Ind. Eng. Chem. Res. 2005, 44, 8909-8921
8909
Extension of Nonrandom Two-Liquid Segment Activity Coefficient Model for Electrolytes Chau-Chyun Chen* and Yuhua Song Aspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141
The nonrandom two-liquid segment activity coefficient model of Chen and Song (Ind. Eng. Chem. Res. 2004, 43, 8354) has shown to be a simple and practical tool for chemists and engineers to correlate and estimate solubilities of organic nonelectrolytes in support of chemical and pharmaceutical process design. In this paper, the model is extended for the computation of ionic activity coefficients and solubilities of electrolytes, organic and inorganic, in common solvents and solvent mixtures. In addition to the three types of molecular parameters defined for organic nonelectrolytes, i.e., hydrophobicity X, polarity Y, and hydrophilicity Z, an electrolyte parameter, E, is introduced to characterize both local and long-range ion-ion and ion-molecule interactions attributed to ionized segments of electrolytes. Successful representations of mean ionic activity coefficients and solubilities of electrolytes, inorganic and organic, in aqueous and nonaqueous solvents are presented. Introduction The pharmaceutical industry screens and develops hundreds of new drug candidates each year. Chemists and engineers are tasked to develop process recipes for these new molecules, and the recipes often involve multiple reaction steps coupled with separation steps such as crystallization or extraction. A critical consideration in the pharmaceutical process design is the choice of solvents and solvent mixtures, from among hundreds of solvent candidates, for reaction, separation, and purification.2 Phase behavior, especially solubility, of the new molecules in solvents or solvent mixtures weighs heavily in the choice of solvents for recipe development, while little, if any, such experimental data is available for these new molecules. Although limited solubility experiments may be taken as part of the process research and development, solvent selection today is largely dictated by researchers’ preferences or prior experiences. Predictive thermodynamic models that allow for estimation of phase behavior are desperately needed. Existing solubility parameter models such as that of Hansen3 offer very limited predictive power, while group contribution models such as UNIFAC4 are rather inadequate because of the missing functional groups or the collapse of the functional group additivity rule with large, complex molecules. Recently Chen and Song1 proposed a nonrandom twoliquid segment activity coefficient (NRTL-SAC) model for fast, qualitative correlation and estimation of the solubilities of organic nonelectrolytes in common solvents and solvent mixtures. Conceptually, the approach suggests that one could account for the liquid phase nonideality of mixtures of small solvent molecules and complex pharmaceutical molecules in terms of predefined conceptual segments with predetermined binary interaction characteristics. Examples of the conceptual segments are the hydrophobic segment, the polar segment, the and hydrophilic segment. The numbers of conceptual segments for each molecule, solvent or * To whom correspondence should be addressed. Tel.: (617) 949-1202. Fax: (617) 949-1466. E-mail: chauchyun.chen@ aspentech.com.
solute, reflect the characteristic surface interaction area and nature of the surface interactions. While loosely correlated with molecular structure, they are identified from true behavior of the molecules in solution, i.e., available experimental phase equilibrium data. The molecular makeup in terms of numbers of conceptual segments, i.e., hydrophobicity X, polarity types Y- and Y+, and hydrophilicity Z, constitutes the molecular parameters for the solvent and solute molecules. Given the molecular parameters for solvent and solute molecules, the model offers a thermodynamically consistent expression for the estimation of phase behavior, including solubilities, for organic nonelectrolytes in pharmaceutical process design. It is estimated that half of all drug molecules used in medicinal therapy are administered as salts.5 Here, the NRTL-SAC model for organic nonelectrolytes is extended to account for the liquid-phase nonideality due to the presence of ionic charges in organic electrolytes. We aim to develop a simple and practical qualitative ionic activity coefficient model to aid in the pharmaceutical process design for organic salts. The extension represents a major advance to the scope of the NRTLSAC thermodynamic framework. In the absence of organic moiety, organic electrolytes become inorganic electrolytes. In the absence of ionized moiety, organic electrolytes become organic nonelectrolytes. The electrolyte extension of the NRTL-SAC model (eNRTL-SAC) must provide a consistent and comprehensive thermodynamic framework for systems ranging from organic nonelectrolytes to organic electrolytes and to inorganic electrolytes. In other words, the NRTL-SAC model becomes a limiting case for the eNRTL-SAC model. NRTL Segment Activity Coefficient Model The NRTL-SAC activity coefficient model for component I is composed of the combinatorial term γCI and the residual term γRI :
ln γI ) ln γCI + ln γRI
(1)
Here, the combinatorial term γCI is calculated from the Flory-Huggins equation for the combinatorial entropy
10.1021/ie0503592 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/06/2005
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Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005
of mixing. The residual term γRI is calculated from the local-composition (lc) interaction contribution γlc I of polymer NRTL.6 The polymer NRTL equation incorporates the segment interaction concept and computes the activity coefficient for component I in a solution by summing up contributions to activity coefficient from all segments that make up component I. The equation is given as follows,
ln γRI ) ln γlc I )
lc,I rm,I[ln Γlc ∑ m - ln Γm ] m
with
lnΓlc m
)
∑j xjGjmτjm ∑k xkGkm
+
xm′Gmm′
∑k
xk,IGkm
+
xm′,IGmm′
(
τmm′ ∑ m′ x G ∑k k,I km′ ∑k xk,IGkm′
xj )
∑I ∑i xIri,I
xj,I )
rj,I
∑i
)
∑j xj,IGjm′τjm′
∑I xIrj,I
)
∑j xjGjm′τjm′
τmm′ ∑ m′ x G ∑k k km′ ∑k xkGkm′
ln Γlc,I m )
∑j xj,IGjmτjm
(
(2)
(3)
(4)
(5)
(6)
ri,I
where I is the component index; i, j, k, m, and m′ are the segment species indexes; xI is the mole fraction of component I; xj is the segment-based mole fraction of segment species j; rm,I is the number of segment species m contained only in component I; Γlc m is the activity coefficient of segment species m; and Γlc,I m is the activity coefficient of segment species m contained only in component I. G and τ in eqs 3 and 4 are local binary quantities related to each other by the NRTL nonrandom factor parameter R:
G ) exp(-Rτ)
(7)
Four predefined conceptual segments were suggested by Chen and Song:1 one hydrophobic (x), two polar (yand y+), and one hydrophilic (z). The model molecular parameters, i.e., hydrophobicity X, polarity types Yand Y+, and hydrophilicity Z, correspond to rm,I (m ) x, y-, y+, and z), the number of various conceptual segments in component I. In the notation used throughout this paper, subscript I (upper case) refers to components while subscript i (lower case) refers to segments. eNRTL Segment Activity Coefficient Model. The extension of the NRTL-SAC model for electrolytes is based on the generalized eNRTL model as summarized by Chen and Song.7 Here, we only present the extended NRTL-SAC model.
