JOURNAL OF
`
`PHARMACEUTICAL
`SCIENCES
`
`MAY 1980
`
`VOLUME 69 NUMBER 5
`
`RESEARCH ARTICLES
`
`Extended Hildebrand Solubility Approach:
`Solubility of Theophylline in Polar Binary Solvents
`
`A. MARTIN ", J. NEWBURGER, and A. ADJEI
`
`Received September 6. 1979, from the Drug Dynamics Institute, College of Pharmacy, University of Texas. Austin, TX '.'r'8}'l2.
`publication November 21, 1979.
`
`Accepted for
`
`Abstract D A quantitative approach is presented for predicting solu-
`bilities of crystalline compounds in binary solvent systems. The solubility
`of theophylline in mixed solvents consisting of dioxane and water was
`determined at 25 1 0.1“. The soiubilities across this range of polar sol-
`vents were baclocalculated using a technique involving an interaction
`energy term. W. This parameter is regressed against a polynomial ex-
`pression in 5;, the solubility parameter for the mixed solvents. Except
`for the endpoints, solubilities were predicted within <12% and with
`considerably better accuracy in most cases. The new approach modifies
`the well—l(nown Hildebrand solubility equation to make it applicable to
`polar systems. Although the method may be used with solutes in pure
`solvents, its greatest application appears to be the prediction of drug
`solubility in binary solvent mixtures.
`
`Keyplirases Cl 'I‘heophylline—solubility in polar binary solvents D
`Solvent systen1a——solubility of theophylline in polar binary solvents
`
`Solubility data on drugs and pharmaceutical adjuncts
`in mixed solvents have wide application in the drug sci-
`ences. Knowledge of interactive forces between solutes and
`solvents are of considerable theoretical and practical in-
`terest throughout the physical and biological sciences.
`
`BACKGROUND
`
`The term regular solution was introduced by Hildebrand (1) to describe
`solutions showing random molecular distribution and orientation as
`found in ideal solutions. There is no entropy change. but heat is absorbed
`when the components of a regular solution are mixed. Although these
`solutions are not ideal. they yield curves of log solubility versus 1/7‘ that
`are quite regular. Other kinds of solutions. those that involve salvation
`or association, produce irregular solubility curves. Modifications of the
`Hildebrand approach for irregular solutions have been reported in the
`field of solution technology by various investigators (2-8).
`The I-lildebrand—Scatchard equation for the solubility of solids in a
`regular solution may be written as (9-13):
`I
`T
`2
`"M
`-he X2=é_:2lOg—'l?+2..’lh??iT
`where X g is the mole fraction solubility of the solute or drug (represented
`by subscript 2): AS-:, is the entropy of fusion of the crystalline drug at its
`melting point. Tm, on the Kelvin scale; 7‘ is the temperature in degrees
`
`(23. — 5,12
`
`(Eq. 2}
`
`Kelvin at which the solubility is determined; R is the molar gas constant:
`‘/2 is the molar volume of the drug; (it; is the volume fraction of the solvent
`(represented by subscript ll: 5-; is the solubility parameter of the solute:
`and 51 is the solubility parameter of the solvent or mixed solvent. Solu-
`bility parameters also are referred to as delta values. The first right-hand
`term of Eq. I frequently is written as {AHl...’2.303RTl{1/T - 117),. l, but
`the AS{.. term of Eq. I. is more correct. as will be discussed later.
`The solubility parameter or delta value of the solvent, 51, is obtained
`as suggested by Hildebrand and Scott U0} using the relationship:
`6 =
`1:2 _ (AH-E _ R1]1;2
`v,
`‘
`v
`where AE'{ is the molar energy of vaporization, AH? is the heat of va-
`porization, and V1 is the molar volume of the solvent. The square of 51,.
`or ELEV V1, is called the cohesive energy density of the solvent. Other
`methods for obtaining 51 were given by Hildebrand and Scott (10).
`The solubility parameter of a solid compound is difficult to obtain, and
`few values are recorded in the literature. The :5’ value for iodine is --14.1
`(10), and the value for naphthalene. phenanthrene, and anthrancene is
`~9.3. Several investigators (12. 14) estimated the solubility parameters
`of benzoic acid and some p-hydroxybenzoic acid esters from the peak
`values obtained from a plot of mole fraction solubility versus delta values
`of solvents. The parameter for benzoic acid also was determined from
`the solubility data in hexane and was found to be 11.5. The solubility
`parameters for barbiturates have been determined (15. 16). Yalkowsky
`at cal. (17) obtained the solubility parameters for p-aminobensoate esters
`from their solubility in hexane. For example, the value for the ethyl ester
`was 12.05. James er al. (18) reported solubility parameters for some
`testosterone esters and related compounds.
