Symmetric Nonrandom Two-Liquid Segment Activity Coefficient Model

5522

Ind. Eng. Chem. Res. 2009, 48, 5522–5529

Symmetric Nonrandom Two-Liquid Segment Activity Coefficient Model for Electrolytes Yuhua Song and Chau-Chyun Chen* Aspen Technology, Incorporated, 200 Wheeler Road, Burlington, Massachusetts 01803

The nonrandom two-liquid segment activity coefficient model (NRTL-SAC) [Chen, C.-C.; Song, Y. Solubility Modeling with a Nonrandom Two-Liquid Segment Activity Coefficient Model. Ind. Eng. Chem. Res. 2004, 43, 8354-8362] has been shown to be a simple and practical thermodynamic model for correlating and predicting phase behavior for complex pharmaceutical molecules. The model was later extended for electrolytes [Chen, C.-C.; Song, Y. Extension of Non-Random Two-Liquid Segment Activity Coefficient Model for Electrolytes. Ind. Eng. Chem. Res. 2005, 44, 8909-8921]. However, the electrolyte model was presented as an unsymmetric activity coefficient model with aqueous phase infinite dilution reference state. Such an unsymmetric model becomes rather cumbersome to use when dealing with nonaqueous solvents. In this work, we present a symmetric electrolyte NRTL-SAC model which simplifies the long-range interaction term. We further examine the utility of the model in predicting salt solubilities in nonaqueous solvents and mixed solvents. Introduction The pharmaceutical industry requires robust predictive thermodynamic models for estimating solubilities of drug molecules. Recently a segment activity coefficient model based on the nonrandom two-liquid theory was proposed1 for fast, qualitative correlation and prediction of solubilities of organic nonelectrolytes in common solvents and solvent mixtures. NRTL-SAC was later extended2 for computation of ionic activity coefficients and solubilities of electrolytes, organic and inorganic, in solvents and solvent mixtures. The model has shown significant advantages over existing solubility parameter models such as that of Hansen3 and group contribution models such as UNIFAC.4 NRTL-SAC provides a simple molecular thermodynamic framework that characterizes molecular surface interactions in terms of numbers of four conceptual segments (hydrophobic segment, solvation segment, polar segment, and hydrophilic segment) or four dimensionless molecule-specific parameters (hydrophobicity (x), solvation strength (y-), polarity (y+), and hydrophilicity (z)). These molecular descriptors for interested compounds can be identified from very limited phase equilibrium data such as compound solubilities in a few reference solvents1 or infinite dilution activity coefficients of reference solvents in the compound or vice versa.5 Given the conceptual segment numbers (x, y-, y+, and z), the model can then be used to predict qualitatively the phase behavior of mixtures containing the compound as long as the conceptual segment numbers for other species in the mixture are also available. The model has been shown recently6-8 to deliver robust qualitative predictions for drug molecule solubilities in solvents and solvent mixtures within a factor of 2. This is an accuracy that is effective for solvent selection and active pharmaceutical ingredient (API) process design in the pharmaceutical industry. A major consideration in the extension of NRTL-SAC for electrolytes is the treatment of reference state for activity coefficient calculations.2 While the conventional reference state for nonelectrolytes in solution is the pure liquid component, the conventional reference state for electrolytes in solution is the infinite dilution aqueous solution, and the corresponding * To whom correspondence should be addressed. Phone: 781-2216420. Fax: 781-221-6410. E-mail: [email protected].