Electrolytes dissociate to ions in solutions. For “strong” electrolytes, they dissociate “completely” to ionic species. For “weak” electrolytes, they dissociate partially to ionic species, while undissociated electrolytes, similar to nonelectrolytes, remain as neutral molecular species. Complexation of ionic species with solvent molecules or other ionic species may also occur. An implication of the electrolyte solution chemistry is that the extended model should provide a thermodynamically consistent framework to compute activity coefficients for both molecular species and ionic species. In the simplest case of a strong electrolyte CA, we may use the following chemical reaction to describe the complete dissociation of the electrolyte:
CA f υCC+zC + υAA-zA
(8)
υCZC ) υAZA
(9)
with
where υC is the cationic stoichiometric coefficient, υA is the anionic stoichiometric coefficient, ZC is the absolute charge number for cation C, and ZA is the absolute charge number for anion A. In applying the segment contribution concept to electrolytes, we introduce a new conceptual electrolyte segment e. This conceptual segment e would completely dissociate to a cationic segment (c) and an anionic segment (a), both of unity charge. We then follow the like-ion repulsion and the electroneutrality constraints imposed by the eNRTL model to derive the activity coefficient equations for ionic segments c and a. All electrolytes, organic or inorganic, symmetric or unsymmetric, univalent or multivalent, are to be represented with this conceptual uni-univalent electrolyte segment e together with the previously defined hydrophobic segment, x, polar segments, y- and y+, and hydrophilic segment, z. Because we introduce only one conceptual electrolyte segment e, the resulting eNRTL-SAC model is much simpler than the generalized eNRTL model proposed earlier.7 A major consideration in the extension of NRTL-SAC for electrolytes is the treatment of the reference state for activity coefficient calculations. While the conventional reference state for nonelectrolyte systems is the pure liquid component, the conventional reference state for electrolytes in solution is the infinite-dilution aqueous solution and the corresponding activity coefficient is “unsymmetric”. Following the eNRTL model, the logarithm of the unsymmetric activity coefficient of an ionic species, ln γ/I , is the sum of three terms: the local-composition ¨ ckel term, ln γ/PDH ; term, ln γ/lc I ; the Pitzer-Debye-Hu I . and the Flory-Huggins term, ln γ/FH I /PDH + ln γ/FH ln γ/I ) ln γ/lc I + ln γI I
(10)
Equation 10 applies to aqueous electrolyte systems where water is a sole solvent within the solution. For mixed-solvent solutions, the Born term, ∆ln γBorn , is I used to correct the change of the infinite-dilution reference state from the mixed-solvent composition to the aqueous solution for the Pitzer-Debye-Hu¨ckel term: /PDH + ln γ/FH + ∆ln γBorn ln γ/I ) ln γ/lc I + ln γI I I
(11)
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005 8911
Since we adopt the aqueous-phase infinite-dilution reference state for γ/I , the Born term correction is required for nonaqueous systems. With the introduction of the conceptual electrolyte segment e and the corresponding conceptual ionic segments c and a, we can rewrite eq 11 in terms of contributions from all conceptual segments, /PDH ln γ/I ) ln γ/lc + ln γ/FH + ∆ln γBorn I + ln γI I I
)
/PDH rm,I(ln Γ/lc ) + rc,I(ln Γ/lc ∑ m + ln Γm c + m
xj )
) + ln γ/FH (12) ∆ln ΓBorn a I
Gex,lc
where r is the segment number, m is the conceptual molecular segment index (i.e., m ) x, y-, y+, and z), and c and a are the cationic and anionic segments, respectively, resulting from the dissociation of the conceptual electrolyte segment e. Also notice that, in eq 12, the Flory-Huggins term, unlike the local-composition term and the long-range ion-ion interaction terms, remains as the component-based contribution. For systems of single electrolyte CA with a segment number re, rc and ra must satisfy electroneutrality and can be computed from re, ZC, and ZA.
nsRT
(13)
ra,A ) re,CAZA
(14)
For systems of multiple electrolytes, the mixing rule is needed to compute segment number rc and ra for each cation C and anion A.
rc,C )
re,CAZC(xAZA/∑xA′ZA′) ∑ A A′
(15)
ra,A )
re,CAZA(xCZC/∑xC′ZC′) ∑ C C′
(16)
)
Gex,lc nRT
)
rc,IxI
with
[ ( ) ( ) ( )]
rm,IxI ∑I ∑ m
∑k xkGkc,ac
xm ∑ m
( )( ) ( ) ∑j xjGjmτjm
+ ra,IxI
ln Γlc j )
∑k xkGka,ca
∑j xjGjc,acτjc,ac
+
∑k xkGkc,ac
∑j xjGja,caτja,ca ∑k xkGka,ca
( )
1 ∂Gex,lc RT ∂nj
(19)
i, j ) m, c, a (20)
T,P,ni*j
Specifically, the activity coefficients from eq 20 for molecular segments, cationic segment, and anionic segment can be carried out as follows:
∑x G j
Γlc m
)
jmτjm
j
+
∑x G k
km
(
k
xcGmc,ac
∑x G k
xm′Gmm′
∑ ∑x G m′
k
ln
Γlc c
)
xmGcm
j
j
τmc,ac -
∑x G k
kc,ac
k
∑x G j
τma,ca -
xkGka,ca
(
∑k xkGkc,ac
+
∑
)
xa
)
ja,caτja,ca
j
∑j xjGjmτjm
-
km′
xkGka,ca
k
τcm ∑ m ∑k xkGkm ∑k xkGkm
∑j xjGjc,acτjc,ac
k
k
+
jc,acτjc,ac
j
(
∑x G
)
)
jm′τjm′
j
τmm′ -
∑x G
xaGma,ca
∑
(
∑x G
km′
k
kc,ac
k
k
(17)
+ xc
∑k xkGkm
+
∑j xjGja,caτja,ca
(18)
where ns is the total number of all segments. Accordingly, the segment activity coefficient can be calculated as follows:
ln
∑k xkGkm
∑j xjGjc,acτjc,ac
∑I ∑i xIri,I
xa
re,CA, the number of conceptual electrolyte segments e in electrolyte CA, becomes the new model parameter for electrolytes. For the sake of brevity, we call re,CA parameter E, the electrolyte segment number. Local-Composition Interaction Contribution. To derive the expression for the local-composition interaction contribution, we simplify the generalized excess Gibbs energy expression of Chen and Song7 for systems with multiple molecular segments m and a single electrolyte segment e. The single electrolyte segment e is then decomposed into a cationic segment c and an anionic segment a:
∑j xjGjmτjm
i, j ) m, c, a
where Gex,lc is the excess Gibbs energy from localcomposition interactions, n is the total mole number, R is the gas constant, and T is the temperature. To derive the segment activity coefficient, we can rewrite eq 17 as follows,
/PDH ln Γ/PDH + ∆ln ΓBorn ) + ra,I(ln Γ/lc + c c a + ln Γa
rc,C ) re,CAZC
∑I xIrj,I
(21)
+
( ) ∑j xjGja,caτja,ca
∑k xkGka,ca ∑k xkGka,ca
(22)
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(
)
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005
ln Γlc a )
xmGam
∑j xjGjmτjm
τam ∑ m ∑k xkGkm ∑k xkGkm