`The first term on the right side of Eq. 1, the ideal solubility term, is for
`the dissociation of the crystal lattice of a solid compound, rendering it
`in the liquid form. In the presence of solvents that form nearly idea] so-
`lutions, the second right—hand quantity. the reguiar solution term. is
`nearly zero and may be omitted.
`The regular solution term, involving solubility parameters. is an activity
`coefficient. log of... used to represent nonideality due to the interaction
`of solute and solvent molecules in a regular solution where only nonpolar
`and weak polar forces exist:
`
`leg as ‘= EH51 m 62)’
`
`(Eq. 3)
`
`where A represents Vg¢'f'f2.303RT and the subscript u stands for van der
`Waala forces.
`Following the suggestion by Crowley et cl. (8), Hansen (5. 6} introduced
`a three—dirnensiona.l system of solubility parameters. The energy of va-
`
`002243549: 80: 0500443750 1.00:0
`© 1930, American Pnmmscsuncaiassocisrmn
`
`Joumalof Pltsmasoauricalsclencesf 48?
`Vol. 69. No. 5. May 1980
`
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 1
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 1
`
`

`
`porisation in Eq. 2 was assumed to he an additive quantity composed of
`three energies representing London dispersion forces [AE§). polar forces
`(AE3). and hydrogen bonding (AE3) in the solvent. Dividing each term
`hy the molar volume of the solvent (V. l. the total cohesive energy density
`was obtained:
`
`. =AE‘:.+AE"E,+AE';.
`V1
`V1
`V1
`V;
`or. in terms of delta values:
`
`{Eq. 4)
`
`(Eq. 5}
`52 = 53 + 63, + at
`where 67‘ is the total cohesive energy density for the liquid. Values for 45¢
`were determined by reference to a corresponding hydrocarbon called a
`homomorph. and 5.. and 6;. were estimated by empirical methods. Hoy
`er al. (19) developed extensive tables of three-dimensional delta values.
`which differ somewhat from the values of Hansen [5, 6}. Hansen noted
`that there is yet no theoretical basis for the new three-dimensional sol-
`ubility parameters. and he used them empirically to interpret the solu-
`bility of polymers and other solutes employed in industry and com-
`tnerce.
`Wcimer and Prausnitz [20] calculated polar and nonpolar solubility
`parameters using the hornomorph concept. and Blanks and Prauanitz
`('7) applied these values to the study of polymer solubility in polar sol-
`vents. These investigators did not consider hydrogen bonding sys-
`terns.
`
`Solubility Dete1-n1iJ1ation—- The solubility of theophylline (52 = 14.0}
`was determined in mixed solvents consisting ofdioxane (5., = 10.01) and
`water [51, = 23.45}. Glass-distilled deionized water was used toprepare
`mixtures with dioxane in concentrations of D—100% by volume of dioxane.
`About 10 ml of the mixture was introduced into screw-capped vials
`containing excess theophylline. The vials were agitated for 96 hr in a
`shaker bath maintained at 25 -.t 0.t°. Preliminary studies showed that
`this period was sufficient to ensure saturation at 25".
`After equilibrium was attained. vials were removed for analysis. The
`solutions were filtered. and aliquots were placed in volumetric flasks and
`brought to the final volume with the solvent mixture in which the drug
`was originally dissolved. The solutions were analyzed in a spectropho-
`tometer‘ at 2"I3.4 nm. Three samples were withdrawn from four separate
`vials and measured at each mixed solvent concentration. The standard
`error for the analysis of individual samples was ($0.? pgfml.
`The densities ofthe solvent mixtures and the filtrates of the saturated
`solutions of theophylline were determined in triplicate at 25 :E 0. 1° using
`a pycnometer.
`Heat of Fusion-—-The heat of fusion of crystalline theophyllina was
`determined experimentally in a differential scanning calorimeter”.
`Therrnograms were run at 100 psi to retard sublimation. and the heat of
`fusion was determined from the area under the curve. using indium metal
`as a standard. The equation employed in calculating the heats of fusion
`from differential scanning calorimetry is:
`
`AH‘.'.. = AHL. {standard} X standard weight X sample peak area X
`
`instrument range for sample X sample mol. wt.
`instrument range for standard >< sample weight
`>( standard moi. wt. X standard peak area
`
`(Eq. Ill
`
`The method presented here‘ allows calculation of the solubility of polar
`and nonpolar solutes in solvents ranging from nonpolar (hexane) to
`aprotic polar le.g., N,.N-dimethylformamide} and highly polar protic
`solvents such as alcohols. acids. and water. Although formulated specif-
`ically in terms of the solubility of a nonelectrolytic solid in liquid solution.
