, ropyleneglyco1:ethanol; € ,g1ycerin:ethano~; watecethanol; D, propylene glyco1:water. 3 C, ata from Lordi et al., 1964.) Dielectric Constant 20 40 60 80 100 " 5 10 155 2 20 Solubility Parameter The solubility of caffeine in a solvent cansisti~gof dioxane and water at 25°C:Top curve (with top abscissa) is solubility versus dielectricconstant, and the lower curve (withthe lower abscissa) is the solubility versus the solubility parameter (ai). (Data from Martinetal., 1961.)

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