∑j xjGja,caτja,ca ∑k xkGka,ca
(
)
+
)
xmGmc,acτmc,ac ∑ m
xc
-
lc ∞lc ln γ/lc I ) ln γI - ln γI
∑k xkGkc,ac
∑k xkGkc,ac
)
)
∑i ri,I ln Γlci
)
rm,I ln ∑ m
+ rc,I ln
Γlc c
+ ra,I ln
ln
) )
∑i ri,I[ln
Γlc i
(24)
- ln
Γlc,I i ]
lc lc,I ln Γlc,I c ] + ra,I[ln Γa - ln Γa ] (25)
is the activity coefficient of the segment i Here, Γlc,I i contained in the symmetric reference state of component I; it can be calculated from eqs 21-23 by setting xI ) 1:
(27)
with
)
(31)
lc ∞lc ln Γ/lc a ) ln Γa - ln Γa
(32)
i ) m, c, a
(33)
where xW is the mole fraction of water in the solution. Long-RangeInteractionContributionfromPitzerDebye-Hu 1 ckel (PDH) Model. To account for the longrange ion-ion interactions, the eNRTL-SAC model uses the unsymmetric Pitzer-Debye-Hu¨ckel (PDH) formula8 on the segment basis,
( )( )
1000 G*ex,PDH )nSRT M hS
4AφIx ln(1 + FI1/2 x ) (34) F
1/2
with
lc ∞lc ln γ/lc I ) ln γI - ln γI
- ln Γlc,I ∑i ri,I[ln Γ∞lc i i ]
lc ∞lc ln Γ/lc c ) ln Γc - ln Γc
(26)
Finally, the unsymmetric convention in eq 11 requires us to compute the infinite-dilution activity coefficient, γ∞lc I , for a component:
ln γ∞lc I )
(30)
Because we adopt the aqueous-phase infinite-dilution reference state, the infinite-dilution activity coefficients of conceptual segments can be calculated from eqs 2123 by setting xW ) 1,
i ) m, c, a
i ) m, c, a
lc ∞lc lnΓ/lc m ) ln Γm - ln Γm
) ln Γlc ln Γ∞lc i i (xW ) 1)
lc,I lc rm,I[ln Γlc ∑ m - ln Γm ] + rc,I[ln Γc m
) ln Γlc ln Γlc,I i i (xI ) 1)
/lc /lc rm,Iln Γ/lc ∑ m + rc,I ln Γc + ra,I ln Γa m
with
Γlc a
However, the activity coefficient by eq 24 needs to be further normalized so that γlc I ) 1 as xI f 1 for any component; this is the so-called symmetric reference state. The normalization can be done as follows:
γlc I
∞lc lc rm,I[ln Γlc ∑ m - ln Γm ] + rc,I[ln Γc m
(29)
i ) m, c, a
Γlc m
i ) m, c, a
lc ∞lc ln Γ∞lc c ] + ra,I[ln Γa - ln Γa ]
(23)
The local-composition term for the logarithm of the activity coefficient of component I is computed as the sum of the individual segment contributions.
ln γlc I )
∑i ri,I[ln Γlci - ln Γ∞lc i ]
i ) m, c, a
lc,I ∞lc rm,I[ln Γ∞lc ∑ m - ln Γm ] + rc,I[ln Γc m ∞lc lc,I ln Γlc,I c ] + ra,I[ln Γa - ln Γa ] (28)
Combining eqs 25 and 28, we can obtain:
(
)( )
hS 1 2πNAd Aφ ) 3 1000 Ix ) 1/2
1/2
Qe2 jSkBT
∑i xizi2
3/2
(35) (36)
where Aφ is the Debye-Hu¨ckel parameter, Ix is the ionic strength (segment mole fraction scale), M h S is the average molecular weight of the mixed solvents, F is the closest approach parameter, NA is Avogadro’s number, d h S is the average density of the mixed solvents, Qe is the electron charge, jS is the average dielectric constant of the mixed solvents, kB is the Boltzmann constant, and zi (zm ) 0; zc ) za ) 1) is the charge number of segmentbased species i. Applying the PDH model to the conceptual segments, the activity coefficient of segment species i can be derived as follows:
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005 8913
ln Γ/PDH ) i
(
) [( )
1 ∂G*ex,PDH RT ∂ni
( )
1000 )M hS
1/2
Aφ
i, j ) m, c, a
T,P,nj*i
2zi2 ln(1 + FI1/2 x ) + F zi2I1/2 x
]
- 2I3/2 x 1/2 + FIx
1
w′S ) (37)
The unsymmetric long-range term for the logarithm of the activity coefficient of component I is the sum of the contributions from its various segments:
) ln γ/PDH I
rm,I ln Γ/PDH + rc,I ln Γ/PDH + ra,I ln Γ/PDH ∑ m c a m
(38)
where
( ) 1000 M hS
)2 ln Γ/PDH m
ln
Γ/PDH c
) ln
Γ/PDH a
x′S )
1/2
AφI3/2 x
( ) [( )
1000 )M hS
(39)
1 + FI1/2 x
1/2
Aφ
FI1/2 x ) +
2 ln(1 + F
]
3/2 I1/2 x - 2Ix
1 + FI1/2 x
(40)
(
)( )
hS 1 2πNAd Aφ ) 3 1000
1/2
Qe2 jSkBT
1 Ix ) (xc + xa) 2
3/2
(41) (42)
∆GBorn
M hS) 1 d hS
)
jS ) with
∑S x′SMS x′S
∑S d
∑S
(43)
)
(
Qe2 1 2kBT jS
) ∆ln ΓBorn i
(
)
( (
) )
1 ∂∆GBorn RT ∂nm 1 ∂∆GBorn RT ∂ni
(47)
∑S MSxS
-
)∑
1
W
xizi2 ri
i
× 10-2
(48)
)0
T,P,nj*m
m ) x, y -, y+, z (49) )
T,P,nj*i
i ) c, a (50)
The Born correction term on the logarithm of the activity coefficient of component I is the sum of contributions from its various segments:
) rc,I ∆ln ΓBorn + ra,I ∆ln ΓBorn ∆ln γBorn I c a
(44) ∆ln
ΓBorn a
( (
) )
(51)
Qe2 1 1 1 × 10-2 2kBT jS W rc
(52)
Qe2 1 1 1 ) × 10-2 2kBT jS W ra
(53)
∆ln ΓBorn ) c
(45)
MSxS
2 Qe2 1 1 zi × 10-2 2kBT jS W ri
S
w′SS
∑S xS
∆GBorn is the Born term correction to the unsymmetric Pitzer-Debye-Hu¨ckel formula, G*ex,PDH; W is the dielectric constant of water and ri is the Born radius of segment species i. Applying eq 48 to all conceptual segments, the corresponding expression for the activity coefficient of segment species i can be derived as follows:
) ∆ln ΓBorn m The Debye-Hu¨ckel theory is based on the infinitedilution reference state for ionic species in the actual solvent media. For systems with water as the only solvent, the reference state is the infinite-dilution aqueous solution. For mixed-solvent systems, the reference state for which the Pitzer-Debye-Hu¨ckel formula remains valid is the infinite-dilution solution with the corresponding mixed-solvent composition. Consequently, the molecular quantities for the single solvent need to be extended for mixed solvents; simple compositionaverage mixing rules are adequate to calculate them as follows,
(46)
where S is a solvent component in the mixture and MS is the molecular weight of the solvent S. It should be pointed out that eqs 43-47 should be used only in eq 37 and M h S, d h S, and jS were already assumed as constants in eqs 34 and 35 when deriving eq 37 for mixed-solvent systems. Table 1 shows the values of dielectric constant at 298.15 K used in this study for the same 62 solvents investigated by Chen and Song.1 These values were compiled from various sources including Internet websites and commercial software Aspen Properties. Born Term Correction to Activity Coefficient. Given that the infinite-dilution aqueous solution is chosen as the reference state, we need to correct the change of the reference state from the mixed-solvent composition to the aqueous solution for the PitzerDebye-Hu¨ckel term. The Born term9,10 on the segment basis is used for this purpose:
nSRT
with
xS
Flory-Huggins Term Correction to Activity Coefficient. Although in most common electrolyte systems, the combinatorial entropy of the mixing term
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Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005
xIrI
Table 1. Dielectric Constant of Solvents at 298.15 K solvent name
dielectric constant at 298.15 K