`the technique should apply as well to liquid—liquid and other equilibrium
`systems.
`Equation 1 ordinarily provides a poor prediction for the solubility of
`a drug or other crystalline compound in a polar solvent. These solutions
`are quite irregular, often involving self-association or solvation. The
`logarithm of the activity coefficient. calculated using Eq. 3. accounts for
`the ncnideality of solutions resulting from the interaction of solute and
`solvent molecules of the physical or van der Waals type. Several inves-
`tigators. including Hildebrand. have cautioned that expressions in the
`form of Eq. 3 are not good representations of nonideality in solutions of
`polymers and various polar and semipolar compounds in polar and hy-
`drogen bonding solvents. For irregular solutions. a total activity coeff-
`cient. org, must be written consisting of the term [Eq. 3) representing
`physical or van der Waals forces and an additional term. log (13. repre-
`senting residual. presumably stronger. forces:
`
`To obtain the ideal solubility. X -'2. Hildebrand et oi. (22) showed that
`entropy of fusion. ASl... can replace heat of fusion to take into account
`the molar heat capacity change. AC9, in going from a solid to a liquid
`solute. The equation for calculating the ideal solubility employing diS{,.
`is (22):
`
`log X‘ ;
`
`T
`AS{,,
`R mg 7'...
`The entropy of fusion is obtained by plotting log X2 versus log '1“ under
`ideal solution conditions. With solubility data. X2. at three or four tem-
`peratures. a linear plot with a slope proportional to ASL. is obtained.
`However, the AH‘; and T... values may be determined more conveniently
`using a differential scanning calorimeter (23). Once these values are ob-
`tained, AS{,,. is calculated from:
`
`(FJq.12)
`
`Solubility Parameter of Mixed Solvents——T'he solubility parameter.
`5., for a mixture of two solvents. a and b. is calculated (2-1., 25) using the
`expression:
`
`(Eq. 13}
`
`log or, = log “U + log on
`The total activity coefficient may be written as:
`
`{Eq. 6)
`
`where:
`
`(Eq. 7)
`log (‘(3 = Alt? + .53 — 2w)
`where W is the interaction energy between the solute and solvent in an
`irregular solution.
`Employing Eqs. 3. 6. and 7. one obtains for the residual term:
`
`log Ora = 2405152 * Wl
`
`(Eq. 8}
`
`The logarithm of the total activity coefficient may be written for irregular
`solutions as:
`
`log 0:: = A(51- 62)” + 2Ala[52 — Wl
`and the modified Hildebrand solubility equation becomes:
`AS’
`T...
`-logX2= ?"'log?+ Aid? +5;-2W)
`
`(Eq. 9)
`
`(Eq. 10)
`
`EXPERIMENTAL
`
`Materials-—-Anhydrous theophylline2 and p-dioxane3 were obtained
`commercially.
`
`l The first report in this series is Ref. 21. It provides a sample calculation for the
`method described here and in subsequent papers.
`3 Knoll Chemicals.
`3 Mallinclrrodt Chemical Works.
`
`433 3 Journal of Pharmacoufical Sciences
`Vol. 6.9. No. 5. May I980
`
`1
`6 = ¢..r5.. + $55!:
`¢a+¢a
`
`do = 95::
`
`'i' do
`
`USE}. 14)
`
`(E1;-15l
`
`in which at». is the total volume fraction of the two solvents and 61. the
`solubility parameter of solvents a and b, is averaged in terms of volume
`fractions.
`Volume Fraction and Mean Molar Volume in Mixed Solvents-—
`The total volume fraction, or, of the solvent mixture is calculated using
`the expression:
`
`dl
`
`=
`
`l1'X2)VI
`(1 - X3)V; + X2Vg
`where X 2 is the mole fraction solubility of the drug in the mixed solvent
`and V; is the mean molar volume of the binary solvent. For each mixed
`solvent composed of solvents o and b in various proportions:
`
`(E9. 16)
`
`V1
`
`= X..M.. + (1 — x.:M.
`:01
`
`U301. 17}
`
`where X . and M. are the mole fraction and molecular weight ofthe par-
`
`“ Model 25 spectrophotometer, Becltrnan.
`5 Model 1B. Perkin-Elmer.
`3 L. '1‘. Grady of the United States Pharmaoopeia Laboratories. Rockville, Md..