activity coefficient is “unsymmetric.” Following the generalized eNRTL model,9 Chen and Song2 proposed that the logarithm of the unsymmetric activity coefficient of an ionic species is the sum of three terms: the local composition term for shortrange interaction contribution, the Pitzer-Debye-Hu¨ckel term for long-range interaction contribution, and the Flory-Huggins term for the combinatorial entropy of mixing, all expressed in unsymmetric formulations. For nonaqueous or mixed solvent solutions, an additional Born term is required to account for the transfer of the infinite dilution reference state from the nonaqueous or mixed solvent solution to the aqueous solution for the Pitzer-Debye-Hu¨ckel term. While the formulation of the unsymmetric electrolyte NRTLSAC model is general, the choice of the infinite dilution aqueous solution reference state for electrolytes has made the use of the unsymmetric model for nonaqueous solutions rather difficult or impossible. Water needs to be introduced into the solutions in order for the unsymmetric model to compute properly the ionic activity coefficients with aqueous phase infinite dilution reference state. In this study, we present a symmetric formulation of the electrolyte NRTL-SAC model with the reference states chosen to be pure liquids for nonelectrolytes and pure fused salts for electrolytes. Due to the simplicity of the symmetric formulation together with the representation of the molecular surface interactions of electrolyte solutions in terms of four molecular conceptual segments plus one electrolyte conceptual segment, the resulting symmetric electrolyte NRTL-SAC offers a simple and practical thermodynamic framework for correlating and predicting phase behavior of electrolytes in solvents and solvent mixtures. Symmetric eNRTL-SAC Model As an extension of NRTL-SAC model to electrolyte systems, eNRTL-SAC introduces an additional conceptual “electrolyte” segment. A component in eNRTL-SAC can be represented by up to five conceptual segments: four molecular segments (hydrophobic, solvation, polar, and hydrophilic) m ) x, y-, y+, z and a single 1-1 electrolyte segment e. The electrolyte segment further dissociates and results in a corresponding cationic segment c and a corresponding anionic segment a by the dissociation reaction for the 1-1 electrolyte segment

10.1021/ie900006g CCC: $40.75  2009 American Chemical Society Published on Web 05/06/2009

Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009 +

efc +a

-

(1)

For a given electrolyte component CA, its dissociation to a cationic component and an anionic component can be defined by the following chemical equation CA f νCC+zC + νAA-zA

(2)

νCzC ) νAzA

(3)

choosing a reference state for any molecular and ionic species, respectively. The reference state for a molecular component is the so-called standard state of pure liquids for molecular components. This is called the symmetric reference state for molecular components. It is defined as follows

with

γI(xI f 1) ) 1

where νC and zC are the cationic stoichiometric coefficient and charge number, respectively, and νA and zA are the anionic stoichiometric coefficient and charge number, respectively. Since an electrolyte component CA can be “measured” by up to five conceptual segments ri,CA (i ) m, e), we can calculate the conceptual segment numbers of a cationic component ri,C (i ) m, c) and an anionic component ri,A (i ) m, a) as follows for a system of a single electrolyte rc,C ) zC re,CA

(4)

rm,C ) zC rm,CA, m ) x, y-, y+, z

(5)

ra,A ) zAre,CA

(6)

rm,A ) zArm,CA, m ) x, y-, y+, z

(7)

The equations above can be extended for multielectrolytes rc,C ) zC

∑Y r

A e,CA

(8)

5523

(15)

where xI is the mole fraction of component I. The standard state of pure liquids is nonexistent for ionic species in electrolyte systems. Instead, the symmetric reference state is defined as the pure fused salt state of each electrolyte component in the system. For an electrolyte component CA, the pure fused salt state can be defined as follows γCA(xCA f 1) ) γ((xCA f 1) ) 1

(16)

xCA ) xC + xA

(17)

where γ( is the mean ionic activity coefficient of the electrolyte component and is related to the corresponding cationic and anionic activity coefficients γC and γA by this expression 1 ln γCA ) ln γ( ) (νC ln γC + νA ln γA) (18) ν where ν ) νC + νA; νC and νA are given by the chemical equation describing the dissociation of the electrolyte as defined in eq 2. Equation 18 can also be written in terms of charge numbers zC and zA

A

rm,C ) zC

∑Y r

A m,CA,

m ) x, y-, y+, z

ln γCA ) ln γ( )

(9)

A

ra,A ) zA

∑Y

C re,CA

∑Y

C rm,CA,

m ) x, y-, y+, z

At the pure fused salt state, the total moles of ionic species (assume one mole of salt) are

(11)

ν ) ν C + νA

C

where YC is a cationic charge composition fraction and YA is an anionic charge composition fraction; they are defined as follows YC )

zCxC

∑z

(12)

C′

YA )

∑z

(13)

A′xA′

A′

where xC and xA are the mole fractions of cation C and anion A, respectively. Note that “electrolyte CA” here is meant to represent a pair of cationic component and anionic component in the system. The “electrolytes” are generated from all possible pairings of cations and anions in the system. The logarithm of the activity coefficient of a component species (molecule or ion), ln γI in eNRTL-SAC, is the sum of three terms: the local composition term, ln γlcI , the Pitzer-DebyeHu¨ckel term, lnγIPDH, and the Flory-Huggins term, ln γIFH. ln γI ) ln γIlc + ln γIPDH + ln γIFH