acetic acid acetone acetonitrile anisole benzene 1-butanol 2-butanol n-butyl acetate methyl tert-butyl ether carbon tetrachloride chlorobenzene chloroform cumene cyclohexane 1,2-dichloroethane 1,1-dichloroethylene 1,2-dichloroethylene dichloromethane 1,2-dimethoxyethane N,N-dimethylacetamide N,N-dimethylformamide dimethylsulfoxide 1,4-dioxane ethanol 2-ethoxyethanol ethyl acetate ethylene glycol diethyl ether ethyl formate formamide formic acid n-heptane n-hexane isobutyl acetate isopropyl acetate methanol 2-methoxyethanol methyl acetate 3-methyl-1-butanol 2-hexanone methylcyclohexane methyl ethyl ketone methyl isobutyl ketone isobutanol n-methyl-2-pyrrolidone nitromethane n-pentane 1-pentanol 1-propanol isopropyl alcohol n-propyl acetate pyridine sulfolane tetrahydrofuran 1,2,3,4-tetrahydronaphthalene toluene 1,1,1-trichloroethane trichloroethylene m-xylene water triethylamine 1-octanol
6.13 20.83 36.97 4.3 2.27 17.7 15.8 5.1 2.6 2.23 5.56 4.7 2.22 2.02 10.19 4.6 4.6 8.9 not available not available 38.3 47.2 2.21 24.11 not available 6.02 41.2 4.26 7.16 109.5 58.5 1.92 1.89 5.6 not available 32.62 not available 6.68 14.7 14.6 2.02 18.5 13.1 17.9 33 6.26 1.84 13.9 20.1 19.9 6 2.3 43.3 7.52 not available 2.36 7.5 3.42 2.24 78.54 2.44 10.3
nRT with
)
() φI
∑I xI ln x
I
(55)
∑J xJrJ
where Gex,FH is the Flory-Huggins term for the excess Gibbs energy, φI is the segment fraction of component I, and rI is the number of all conceptual segments in component I:
rI )
rm,I + rc,I + ra,I ∑ m
(56)
The activity coefficient of component I from the combinatorial term can be derived from eq 54:
()
ln γFH ) ln I
φI xI
+ 1 - rI
φJ
∑J r ln
)
J
( ) rI
∑J
+1-
xJrJ
rI
∑J
(57)
xJrJ
The infinite-dilution activity coefficient of a component in water is
) ln ln γ∞FH I
()
rI rI +1rW rW
(58)
In both NRTL-SAC and eNRTL-SAC, water is selected as the reference for the hydrophilic segment z. Therefore, we can set rW ) 1. Thus, we have
ln γ∞FH ) ln rI + 1 - rI I
(59)
We can then compute the unsymmetric activity coefficient from the Flory-Huggins term as follows:
) ln γFH - ln γ∞FH ) ln γ/FH I I I
∑J
rI - ln(
xJrJ) -
rI
∑J xJrJ
(60)
Model Parameters
is much smaller than that of the residual term, we may still want to include it in a general model. We follow the polymer NRTL model6 and use the Flory-Huggins term to describe the combinatorial term:
Gex,FH
φI )
(54)
NRTL Binary Parameters. In eqs 3 and 4 for NRTL-SAC, the model formulation requires the asymmetric interaction energy parameters, τ, and the symmetric nonrandom factor parameters, R, for each binary pair of the conceptual segments. In eqs 21-23 for eNRTL-SAC, we need additional binary parameters of τ and R between conceptual molecular segments, m, and ionic segments, c or a. In practice, we fix the values of R’s for the binary pairs of molecular segment and ionic segment to the single value of 0.2, while the values of τ for the binary pairs of molecular segment and ionic segment are calculated from the τ’s for the binary pairs of molecular segment and electrolyte segment. Following the same scheme in eNRTL,7 we can calculate these binary interaction energy parameters as follows:
τcm ) τam ) τem
(61)
τmc,ac ) τma,ca ) τme
(62)
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005 8915 Table 2. NRTL Binary Interaction Parameters segment (1) segment (2)
x y-
x z
yz
y+ z
x y+
τ12 τ21 R12 ) R21
1.643 1.834 0.2
6.547 10.949 0.2
-2.000 1.787 0.3
2.000 1.787 0.3
1.643 1.834 0.2
segment (1) segment (2)
x e
ye
y+ e
z e
τ12 τ21 R12 ) R21
15 5 0.2
12 -3 0.2
12 -3 0.2
8.885 -4.549 0.2
Following the treatment of NRTL-SAC, we identify a reference electrolyte for the conceptual electrolyte segment e. In searching for the reference electrolyte, we choose one elemental electrolyte that has abundant literature data. In this study, NaCl is used as the reference electrolyte for e. The ionic radii for sodium ion and chloride ion are 1.680 × 10-10 m and 1.937 × 10-10 m, respectively.7 With NaCl as the reference electrolyte, the energy parameters for the z-e pair are set to (8.885, -4.549), as reported by Chen et al.11 for the waterNaCl pair. The energy parameters for the x-e pair are set to (15, 5), in line with the parameters identified earlier for the C2H4-NaCl pair by Chen and Song.7 The energy parameters for the y-e pairs are set to (12, -3) after limited trials to optimize the performance of the model in this study. The complete set of NRTL binary interaction energy parameters is given in Table 2. While the choice of the reference electrolyte and parameter values should be further investigated and optimized, the manuscript reports the general behavior of the eNRTLSAC model based on the parameters reported in Table 2. The electrolyte segment e is the only extra molecular descriptor, and the electrolyte parameter E is the only extra molecular parameter, for all electrolytes, inorganic or organic. All local and long-range interactions derived from the existence of cationic and anionic species of various ionic charge valence, radius, chemical makeup, etc., are to be accounted for with this extra molecular descriptor for electrolytes, together with combinations of conceptual molecular segments, i.e., hydrophobicity, polarity, and hydrophilicity. In other words, every electrolyte, organic or inorganic, will be modeled as combinations of E, X, Y, and Z. As such, electrolytes are recognized as “hydrophobic” electrolytes, “polar” electrolytes, “hydrophilic” electrolytes, and their various combinations. Likewise, the ionic activity coefficient of each ionic species will be computed from its share of E, X, Y, and Z. The ions are to be considered as “hydrophobic” ions, “polar” ions, or “hydrophilic” ions. Figures 1-5 show the effects of the molecular parameters on the mean ionic activity coefficients (mole fraction scale) of the reference electrolyte, i.e., electrolyte with E ) 1. As shown in Figures 1-5, the hydrophobicity parameter X brings down the mean ionic activity coefficient at a low electrolyte concentration, but in a rather nonlinear way. Polarity parameter Y- raises the mean ionic activity coefficient, while polarity parameter Y+ lowers the mean ionic activity coefficient. Hydrophilicity parameter Z has a relatively slight downshift effect on the mean ionic activity coefficient. Electrolyte parameter E brings down the mean ionic activity coefficient at low electrolyte concentration and pushes up the mean ionic activity coefficient at high electrolyte concentration.