`and S. H. Yallmwsky. The Upjohn Co. Kalamazoo. Mich.. provided independent
`measurements of thoophylline, theobmrnine. and caffeine In their laboratories.
`
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 2
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 2
`
`

`
`Table I—Mole Fraction Solubility of Theopbylline in Dioxsne-Water Mixtures at 25°
`
`Diorane.
`%
`0
`5
`10
`15
`20
`25
`30
`35
`40
`45
`55
`60
`62
`66
`70
`75
`77
`80
`35
`90
`100
`
`V;
`18.063
`21.577
`24.880
`28.232
`31.566
`34.875
`38.158
`41.450
`44.713
`48.018
`54.589
`57.907
`59.228
`64.630
`64.630
`68.011
`69.385
`71.464
`74.956
`78.459
`85.663
`
`5.
`23.45
`22.78
`22.11
`21.43
`20.76
`20.09
`19.42
`18.75
`18.07
`17.40
`16.06
`15.39
`15.12
`14.58
`14.04
`13.37
`13.10
`12.70
`12.03
`11.35
`10.01
`
`Solution
`Density
`0.9988
`0.9983
`1.0058
`1.0105
`1.0148
`1.0190
`1.0232
`1.0265
`1.0300
`1.0321
`1.0362
`1.0374
`1.0379
`1.0379
`1.0379
`1.0379
`1.0375
`1.0368
`1.0352
`1.0367
`1.0286
`
`6;
`0.99493
`0.99390
`0.99228
`0.99082
`0.98916
`0.98761
`0.98494
`0.98320
`0.98168
`0.97877
`0.97438
`0.97352
`0.97287
`0.97277
`0.97293
`0.97428
`0.97480
`0.97617
`0.97816
`0.98162
`0.99625
`
`X2 (oba.}“
`0.0007414
`0.0010668
`0.0015583
`0.0021046
`0.0027 831
`0.0035158
`0.0046818
`0.0056783
`0.0066856
`0.0083295
`0.0114449
`0.012541 1
`0.0131436
`0.0143803
`0.0142926
`0.014271 1
`0.0142592
`0.0138736
`0.0331770
`0.01 17074
`0.0025959
`
`A
`0.08987
`0.08978
`0.09849
`0.08922
`0.08925
`0.08865
`0.08817
`0.08786
`0.08759
`0.08707
`0.08629
`0.08614
`0.08602
`0.08600
`0.08603
`0.08627
`0.08636
`0.08661
`0.08696
`0.08758
`0.09020
`
`L08 flfs
`1.40772
`1.24969
`1.08512
`0.95460
`0.83325
`0.73175
`0.60736
`0.52356
`0.45263
`0.35716
`0.21916
`0.17944
`0.15906
`0.12001
`0.12266
`0.1 2832
`0.12368
`0.13559
`0.15334
`0.20931
`0.86349
`
`Log au-
`8.03426
`6.92101
`5.88581
`4.92565
`4.06366
`3.28777
`2.59010
`1.98230
`1.45084
`1.00650
`0.36617
`0.16642
`0.10791
`0.02893
`0.00014
`0.03424
`0.06995
`0.14636
`0.33748
`0.61500
`1.43607
`
`Log on
`-6.62654
`-5.67 132
`—-1.80069
`- 3.97105
`—3.2304l
`-2.55602
`— 1.98274
`— 1 .45 874
`-0.99 82 1
`—0.64934
`—0. 14701
`0.01302
`0.051 15
`0.09106
`0.12252
`0.08908
`0.05373
`-0.01077
`-0.18414
`—0.40569
`-0.57258
`
`W (Eq. 7)
`365.128
`350.504
`336.363
`322.273
`308.804
`295.677
`283.124
`270.802
`258.679
`247.329
`225.692
`215.384
`211.383
`203.590
`195.848
`186.664
`183.089
`177.862
`169.479
`161.216
`143.314
`
`3 Mole fraction solubilities are obtained at best to five Figures to
`llowing the decimal point. Two additional positions have been retained to provide four to six significant
`figures and thus facilitate comparison with calculated solubility values and to compute percentage differences.
`
`The original Hildebrand equation for regular solution behavior cannot
`be used to represent solubility in these polar solvents. However. Eq. 11].
`which involves the interaction term. W. does reproduce exactly the sol-
`ubility of theophyllins in dioxane, water, and the mixed solvent systems.
`Figure 1 shows that the peak solubility. although lower than ideal. oc-
`curred at a 51 value of ~14.0, which was taken as the 53 value of theo-
`phylline. The Fedora method (26) of calculating 6 values from molecular
`group and fragment constants gives essentially the same value (14.1).