(20)

therefore

C′xC′

zAxA

(19)

(10)

C

rm,A ) zA

zA ln γC + zC ln γA zC + z A

xC )

zA νC ) ν zC + zA

(21)

xA )

zC νA ) ν zC + z A

(22)

xCA ) xC + xA ) 1

(23)

Segment-Based Local Composition Interaction Contribution. The segment-based excess Gibbs free energy of the local interactions for systems with multiple molecular segments m and single-electrolyte segment e (with the corresponding cation segment c and anion segment a) can be written as follows Gex,lc ) nsRT

∑

( )( ) ( ) ∑xG i

xm

m

imτim

i

∑xG i

∑xG τ i

+ xc

ic ic

i*c

im

i

∑xG i

+ xa

ic

i*c

∑xG

iaτia

i

i*a

∑xG i

(14)

We first define the reference states for the species, and then each of the three terms will be presented separately in the following sections. Reference States in Electrolyte Systems. The activity coefficient calculated by eq 14 needs to be normalized by

(24)

ia

i*a

∑xr

I m,I

xm )

I

∑ ∑xr

, i ) m, c, a,

m ) x, y-, y+, z

I i,I

I

i

(25)

5524

Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009

∑xr

xc )

, i ) m, c, a

∑ ∑xr

(26)

i

∑xr

I a,I

xa )

∑ ∑xr

, i ) m, c, a

(27)

I i,I

I

+

ns )

∑ ∑nr

I i,I

I

∑n,

)

i

ni )

i ) m, c, a

i

m′′Gm′′m′τm′′m′

+ xcaGca,m′τca,m′

∑x

+ xcaGca,m′

m′′

m′′Gm′′m′

xcGm,ca

∑x

+ xa

m′Gm′,ca

m′

(

+

(29)

I i,I

∑x m′

(37)

xmGca,m

∑

∑x

(

Since there is only a single 1-1 electrolyte segment, the binary interaction parameters between a molecular segment and the electrolyte segment can be simplified as follows (30)

τmc ) τma ) τm,ca

(31) (32)

τca,m -

Gmc ) Gma ) Gm,ca

+

Gex,lc ) nsRT

(

+ xcaGca,mτca,m

∑x

+ xcaGca,m

m′

m

m

+ xc

with

(

∑x

m′Gm′mτm′m

∑x

τm,ca -

m′Gm′m

m′

∑x G m

m,caτm,ca

m

∑x G m

m,ca

+ xa

m

)( + xa

∑x G

)

m,caτm,ca

m

∑x G m

m,ca

+ xc

m

xca ) xc + xa

)

+ xcaGca,m

xa

-

m,ca

∑x G

+ xa

m

m,ca

+ xc

m

( )

1 ∂G ln Γlci ) RT ∂ni

, i, j ) m, c, a

ln

)

(

(35)

(

∑x

m

∑x G m

∑x G

+ xcaGca,mτca,m

∑x

+ xcaGca,m

xc

-

m,ca

∑x G

+ xc

m

m

m,ca

m

+ xc

)

(38)

m′Gm′mτm′m

m,caτm,ca

m,ca

m

×

m′

m

m,caτm,ca

m

m′

m′Gm′m

)

∑x G

m′

∑x G m

+ xcaGca,m

m′Gm′m

m

+ xa

(

)

∑x G m

m,caτm,ca

m

∑x G m

m,ca

m

+ xa

)

(39)

The local composition term for the logarithm of the activity coefficient of component I, before normalization to a chosen reference state, is computed as the sum of the individual segment contributions

∑r

ln γIlc )

i,I

ln Γlci , i ) m, c, a

(40)

i

(36)

T,P,nj*i

The local composition contributions to the activity coefficients for molecular segments, the cationic segment, and the anionic segment can be carried out as follows

Γmlc

∑x

τca,m -

where m and m′ are molecular segment indices. Finally, in eqs 24 and 34, G and τ are local binary quantities related to each other by the NRTL nonrandomness factor parameter R: G ) exp(-Rτ). The local composition contribution to the segment activity coefficient can be calculated as follows ex.lc

xmGca,m

∑

+ xc

×

∑x

m

+

m′Gm′,ca

m′

+ xcaGca,mτca,m

m,caτm,ca

∑x G

(34)