Figure 1. Effect of hydrophobicity parameter X on the natural logarithm of mean ionic activity coefficient of aqueous electrolytes with E ) 1.
Figure 2. Effect of polarity parameter Y- on the natural logarithm of mean ionic activity coefficient of aqueous electrolytes with E ) 1.
Figure 3. Effect of polarity parameter Y+ on the natural logarithm of mean ionic activity coefficient of aqueous electrolytes with E ) 1.
Experimental data for ionic activity coefficients are not readily available, though they are emerging.12 Given the fact that existing experimental data are limited to mean ionic activity coefficients for neutral electrolytes, we are not able to directly identify the molecular parameters for ionic species. In preparing Figures 1-5 discussed above and the subsequent studies reported in the Model Applications Section, we use eqs 13 and 14 to determine from electrolyte parameter E the ionic segment numbers for the ions and we arbitrarily assign molecular segment parameters (X, Y-, Y+, and Z) only to the anion. This practice is acceptable, since virtually all electrolytes investigated in this study are electrolytes with elemental cations. This practice should be revised if and when single-ion activity coefficients can be routinely and reliably measured.
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Figure 4. Effect of hydrophilicity parameter Z on the natural logarithm of mean ionic activity coefficient of aqueous electrolytes with E ) 1.
Figure 5. Effect of electrolyte parameter E on the natural logarithm of mean ionic activity coefficient of aqueous electrolytes.
Model Applications Limited amounts of mean ionic activity coefficient data are available in the public literature for aqueous electrolytes. We test the eNRTL-SAC model against mean ionic activity coefficient data of aqueous electrolyte systems. In addition, we test the model against salt solubility data in multiple solvents for a number of inorganic and organic electrolytes. To the best of our knowledge, public literature data is very scarce for such salt solubility data. Proprietary solubility data from our industrial collaborators was also used to test the applicability of the model. However, results with such proprietary solubility data are not included in this manuscript. Mean Ionic Activity Coefficients in Aqueous Systems. For an electrolyte CA that dissociates to cation C and anion A, the mean ionic activity coefficients γ/( are related to individual ionic activity coefficients as follows,
1 ln γ/( ) (νC ln γ/C + νA ln γ/A) ν
(63)
where ν ) νC + νA. Equation 63 gives the mean ionic activity coefficient on the mole fraction scale, and it can be converted to the molality scale, / ) ln γ/( - ln(1 + νmMS/1000) ln γ(m
(64)
/ is the mean ionic activity coefficient on the where γ(m molality scale, m is the molality of the salt (mol/(kg of solvent)), and MS is the molecular weight of the solvent (g/mol).
Table 3 shows the fit to molality scale mean ionic activity coefficient data and the identified electrolyte and molecular parameters for the aqueous inorganic and organic electrolytes at 298.15 K, as compiled by Robinson and Stokes.9 All mean ionic activity coefficient data are assumed to have a standard deviation of 5%. The data for C5 and higher sodium carboxylates were excluded from the fit because these organic electrolytes were known to form micelles at high electrolyte concentrations.13 With a few exceptions such as LiBr, most uni-univalent and uni-bivalent electrolytes are well represented as combinations of E and Y- or Y+ parameters. Most uni-univalent electrolytes have an E parameter ∼1, while higher E values are found for higher valent electrolytes. We also found that the fit seems to deteriorate for electrolytes with higher E values. This observation is consistent with the understanding that higher valent electrolytes are known to be prone to the formation of hydrated species or other complexation species. The relatively poor representation of these electrolytes with the model reflects the inadequate assumption of complete dissociation for such electrolytes.14 As a derived property, the mean ionic activity coefficient becomes meaningless if the complete dissociation assumption of electrolytes does not hold true. To illustrate the quality of the fit, Figure 6 shows the comparison of experimental and calculated molality scale mean ionic activity coefficients for five aqueous electrolytes at 298.15 K. The solid lines are the calculated values from the model. It shows that the eNRTLSAC model provides reasonable qualitative representation of the data, while the original eNRTL model11 achieves excellent quantitative representation of the data. Salt Solubility in Mixed-Solvent Systems. At the solubility limit of nonelectrolytes, the solubility product constant, Ksp, can be written in terms of the product of the solute concentration and the solute activity coefficient at the saturation concentration:
Ksp ) xIγI
(65)
At the solubility limit of an electrolyte, ionic species precipitate to form salt.
vCC+ZC + vAA-ZA f CvCAvA(s)
(66)
The corresponding solubility product constant can be defined as follows. vA /vA C Ksp ) xvCC γ/v C x A γA
(67)