`When solubility was plotted as moles per liter instead of as mole
`fraction concentration. a slightly different shape than the curve of Fig.
`1 was obtained. Peaks and valleys were not obtained in the curve of
`theophylline in dioxane -water mixture as reported by Paruta at oi. (27).
`However, two small plateaus were found. These plateaus possibly were
`overlooked because the solubility measurements were not spaced as
`closely in the solvent composition as in the results of‘Paruta et oi. Ongoing
`work in this laboratory with caffeine in dioxene—water has reproduced
`the two-peak maximum reported by Paruta et al. (27).
`
`LDEAL SOL IJIILITV L W!
`
`.
`
`9 ca & an
`
`
`
`
`
`MOLEFRACTIONSOLUBILITYX .0 35
`
`E
`
`K-..._ CALCULIJED SOLUBILITY
`LINE
`
`ticular solvent in the mixture. respectively. and p; is the density of the
`solvent mixture at the experimental temperature.
`_
`Molar Volume and Solubility Parameter for Solute—Tl1e molar
`volume of theophylline taken as a supercooled liquid at 25'' is calculated
`using the group contribution approach of Fedora (26). The solubility
`parameter. 5;, for theophyllins is obtained at the peak solubility where
`the 61 value of the solvent should equal 52 as required by Eq. 1. This
`principle was discussed previously (12). The 02 value of theophylline also
`may be calculated by the Fedora method (26).
`Calculations of Ideal Solubility. Activity Coefficients. and
`W—-The method begins with a calculation of the ideal solubility,
`5. of
`theophylline. employing the first right-hand term of Eq. 1 or 10. The
`logarithmic ideal solubility, together with the logarithm of the experi-
`mentally deten-nined solubility, yields the logarithm of the solute activity
`coefficient:
`
`(Eq.18l
`logxl-logX2=logor2
`Log or, is obtained from the application of Eq. 3. and log on is obtained
`from Eq. 6 or 8. Values for W. the solute—solvent interaction energy. are
`calculated with Eq. 7.
`
`RESULTS AND DISCUSSION
`
`The experimentally determined solubilities of theophylline at 25° in
`dioxane—water mixtures are found in Table I together with the compo-
`sition and densities of the solutions. The densities of solutions are in-
`cluded in reporting solubility data to allow conversion from mole fractions
`to molar concentrations. to assist in obtaining partial molar volumes, and
`to permit the calculation of other quantities. The calculated log org. log
`or”, log cm, and W values also are found in Table I.
`By employing the procedure described under Experimental to yield
`ideal solubility. a M-15,, value of 7097 cslfmolel and T... value of 547.7°1(
`were obtained. Then ASL was calculated using Eq. 13 to yield 8 value of
`12.96 calfmolefdegree and a mole fraction ideal solubility for tlieophylline
`of 0.01896 (log X § = —1.7222). The molar volume of theophylline is
`124.
`The experimental solubilities, expressed as mole fractions, are plotted
`in Fig. 1 against the solubility parameter. 5;, of the various mixed solvents.
`Also shown in Fig. 1 is the ideal solubility level (horizontal line at a mole
`fraction of 0.01896). The regular solution line of Fig. 1 is a curve ex-
`pressing solubilities of theophylline. with the assumption that the mix-
`tures lollow regular solution theory. The solubility of theophylline (52
`'= 14.0) in pure dioxane (51 = 10.01). in pure water {:51 = 23.45), and in the
`binary solutions composed of these two solvents did not approach the
`level of ideality. namely X 2 = 0.019. and did not coincide with regular
`solution behavior except where the experimental curve. by chance.
`crossed the regular solution line.
`
`"' L. T. Grady and W. H. ‘falkosvsky (personal communications) obtained values
`of on-;, varying between 5940 and 7225 csllrnols.
`
`22
`20
`‘I8
`16
`14
`SOLUBILITY PARAMETER. 6,
`Figure 1—Solul:il:'ty of theophylline in diorama. water, and dioram-
`woter mixtures at 25°. Key: I experimental sotubilities; and —
`back—colculored solubilities from Eq. .:'0.
`
`24
`
`Journal of Pharrnaceuflcaf Sciences! 489
`Vol. 69, No. 5. May 7980
`
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 3
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 3
`
`

`
`Table II——CalcuIated Solubilities of Theophylline in Dioxanenwator Systems at 25°
`
`Dioxane, %
`0
`5
`10
`15
`20
`25
`30
`35
`40
`45
`55
`30
`62
`66
`70
`75
`77
`80
`65
`90
`100
`
`5.