∑x

m′Gm′mτm′m m′Gm′m

m

m

m

m′

m′

∑x G

ln Γlca )

∑x

)

m′Gm′,caτm′,ca

We can then rewrite the excess Gibbs free energy as follows

∑x

+ xa

m′Gm′,ca

m′

m′

m

(33)

(

∑x

m′

m

Gcm ) Gam ) Gca,m

+ xcaGca,m

m′Gm′m

m

τcm ) τam ) τca,m

+ xc

m′Gm′,ca

ln Γlcc )

m′

τm,ca -

i

I

∑x

)

m′Gm′,caτm′,ca

xaGm,ca

(28)

∑nr

×

+ xcaGca,m′

∑x

m′′

i

where I is the component index, i is the segment index, ri,Iis the number of segments i in component I, xI is the mole fraction of component I, xi is the segment fraction of segment i, ns is the total number of all segments in the system, and nI is the number of moles of component I

m′′Gm′′m′

m′′

τmm′ -

I

∑x

m′

(

I i,I

I

xm′Gmm′

∑

+

I c,I

I

Specifically, for nonelectrolyte (molecular) components, the local composition terms of the activity coefficients are given as follows ln γIlc )

∑r

m,I

ln Γmlc, m ) x, y - , y + , z

(41)

m

∑x

m′Gm′mτm′m

+ xcaGca,mτca,m

∑x

+ xcaGca,m

For a cationic component, we have

m′

m′Gm′m

m′

ln γlcC )

∑r

i,C

i

ln Γlci , i ) m, c

(42)

)


For an anionic component, we have

∑

ln γAlc )

(43)

i

Applying eq 15, the normalization to the pure liquid reference state for molecular components can be done as follows ln γIlc )

∑r

m,I[ln

(

1 ∂G RT ∂ni

ln ΓPDH ) i

ri,A ln Γlci , i ) m, a

ex,PDH

ln

2AφIx/2

)

1

1 + FIx/2

(45)

The local composition contributions to the symmetric activity coefficients for ionic components can be carried out as follows ln γlcC )

∑r

i,C[ln

Γlci - ln Γi ], i ) m, c

∑r

i,A[ln

x Γlci - ln Γlc,I i ], i ) m, a

lc,I0x

(46)

i

ln γlcA )

0

(47)

i

0

x ) ln Γlci (xI f 0) ln Γlc,I i

[

1

4AφIx 1 + FIx/2 Gex,PDH )ln 1 nsRT F 1 + F(I0x ) /2

]

(49)

with

( )( 1/

1 2πNA Aφ ) 3 V Ix )

1 2

∑xz , 2 i i

2

Q2e εkBT

)

3/

i ) c, a

]

1

1

3

}

The original Debye-Hu¨ckel theory is based on a single electrolyte with water as the solvent. The molar volume V and the dielectric constant ε for the single solvent water need to be extended for mixed solvents based on the molecular solvent properties; a simple composition-average mixing rule is proposed to calculate them as follows

∑x V

S S

S

V)

(57)

∑x

S

S

∑x M ε S

S S

S

ε)

(58)

∑x M S

S

S

where S is a solvent component, MS is the solvent molecular weight, and each sum is over all solvent components in the solution. The long-range interaction term for the logarithm of the activity coefficient of component I is computed as the sum of the individual segment contributions

∑r

i,I

ln ΓPDH , i ) m, c, a i

(59)

i

(50) (51)

For molecular components, the PDH terms of the activity coefficients are given as follows ln γIPDH )

∑r

m,I

ln ΓmPDH, m ) x, y - , y + , z

(60)

m

From eq 55, it is easy to show that the PDH terms of the activity coefficients for all molecular components are normalized, that is γIPDH(xI f 1) ) 1

(61)

where I applies to all molecular components in the system. The contribution to the symmetric activity coefficients for ionic components from the long-range interaction can be carried out as follows

(52) ) ln γPDH C

For the symmetric reference state 1 2

{( ) [

) -Aφ

ln γIPDH )

where ns is the total segment number of the solution, R is the gas constant, Aφ is the Debye-Hu¨ckel parameter, Ix is the segment-based ionic strength, F is the closest approach parameter, NA is Avogadro’s number, V and ε are the molar volume and dielectric constant of the solvent, Qe is the electron charge, kB is the Boltzmann constant, zi is the charge number of segment i, and Iox represents Ix at the reference state. Since the “single 1-1 electrolyte segment e ) c+ + a-” is defined in the model, we can obtain