Equations 66 and 67 can be expanded to include solvent molecules and other species if the solid polymorph involves hydrates, other solvent-containing salts, double salts, triple salts, and others. We tested the applicability of the eNRTL-SAC model with the very limited public literature data and some proprietary data on the solubilities of a number of inorganic and organic electrolytes in various solvents. This manuscript presents the results with solubility data from public literature. To bring certain consistency to the data treatment, we convert all solute solubility data to mole fraction (except for sodium chloride and sodium acetate). We also assign a standard deviation of 10% to all solute solubility data within the range of
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005 8917 Table 3. Results of Fit for Molality Scale Mean Ionic Activity Coefficient Data of Aqueous Electrolytes at 298.15 K (Data from Reference 9) Y+
σa
max. molality
1.758
0.519 0.466 0.520 1.611 0.952 1.231 1.692
0.050 0.011 0.013 0.014 0.012 0.005 0.002 0.034 0.087 0.136 0.035 0.005 0.009 0.011 0.002 0.010 0.003 0.019 0.004 0.022 0.011 0.006 0.011 0.027 0.058 0.026 0.002 0.116 0.084 0.047 0.033 0.022 0.022 0.014 0.005 0.008 0.010 0.009 0.017 0.011 0.010 0.006 0.002 0.013 0.019 0.010 0.020 0.012 0.029 0.039 0.006 0.026 0.019 0.043 0.011 0.016 0.012 0.014 0.038 0.033 0.000 0.003
6.0 3.5 5.0 6.0 3.0 1.4 1.0 3.0 6.0 6.0 3.0 3.0 3.5 5.5 0.5 4.5 0.7 5.0 4.0 5.0 4.5 1.8 4.5 3.5 6.0 3.5 4.0 6.0 6.0 4.0 3.0 6.0 4.0 4.5 3.5 4.0 2.5 3.5 6.0 3.5 6.0 4.0 1.0 3.5 5.0 5.0 6.0 3.5 6.0 6.0 3.0 4.0 6.0 6.0 3.5 5.0 5.0 5.0 4.5 6.0 0.5 0.4
1.161 1.048 1.386 1.138 1.091 1.259 1.202 0.988 1.071 1.006 1.150
2.568 2.738 2.475 2.177 2.443 1.770 2.699 3.273 2.709 3.477 2.743
0.050 0.075 0.021 0.051 0.051 0.041 0.075 0.090 0.064 0.118 0.052
1.8 3.5 0.7 3.0 4.0 2.0 3.0 4.0 3.5 4.0 1.8
1.328
4.996
0.101
1.4
E 1-1 electrolytes AgNO3 CsAc CsBr CsCl CsI CsNO3 CsOH HBr HCl HClO4 HI HNO3 KAc KBr KBrO3 KCl KClO3 KCNS KF KH malonate KH succinate KH2PO4 KI KNO3 KOH K Tol LiAc LiBr LiCl LiClO4 LiI LiNO3 LiOH LiTol NaAc NaBr NaBrO3 Na butyrate NaCl NaClO3 NaClO4 NaCNS NaF Na formate NaH malonate NaH succinate NaH2PO4 NaI NaNO3 NaOH Na propionate Na Tol NH4Cl NH4NO3 RbAc RbBr RbCl RbI RbNO3 TlAc TlClO4 TlNO3 1-2 electrolytes Cs2SO4 K2CrO4 K2SO4 Li2SO4 Na2CrO4 Na2 fumarate Na2 maleate Na2SO4 Na2S2O3 (NH4)2SO4 Rb2SO4 1-3 electrolytes K3Fe(CN)6
Y-
0.738 1.002 0.950 0.948 0.956 0.981 0.942 1.135 1.324 1.476 1.117 0.971 0.998 0.910 0.968 0.920 0.958 0.876 0.987 0.846 0.912 0.970 0.903 0.856 1.236 0.750 0.962 1.422 1.282 1.145 1.058 1.050 1.028 0.881 0.978 0.992 0.923 0.989 1.000 0.891 0.894 0.925 0.976 0.905 0.878 0.924 0.864 1.009 0.825 1.080 0.992 0.793 0.884 0.813 1.012 0.914 0.929 0.925 0.815 0.864 1.020 1.069
0.438 0.678 0.643 0.719 1.328 0.354 0.654 0.524 0.569 0.824 0.211 0.386 0.311 1.141 0.370 1.053 0.477 0.042 0.920 0.665 1.362 0.168 1.461 0.344 1.296 0.097 0.526 0.436 0.681 0.712 0.294 0.652 0.392 0.301 0.115 0.802 0.566 0.507 0.267 0.128 0.425 0.094 0.664 0.495 1.256 0.266 0.842 0.109 0.448 0.920 0.424 1.128 0.416
/exp [∑N i ((γ(i
/cal /exp 2 γ(i )/γ(i ) /N]1/2
σ is defined to be of data points used in the correlations. a
Y+
σa
max. molality
1.449
9.448
0.146
0.9
1.016 1.267 1.227 1.305 1.354 1.435 1.969 1.701 2.021 1.419 1.108 1.324 1.052 1.780 1.176 1.779 1.397 2.260 1.444 1.033 1.409 1.319 1.192 1.941 1.745 1.988 2.237 1.493 1.273 1.533 1.549 1.129 1.330 1.401 1.742 1.384 0.978 1.277 2.854 1.392 0.906 0.953 2.045 0.868 1.518
0.997 0.358 0.585 0.261
0.214
0.128 0.018 0.029 0.049 0.017 0.008 0.495 0.283 0.431 0.036 0.053 0.294 0.315 0.337 0.037 0.218 0.046 0.488 0.113 0.069 0.117 0.011 0.059 0.347 0.275 0.303 0.470 0.140 0.020 0.123 0.184 0.083 0.023 0.082 0.261 0.030 0.091 0.017 0.883 0.036 0.088 0.065 0.318 0.116 0.176
3.5 2.0 1.8 5.0 2.0 0.4 6.0 6.0 6.0 2.0 6.0 4.0 6.0 2.5 2.5 5.0 4.0 6.0 5.0 6.0 6.0 2.0 4.0 5.0 5.0 4.0 5.0 5.0 6.0 5.0 6.0 2.0 2.0 4.0 6.0 2.0 4.0 3.0 5.5 5.5 6.0 6.0 4.0 6.0 6.0
1.376 1.380 1.287 1.398 1.587 1.339 1.295 1.215
4.077 4.206 4.460 4.381 4.114 4.417 4.547 4.528
0.233 0.238 0.271 0.220 0.154 0.242 0.271 0.309
4.0 3.0 4.0 2.5 1.4 3.5 3.5 6.0
1.730 1.562 1.589 1.551 1.586 1.553 1.575 1.562 1.636 1.581 1.629
0.579 0.883 0.641 0.761 0.820 0.877 0.882 0.892 0.709 0.843 0.807
0.087 0.047 0.022 0.036 0.049 0.042 0.045 0.042 0.041 0.046 0.057
1.8 1.8 1.2 1.4 2.0 2.0 2.0 2.0 1.8 2.0 2.0
1.354 1.257
4.886 4.549
0.222 0.218
1.0 1.2
1.273
1.251
0.056
5.0
E 1-4 electrolytes K4Fe(CN)6 2-1 electrolytes BaAc2 BaBr2 BaCl2 Ba(ClO4)2 BaI2 Ba(NO3)2 CaBr2 CaCl2 Ca(ClO4)2 CaI2 Ca(NO3)2 CdBr2 CdCl2 CdI2 Cd(NO3)2 CoBr2 CoCl2 CoI2 Co(NO3)2 CuCl2 Cu(NO3)2 FeCl2 MgAc2 MgBr2 MgCl2 Mg(ClO4)2 MgI2 Mg(NO3)2 MnCl2 NiCl2 Pb(ClO4)2 Pb(NO3)2 SrBr2 SrCl2 Sr(ClO4)2 SrI2 Sr(NO3)2 UO2Cl2 UO2(ClO4)2 UO2(NO3)2 ZnBr2 ZnCl2 Zn(ClO4)2 ZnI2 Zn(NO3)2 2-2 electrolytes BeSO4 MgSO4 MnSO4 NiSO4 CuSO4 ZnSO4 CdSO4 UO2SO4 3-1 electrolytes AlCl3 CeCl3 CrCl3 Cr(NO3)3 EuCl3 LaCl3 NdCl3 PrCl3 ScCl3 SmCl3 YCl3 3-2 electrolytes Al2(SO4)3 Cr2(SO4)3 4-1 electrolytes Th(NO3)4
Y-
0.028 1.268 0.171 0.309 0.131 0.875 3.164 3.047 3.820 0.500 0.194 0.425
0.296 1.217 0.416 0.255 0.946 0.144
0.162 0.198 0.343 0.189 0.236 1.964 0.183 0.357 0.034 0.076 1.250 0.024 0.372
0.490 0.337 0.971
0.130 0.132
where γ/( is the mean ionic activity coefficient of the electrolyte and N is the number
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Figure 6. Comparison of experimental and calculated molality scale mean ionic activity coefficients of representative aqueous electrolytes at 298.15 K.
Figure 7. Model results for sodium chloride solubility at 298.15 K.