`23.45
`22.75
`22.1 1
`21.43
`2036
`20.09
`19.42
`18.'l5
`18.0‘?
`17.40
`16.06
`15.39
`15.12
`14.58
`14.04
`13.37
`13. 10
`l2.T0
`12.03
`1 1.35
`10.01
`
`Wt!
`
`365.501
`350.581
`336.156
`322.012
`308.544
`295.527
`282.947
`270.790
`258.868
`247.517
`225.934
`215.672
`2 1 1 .632
`203.71 1
`195.994
`186.694
`133.028
`177.683
`168.946
`160.345
`1 44.1 18
`
`Log o:2)'Al'
`14.901
`13.766
`12.536
`11.221
`9.891
`6.554
`“L241
`5.982
`4.768
`3.726
`2.056
`1.508
`1.350
`1.155
`1.133
`1.369
`1.553
`1.925
`2.829
`4.133
`7.962
`
`X-3 (calc.)
`0.0008554
`0.001 1013
`0.0014321
`0.0018904
`0.0025018
`0.0033073
`0.0043582
`00056522
`0.00‘? 21 'i4
`0.0089824
`0.01 26004
`0.01 40577
`0.01 45088
`0.0 1 50604
`0.0151452
`0.0144424
`0.0139202
`0.01 29 1 4'?
`0.0107594
`0.0082371‘
`0.0036262
`
`X2 (Dl)s.) - X2 (calc.)
`Difference
`
`0.00012 (16.2%)
`0.00003 (2.8%)
`000013 (8.3%)
`0.00021 (10.0%)
`0.00026 (10.1%)
`0.00021 (6.0%)
`0.00032 (6.8%)
`0.00003 (0.5%)
`0.00053 (7.9%)
`0.00065 (18%)
`0.00116 (10.1%)
`0.00152 (12.1%)
`0.0013’? (10.4%)
`0.00070 (4.9%)
`0.00085 (59%)
`0.00017 (1.2%)
`000034 (2.4%)
`0.00096 (69%)
`0.00256 (19.2%)
`0.0034’? (29.6%)
`000103 (39.7%)
`
`" Baclwsalculated by Eq_ 19. ° Back-calculated by Eq. 2!.
`
`Figure 2 shows the three activity coefficients. log or... log cm, and log
`022. which represent the van der Waals interactions between the solute
`and solvent, the residual term that accounts for stronger interactions,
`and the total solute activity coefficient, respectively. As expressed by Eq.
`6, log org is the sum of log in. and log on. As noted in Fig. 2, log on, is
`plotted using a positive vertical axis (left side), while log on is plotted
`with reference to a negative (right) axis. The positive and negative values
`almost balance each other so that their composite values. represented
`by log {(2. yield only a moderately bowed curve across the range of 51
`values (horizontal axis). This result demonstrates that the nonregularity
`in mixed solvents is not large and. when contrasted to individual solvents.
`provides a greater possibility of predicting solubilities by back-calculation
`as described.
`The usefulness of a theoretical approach is the ability to calculate
`solubilities of a drug in mixed or pure solvents, using only fundamental
`physicochemicsl properties of the solute and solvent. Unfortunately, W
`is not a property that is readily and accurately back-calculated by inde-
`pendent means. The method could be useful for predicting solubilities,
`however. if a procedure were found for estimating W in this range of
`mixed solvents. Then, with Eq. 10, the solubility of theophylline could
`
`be estimated in pure dinxane, pure water, and mixed dioxane—water
`solvents for which the 5,, values were known.
`When W values. obtained from Eq. 7. are plotted against 5:. a curved
`line results. as shown in Fig. 3 for theophylline in dioxane-water. This
`curve suggests that W should be regressed against a polynomial in 61 for
`as many solutions for which accurate experimental solubilities are
`available. With the data of Table 1, the following third~degree (cubic)
`equation was obtained:
`
`(Eq. 19)
`W = 42.12136? + 9312401251 - 0.0052425? + 0.0081635?
`The W values calculated by the cubic expression (Eq. 19) are shown
`in Table Hand are comparable to the original W values (Table 1) calcu-
`lated by Eq. 7. The W values obtained from the cubic polynomial are
`substituted into Eq. 10 to predict the solubility of theophylline in mixed
`solvents. The back-calculated solubilities are recorded in Table II.
`The solid line, passing through the experimental points in Fig. 1, was
`obtained by this procedure. The solubilities are faithfully reproduced
`for solvent mixtures of high 61 values. At the peak of the curve, the ex-
`perimental points fall below the solubility predicted by the theoretical
`line. but the error is not great [<~12%). Solubilitiea represented by the
`points to the left of the peak values are reproduced less well than to the
`right of the peak. The solubilities of theophylline in pure dioxane and
`in pure water are predicted within an error of (40% by this method.