I0x )

(55)

Ix/2 - 2Ix/2 1 + FIx/2 2 ln + , i ) c, a 1 1 F 1 + F(I0x ) /2 1 + FIx/2 (56)

2

i

1 Ix ) (xc + xa) 2

ΓPDH i

(48)

where I applies to all molecular components in the solution and I0x is the ionic strength at the symmetric reference state. In the case that electrolytes are made up of only the conceptual 1-1 electrolyte segment and none of the molecular segments, this reference state is equivalent to the molten state of the conceptual 1-1 electrolyte segment e f c+ + a-. Segment-Based Long-Range Interaction Contribution from the PDH Model. To account for the long-range ion-ion interactions, the model uses the symmetric Pitzer-Debye-Hu¨ckel (PDH) formula10 modified to the segment basis

, i)m

The long-range term of the activity coefficient for the univalent cationic or anionic segment can be derived as follows10 ln

ln Γmlc,I ) ln Γmlc(xI ) 1)

(54)

3

ΓPDH i

m

where Γmlc,I is the local composition contribution to the activity coefficient of the molecular segment m contained in component I

i, j ) m, c, a

T,P,nj*i

From eq 54, the long-range term of the activity coefficient for a molecular segment can be derived as follows10

Γmlc - ln Γmlc,I], m ) x, y - , y + , z (44)

)

5525

∑ r [ln Γ i,C

PDH i

0

x - ln ΓPDH,I ], i ) m, c i

i

(62) (53)

The long-range term of the activity coefficient of segment i can be derived as follows

ln γPDH ) A

∑ r [ln Γ i,A

PDH i

0

x - ln ΓPDH,I ], i ) m, a i

i

(63)

5526

Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009 0

x ln ΓPDH,I ) ln ΓPDH (xI f 0) i i

(64)

where I applies to all molecular components in the solution. Component-Based Flory-Huggins Term Correction to Activity Coefficient. We use the Flory-Huggins term to describe the combinatorial term Gex,FH ) nRT φI )

∑ x ln I

I

xIrI

φI xI

(65)

nIrI

)

∑x r

()

(66)

∑n r

J J

J J

J

J

where Gex,FH is the Flory-Huggins term for the excess Gibbs energy, φI is the segment fraction of component I, rI is the number of all conceptual segments in component I, J is the component index, and nI is the number of moles of component I rI )

∑r

i ) m, c, a

i,I,

(67)

i

∑n

n)

(68)

I

I

The contribution to the activity coefficient of component I from the combinatorial term is ln γIFH ) ln

()

φI + 1 - rI xI

φJ

∑r J

) ln

J

( ) rI

∑x r

+1-

J J

J

rI

∑x r

(69)

J J

J

It is easy to show that contributions to activity coefficients for molecular components from the Flory-Huggins term are normalized, that is γIFH(xI f 1) ) 1

(70)

where I applies to all molecular components in the solution. The contributions to symmetric activity coefficients for ionic species from the Flory-Huggins term can be carried out as follows 0

FH,Ix x ln γFH - ln γFH,I C ) ln γC C

(71)

0

ln γAFH ) ln γAFH,Ix - ln γAFH,Ix with x ln γFH,I ) ln C

( ) ( ) rC

∑xr

+1-

I I

ln γAFH,Ix ) ln

∑xr

+1-

I

ln

) ln

∑xr

(73)

I

I I

0 x γFH,I C

rC I I

I

rA

(72)

x γFH,I (xI C

rA

∑xr

(74)

I I

I

f 0)

(75)

ln γAFH,Ix ) ln γAFH,Ix(xI f 0)

(76)

0

where I applies to all molecular components in the solution. The symmetric model presented as eq 14 can be converted to the unsymmetric model by choosing the aqueous solution infinite dilution reference state. In doing so, it will be necessary to apply the Born term correction for the Pitzer-Debye-Hu¨ckel