1-0.1, a standard deviation of 20% to all solute solubility data within the range of 0.1-0.01, a standard deviation of 30% to data within the range of 0.01-0.001, and so on. Solubility data of sodium chloride in 12 different solvents at 298.15 K15 were successfully fitted with the model. (Note that the temperature for the acetone data is 291.15 K and the temperature for the ethyl acetate data is 292.15 K. However, they are included as if they were data at 298.15 K.) The sodium chloride solubilities in the 12 solvents vary by 6 orders of magnitude. The satisfactory fit of the data for 10 solvents (formic acid and ethyl acetate excluded) is shown in Figure 7. The model predicts 1 order-of-magnitude higher solubility for sodium chloride in formic acid and virtually no solubility for sodium chloride in ethyl acetate, while the data suggests very low but measurable solubility for sodium chloride in ethyl acetate. The molecular parameters and the solubility product constant were adjusted simultaneously to provide the best fit to the data, and the identified values are given in Table 4. It is worth noting that the electrolyte parameter E for sodium chloride is ∼1, similar to the parameters reported in Table 3 for sodium chloride. Solubility data of sodium acetate in five different solvents15 was also fitted successfully with the model. The solubilities in the five solvents vary by 4 orders of magnitude. The fit of the data is shown in Figure 8. The solid phase for the solubility measurements is anhydrous sodium acetate. Note that the data for methanol and acetone was taken at 291.15 K while the data for
water and ethylene glycol was taken at 298.15 K. The temperature for the 1-propanol data is not known. In fitting the data, we treated all data as if it was 298.15 K data. The identified molecular parameters and the solubility product constant are given in Table 4. As an organic electrolyte, the electrolyte parameter E for sodium acetate is found to be ,1. Parts a and b of Figure 9 show satisfactory representations of the solubility data of benzoic acid in 26 solvents16 and the solubility data of sodium benzoate in 10 solvents,17 respectively. These solvents are chosen in this study because of the availability of the NRTLSAC parameters for the solvents from our prior work.1 The identified molecular parameters for the two solutes were given in Table 4. It is interesting that the molecular parameters identified for benzoic acid with 26 solvents in this study are quite similar to the molecular parameters identified for benzoic acid with 7 solvents in our earlier study.1 We also noted that the solubility range expands as benzoic acid is converted to sodium benzoate. Furthermore, the molecular parameters have changed from a hydrophobic/polar/hydrophilic combination (benzoic acid) to a polar/hydrophilic/electrolytic combination (sodium benzoate). Solubility data of sodium benzoate in 7 other solvents (chloroform, benzene, dioxane, cyclohexane, ethyl acetate, heptane, and chlorobenzene) is excluded from Figure 9b because the model predicts virtually no solubility for sodium benzoate in these solvents, while the data suggests very low but measurable solubility. It is probable that the molecular form of sodium benzoate may be present in such highly hydrophobic solvents. However, because of their low concentrations, we chose to ignore these low solubility solvents in this study, although the current thermodynamic framework can be used to account for the two solubility routes, i.e., eqs 65 and 67, individually or simultaneously. Parts a and b of Figure 10 show successful representations of the solubility data of salicylic acid in 18 solvents and the solubility data of sodium salicylate in 13 solvents,18 respectively. Their molecular parameters were given in Table 4. Like the molecular parameters for benzoic acid and the sodium salt, the molecular parameters have changed from a hydrophobic/polar/ hydrophilic combination (salicylic acid) to a polar/ hydrophilic/electrolytic combination (sodium salicylate). Solubility data of sodium salicylate in benzene, cyclohexane, and heptane is excluded from Figure 10b, again because the model predicts virtually no solubility of sodium salicylate in these three solvents, although the data suggests very low but measurable solubility. Acetic acid is the only outlier among solvents with significant solubility for sodium salicylate. The model prediction for the solubility of sodium salicylate in acetic acid is ∼1 order of magnitude too high. Acetic acid is not included in the 13 solvents shown in Figure 10b. Model results for the solubility data of p-aminobenzoic acid in 19 solvents and sodium p-aminobenzoate in 12 solvents18 are given in parts a and b, respectively, of Figure 11. Again, low solubility solvents (benzene, cyclohexane, and heptane) are excluded from Figure 11b for sodium aminobenzoate. Acetone and DMF are two outliers for sodium aminobenzoate, and they are also excluded from Figure 11b. The model predicts 2 orders of magnitude higher solubilities in these two solvents. The solubility data and model calculations for ibuprofen in 19 solvents and sodium ibuprofen in 11
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005 8919 Table 4. eNRTL-SAC Model Parameters for Solutes solute acidb
benzoic salicylic acidb p-aminobenzoic acidb ibuprofenb diclofenacb sodium chloridec sodium acetatec sodium benzoatec sodium salicylatec sodium p-aminobenzoatec sodium ibuprofenc sodium diclofenacc sodium chlorided sodium acetated sodium benzoated sodium salicylated sodium p-aminobenzoated sodium ibuprofend sodium diclofenacd (∑N i (ln
xexp i
no. of solvents
X
Y-
26 18 19 19 16 10 5 10 13 12 11 10 10 5 10 13 12 11 10
0.494 0.726 0.552 1.038 0.158
0.176 0.423 0.051
0.750 1.819 0.409
0.270 0.454
Y+
Z
0.336
0.468 0.749 0.881 0.318 0.451
0.594 0.028 1.678 1.444 1.417 1.685 0.845 2.299
1.743
0.125 0.394
3.558 1.060 0.249 0.179 0.373 0.649 0.124
2.201 2.417 2.387 2.362 3.486 2.200 0.679 1.825 1.572 1.895 0.823 2.493
E
0.994 0.521 0.539 0.090 0.192 0.150 0.161
ln Ksp
σa
-1.714 -1.624 -3.348 -1.423 -3.560 -6.252 -6.355 -7.312 -4.889 -8.293 -17.844 -14.202 -3.540 -2.277 -2.978 -2.153 -3.247 -2.364 -4.405
0.292 0.774 1.206 1.055 0.991 0.783 0.241 0.493 0.771 1.258 0.886 0.858 0.923 0.281 0.699 1.058 1.904 1.685 1.473
2 1/2 xcal i ) /N)
σ is defined to be - ln where x is the solubility of solute, i.e., mole fraction (note that it is mass fraction for sodium chloride and sodium acetate), and N is the number of data points used in the correlations. b Nonelectrolytes. c Electrolytes. d Treated as nonelectrolytes. a
Figure 8. Model results for sodium acetate solubility at 298.15 K.
solvents17 are given in parts a and b, respectively, of Figure 12. In comparison to other organic solutes, the model provides a rather poor fit to the ibuprofen data. We did notice that the ibuprofen solubility data from Bustamante et al. are significantly different from those reported by Gracin and Rasmuson19 for certain common solvents including methanol, ethanol, acetone, and ethyl acetate. No attempt was made to reconcile the differences between the Bustamante data and the Gracin and Rasmuson data. The model fit to the sodium ibuprofen solubility data appears to be more satisfactory. Again, the 11 solvents reported in Figure 12b do not include low solubility solvents (benzene, cyclohexane, heptane, and chlorobenzene). Similarly, acetone and DMF are two outliers for sodium ibuprofen, and they are also excluded from Figure 12b. The model predicts 2 orders of magnitude higher solubilities in these two solvents than the available data. Bustamnate et al.17 reported high water content of the ibuprofen sample (3.3 wt % water) and the sodium ibuprofen sample (13 wt % water). It is not clear how such high water contents in the samples could impact on the solubility measurements. The solubility data for diclofenac in 16 solvents and sodium diclofenac in 10 solvents18 are fitted and reported in parts a and b, respectively, of Figure 13. The model significantly overestimates the solubilities of
Figure 9. Model results for (a) benzoic acid solubility at 298.15 K and (b) sodium benzoate solubility at 298.15 K.
diclofenac in acetic acid, formamide, and ethylene glycol. These 3 solvents are excluded from the 16 solvents shown in Figure 13a. Data for low solubility solvents (benzene, cyclohexane, ethyl acetate, heptane, and chlorobenzene) for sodium diclofenac are excluded from Figure 13b. Acetic acid and acetone are two outliers with
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Figure 11. Model results for (a) p-aminobenzoic acid solubility at 298.15 K and (b) sodium p-aminobenzoate solubility at 298.15 K.