`Solubilities in these pure solvents are quite small. and this percentage
`error is not excessive.
`The drug solubility obtained by this method is expressed in mole
`fraction concentration. It can be converted to molal concentration or to
`grams of solute per gram of solvent. Since the various solution densities
`are known (Table I). solubility also may be expressed in molarity or in
`
`UCO
`
`MHD
`
`fi mI3
`.1 NO /'
`
`
`
`INTERACTIONENERGY,W El) 0
`
`
`
`
`
`ACTIVITYCOEFFICIENT.loga
`
`13
`
`‘I5
`
`11
`
`19
`
`21
`
`23
`
`
`
`HESIDUALACTIVITYCOEFFICIENT,logop
`
`
`
`
`
`
`
`SOLUBI LITY PARAMETER, 5!
`Figure 2— Values for theophylline activity coefficients. log or... log rig,
`and line H2, over the range of solubility parameter values of the mixed
`a'ioxnn:=—u atvr mlt-enr system. Log av and log org are plotted with refv
`create to the vertical axis on the left side of the figure: log on is plotted
`with ref!-'r‘t-‘P'1t'F-'lf1ll'lt’ right‘ side
`
`°s1sn13151r192123
`SOLUBILITY PARAMETER, 5,
`Figure 3——'l'rocing of a computer plot (Eq. 19) of W values against the
`solubility parameter, 5., for theophylline solutions in dioxoncmiater
`mixtures. Paints represent W ualucs calculated from experimental
`solubility data using Eq. F’.
`
`490 I Journal or Pharrrlaceufical Sciences
`Vol. 69. No. 5. May 1980
`
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 4
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 4
`
`

`
`grams of solute per liter or per milliliter of solution.
`The interaction value, W, may be bypassed and log o-JA may be
`baclr-calculated directly. The removal of W occurs by observing from Eq.
`7 that:
`
`2W = I5E+ 5§— log cc-y"A
`
`(Eq. 20)
`
`Substituting Eq. 20 for W in Eq. 19 yields:
`
`log a,;.n. = 111357266 — 13.84502-as.
`+ 1.010-1345? — 0.0163275;
`
`(Eq. 21)
`
`Equations 19 and 21 are analogous and yield identical results except
`for rounding-off errors. The back-calculated log or: values are found in
`Table 11 and may be compared with the original values obtained from
`experimental solubilities found in Table I.
`This method for adapting the Hildebrand approach to polar systems
`has advantages and drawbacks. and certain precautions should be taken
`in its use. The beat 51 values should be used for pure solvents and should
`be accurate to two decimal points where possible. Bagley 2! ol. (28) and
`Nisbet (29) discussed methods for obtaining accurate solvent delta
`values.
`Solute delta values. 52. and molar volumes. V-2. for solids ordinarily are
`not recorded in the literature and are difficult to determine. Ar: inter-
`esting result of the new approach is that solubility predictions do not
`depend on the choice of 15 or V of the solute. Whatever values for these
`quantities were used originally to obtain the W values will. of course.
`remain unchanged in the back-calculation and will not affect the accuracy
`of solubility predictions. However, the investigator must make every
`effort to obtain reasonable values for 52 and V2 and to employ the same
`values each time a solubility analysis is conducted for a particular solute.
`The best possible 52 and V2 values must be estimated and used uniformly
`from one laboratory to another if consistent and reproducible data are
`to be recorded in the literature.
`
`CONCLUSION
`
`The present technique is an extension of the Hildebrand method for
`expressing the solubility of solids in liquid solvents. It should also find
`use in related equilibria studies. The new method extends the Hildebrand
`approach from regular solutions. where van der Waals forces predomi-
`nate, to irregular systems involving stronger solut.e—eolvent interactions
`such as hydrogen bonding and other acid—base interactions.
`The method is not a new physical theory but rather is a technique
`partly based on polynomial regression for back-calculating solubilities
`of drugs and other solutes in polar and nonpolar liquids. In a previous
`report (21) and in cases to be treated later. the procedure may be used
`to reproduce solubilities of drugs in a range of pure solvents. most satis-
`factorily in a particular class of solvents; however, it appears to be con-
`siderably more successful in predicting solubilities in mixed solvent
`systems.
`
`REFERENCES
`
`(1) J. H. Hildebrand, J. Am. Chem. Soc.. 51,66 (1929).