Figure 10. Model results for (a) salicylic acid solubility at 298.15 K and (b) sodium salicylate solubility at 298.15 K.
the model estimating 1-3 orders of magnitude higher solubilities for sodium diclofenac. The 2 solvents are not included in Figure 13b. The solubility data treatment above assumes complete dissociation of electrolytes and considers the solubility problem as formation of salts from ionized species of electrolytes, i.e., eq 67. One may argue that electrolytes do not dissociate completely into ionic species, especially in organic solvents of low dielectric constant. In the absence of dissociation to ionic species, the solubility relationship can be described by eq 65, and the eNRTL-SAC model reduces to the NRTL-SAC model. We have treated the electrolyte systems above as nonelectrolytes (i.e., no dissociation to ionic species) with NRTL-SAC, and the model results are also included in Table 4. With the absence of the electrolyte parameter, the representation of the solubility data deteriorates substantially. We also noted that the identified molecular parameters (X, Y-, Y+, and Z) with the complete dissociation treatment are roughly twice as large as those reported with the nondissociation treatment. This finding is consistent with the fact that we only assign the molecular parameters (X, Y-, Y+, and Z) to the anion. Very limited solubility data on organic electrolytes is available in the public domain. Given the uncertainties related to solute purities and solid polymorphs and the interference of solvents on experimental readings, the quality of the available solubility data may also be called into question. Nevertheless, we feel the results obtained
Figure 12. Model results for (a) ibuprofen solubility at 298.15 K and (b) sodium ibuprofen solubility at 298.15 K.
from this preliminary investigation are satisfactory. We are particularly encouraged by the finding that the model fits for the solubilities of organic electrolytes are comparable to the fits for the solubilities of organic nonelectrolytes. Further refinements to the model may
Ind. Eng. Chem. Res., Vol. 44, No. 23, 2005 8921
comments on the manuscript and warm encouragement through the years. Literature Cited
Figure 13. Model results for (a) diclofenac solubility at 298.15 K and (b) sodium diclofenac solubility at 298.15 K.
be needed after more extensive validation of the model with available data. Conclusions The NRTL-SAC model, a practical thermodynamic framework for solubility modeling of organic nonelectrolytes, has been extended for electrolytes. The electrolyte NRTL-SAC model requires only one additional component-specific electrolyte parameter over the three types of molecular parameters associated with the NRTL-SAC model. For solute molecules, these parameters are identified from solubility measurements of the solute in a few representative solvents, i.e., hydrophobic, hydrophilic, and polar solvents. While scarcity of public literature solubility data on organic electrolytes has hampered extensive testing, we have shown the extended model to be a promising tool for qualitative correlation and estimation of the solubility of electrolyte systems, including systems with large, complex pharmaceutical organic electrolytes. Acknowledgment The authors are grateful to Hsien-Hsin Tung, Daniel E. Bakken, and Jose E. Tabora of Merck and Peter A. Crafts of Astra Zeneca for their critical evaluation and constructive feedback to our solubility modeling efforts. We also thank Prof. John M. Prausnitz for his insightful
(1) Chen, C.-C.; Song, Y. Solubility Modeling with Nonrandom Two-Liquid Segment Activity Coefficient Model. Ind. Eng. Chem. Res. 2004, 43, 8354. (2) Frank, T. C.; Downey, J. R.; Gupta, S. K. Quickly Screen Solvents for Organic Solids. Chem. Eng. Prog. 1999, December, 41. (3) Hansen, C. M. Hansen Solubility Parameters: A User’s Handbook; CRC Press: Boca Raton, FL, 2000. (4) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086. (5) Stahl, P. H., Wermuth, C. G., Eds. Handbook of Pharmaceutical Salts: Properties, Selection and Use; Verlag Helvetica Chimica Acta: Zu¨rich, Switzerland, 2002. (6) Chen, C.-C. A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions. Fluid Phase Equilib. 1993, 83, 301. (7) Chen, C.-C.; Song, Y. Generalized Electrolyte NRTL Model for Mixed-Solvent Electrolyte Systems. AIChE J. 2004, 50, 1928. (8) Pitzer, K. S. Electrolytes: From Dilute Solutions to Fused Salts. J. Am. Chem. Soc. 1980, 102, 2902. (9) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth: Woburn, MA, 1970. (10) Rashin, A. A.; Honig, B. Reevaluation of the Born Model of Ion Hydration, J. Phys. Chem. 1985, 89, 5588. (11) Chen, C.-C.; Britt, H. I.; Boston, J. F.; Evans, L. B. Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. AIChE J. 1982, 28, 588. (12) Wilczek-Vera, G.; Rodil, E.; Vera, J. H. On the Activity of Ions and the Junction Potential: Revised Values for All Data. AIChE J. 2004, 50, 445. (13) Chen, C.-C.; Bokis, C. P.; Mathias, P. M. Segment-Based Excess Gibbs Energy Model for Aqueous Organic Electrolytes. AIChE J. 2001, 47, 2593. (14) Chen, C.-C.; Mathias, P. M.; Orbey H. Unification of Hydration and Dissociation Chemistries with the Electrolyte NRTL Model. AIChE J. 1999, 45, 1576. (15) Linke, W. F. Solubilities, Inorganic and Metal-Organic Compounds, 4th ed.; American Chemical Society: Washington, D. C., 1958. (16) Beerbower, A.; Wu, P. I.; Martin, A. Expanded Solubility Parameter Approach. I. Naphthalene and Benzoic Acid in Individual Solvents. J. Pharm. Sci. 1984, 73, 179. (17) Bustamante, P.; Pena, M. A.; Barra, J. The Modified Extended Hansen Method to Determine Partial Solubility Parameters of Drugs Containing a Single Hydrogen Bonding Group and Their Sodium Derivatives: Benzoic Acid/Na and Ibuprofen/Na. Int. J. Pharm. 2000, 194, 117. (18) Barra, J.; Pena, M. A.; Bustamante, P. Proposition of Group Molar Constants for Sodium to Calculate the Partial Solubility Parameters of Sodium Salts Using the van Krevelen Group Contribution Method. Eur. J. Pharm. Sci. 2000, 10, 153. (19) Gracin, S.; Rasmuson A. C. Solubility of Phenylacetic Acid, p-Hydroxyphenylacetic Acid, p-Aminophenylacetic acid, p-Hydroxybenzoic acid, and Ibuprofen in Pure Solvents. J. Chem. Eng. Data 2002, 47, 1379.
Received for review March 17, 2005 Revised manuscript received August 2, 2005 Accepted August 16, 2005 IE0503592