`(2) H. Burrell. Interchem. Reuu, 14.31 (1955).
`(3) H. Burrell, in “Polymer Handbook."J. Brandrup and E. H. Im-
`mergut. Eda, Wiley. New York. N.Y.. 1915. p. IV-337.
`
`(<1) P. A. Small, J. Appi. Chem. 71, 1953.
`(5) C. M. Hansen, Ind. Eng. Chem, Prod. Res. .Deo.. 8, 2 (1969).
`(6) C. M. Hansen and A. Beerbower, in "Encyclopedia of Chemical
`Technology." suppl. vol., 2nd ed.. A. Standen, Ed., Wiley. New York.
`N.Y., 1971.
`'
`('1') R. G. Blanks and J. M. Prausnitz, Ind. Eng. Chem. Fund.. 3. 1
`(1964).
`(B) J. D. Crowley. G. S. Teague. and J. W. Lowe. J. Point Technol,
`39. 293 (1965).
`(9) G. Scatchard, Chem. Red. 3. 321 (1931).
`(10) J. H. Hildebrand and R. L. Scott. "The Solubility of Nonelec-
`trolytes." 3rd ed.. Dover. New York, N.Y.. 1964.
`(11) J. H. Hildebrand. J. M. Prausnitz. and R. L. Scott. “Regular and
`Related Solutions." Van Noetrand Reinhold. New York, N.Y.. 19'i’0.
`(12) M. J. Cbertkoff and A. Martin. J. Am. Phorrn. Assoc, Sci. Ed..
`49. 444 (1960).
`(13) A. Martin. J. Swarbrick, and A. Cammarata. “Physical Phar-
`macy.” 2nd ed., Lea & Febiger, Philadelphia, Pa., 1969. chap. 12.
`(14) F. A. Restaino and A. Martin. J. Phorm. Soil, 53. B36 (1964).
`(15) S. A. Khalil and A. Martin. ibtd., 55, 1225 (1967).
`(16) S. A. Khalil. M. A. Moustafa. and 0. Y. Abdullsh. Can. J. Pharm.
`Sci. l.l,121(1976).
`(17) S. H. Yslkowsky. G. L. Flynn. and T. G. Slunick, J. Phcrm. Sci..
`51. 352 (IQT2).
`(18) K. C. James. C. '1‘. N3. and P. R. Noyce. ibid, 66. 658 (1976).
`(19) K. Hoy. B. A. Price. and R. A. Martin. "Tables of Solubility Pa-
`rameters.” Union Carbide, Tarrytown. N.Y.. 1975.
`(20) F. Weimer and J. M. Prausnitz. Hydrocarbon PrV.‘.Ic'ess._ -14. 237
`(1965).
`(21) A. Martin. J. Newburger. and A. Adjei, J. Pharm. Sci, 68. iv (Oct
`1979).
`(22) J. H. Hildebrand, J. M. Prausnitz. and R. L. Scott. “Regular and
`Related Solutions." Van Noetrancl Reinhold. New York. N.Y.. 1971], p.
`22.
`(23) P. A. Schwartz and A. N. Parute. J. Pharm. Sci, 65. 252
`(1976).
`(24) R. L. Scott and M. Magat. J. Polym. Sci. -1.. 555 (1949).
`(25) L. J. Gordon and R. L. Scott. J. Am. Chem. Soc.. 74. 4138
`1952).
`(25) H. F‘. Fedora, Polym. Eng. Sci. 14. 147 (1974).
`(27) A. N. Paruta, B. J. Sciarrone, and N. G. Lordi, J. Phai-m. Sci, 54,
`833 (1965).
`(28) E. B. Bagley. T. P. Nelson. J. W. Barlow. and S. A. Chen.. I.E.C.
`Fund, 9. 93 (1970).
`(29) K. D. Nisbet. in "Structurea°_.olubility Relationships in Poly-
`mers," F‘. W. Harris and R. B. Seymour. Eds.. Academic. New York. N.Y.,
`197?. chap. 4.
`
`I
`
`ACKNOWLEDGMENTS
`
`Supported in part by the endowed professorship provided to A. Martin
`by Mr. Coulter R. Sublett.
`The authors are grateful to Dr. S. Yalliowsky. The Upjohn Co.; Dr. J.
`M. Prausnitz. University of California; Dr. A. Cammarata. Temple
`University; and P. W. M. John. University of Texas. for valuable sug-
`gestions.
`
`Journal of Phamiaoeuricai Sciences I 491
`Vol. 69. No. 5, May 1930
`
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 5
`MYLAN PHARMS. INC. EXHIBIT 1047 PAGE 5