Table 1. NRTL Binary Interaction Parameters for Segment-Segment Binary Pairs segment (1) segment (2) τ12 τ21 R21 ) R21 segment (1) segment (2) τ12 τ21 R21 ) R21

x y1.643 1.834 0.2 x e 15 5 0.2

x z 6.547 10.949 0.2 ye 12 -3 0.2

yz -2.000 1.787 0.3 y+ e 12 -3 0.2

y+ z 2.000 1.787 0.3 z e 10.089 -5.212 0.2

x y+ 1.643 1.834 0.2

expression. Although a different reference state has been chosen, the symmetric model is consistent with the unsymmetric model presented earlier.2 Model Applications The symmetric model requires NRTL binary interaction parameters for the various segment-segment binary pairs. These binary parameters and their corresponding values used in this study are summarized in Table 1. With the exception of the e-z binary interaction parameters, their values are identical to those used in the unsymmetric model.2 In this work, we use the binary parameters of the HCl-water binary pair11 for the e-z binary parameters, while the earlier work for the unsymmetric model uses the binary parameters of the NaCl-water binary pair. HCl is known to be a very strong acid and therefore could be considered as a better reference for “completely dissociated” 1,1-electrolytes. While we foresee a need to find optimal values for the various e-m NRTL binary interaction parameters, we aim to validate the utility of this segment-based model for electrolytes in this early study. The optimization of the NRTL binary interaction parameters for the various pairs will be pursued at a later stage. In the following sections, we examine the behavior of the symmetric model and focus on use of the model for prediction of salt solubilities in nonaqueous solvents and mixed solvents for which the symmetric model is expected to offer significant advantages over the unsymmetric model. In contrast to the relative abundance of solubility data for nonelectrolytes, there is scarcity of public literature data for salt solubilities in nonaqueous solvents and mixed solvents. After an extensive search, we were able to locate limited public literature data for solubilities in mixed solvents for sodium bromide, sodium acetate, and sodium salicylate. We hope the modeling results for these three salts would be indicative of the potential usability of the model for inorganic salts (i.e., sodium bromide), organic salts (i.e., sodium acetate), and pharmaceutical salts (i.e., sodium salicylate). Also, to limit the scope of our study, we do not consider the temperature effect and focus on solubilities at 298.15 K. In investigating these electrolyte systems, the conceptual segment parameters for all the solvents considered are taken from earlier studies.6 Only the conceptual segment parameters for electrolytes are fitted against available data. Note that for each electrolyte there are five conceptual segments including four molecular segments and one electrolyte segment. Similar to the use of the NRTL-SAC model for nonelectrolytes,1 numbers of conceptual segments for electrolytes are best identified from data on phase behaviors, i.e., solubilities, of the electrolyte in either pure solvents or mixed solvents with strong characteristics of such conceptual segments. For example, number of conceptual hydrophilic segment for an electrolyte is best identified from solubility data of the electrolyte in hydrophilic solvents. For number of conceptual electrolyte segment data on the phase behaviors of the electrolyte in solvents of varying dielectric constants will be required. In addition, for


5527

Table 2. eNRTL-SAC Parameters for Sodium Bromide, Sodium Acetate, and Sodium Salicylate electrolytes

x

y-

sodium bromide sodium acetate

0 0

0 0

sodium salicylate

0

0

y+

z

e

lnKsp

Solid Phase

0 0

2.140 1.604

0.717 0.299

0.435

1.551

0.069

-5.664 -4.144 -6.005 -5.153

NaBr NaAc NaAc · 3H2O NaSa

solubility studies, it will be necessary to identify the corresponding solubility product constant for each solid polymorph from available solubility data. It should be pointed out that it is not possible to uniquely identify conceptual segment numbers for individual ions from conceptual segment numbers for electrolytes. In this work, after conceptual segment parameters for electrolytes are identified from experimental data, we apply mixing rules, i.e., eqs 8-11, to determine the five conceptual segment parameters for both the cations and the anions that make up the electrolytes. This treatment is different from the earlier unsymmetric model study2 in which the mixing rules were only applied to the conceptual electrolyte segment parameter. The conceptual molecular segment parameters for the anions were set to be same as those of the electrolytes, while the conceptual molecular segment parameters for the cations were arbitrarily set to zero. NaBr Solubilities in Water-Methanol-Ethanol Mixed Solvents. Recently Pinho and Macedo12 reported solubilities of NaCl, NaBr, and KCl in the solvents water, methanol, ethanol, and methanol + ethanol as well as those of NaBr in water + methanol and water + ethanol mixed solvents in the range between 298.15 and 348.15 K. In addition, the Linke solubility compilation13 reported solubilities of NaBr in various alcohols at 298.15 K. The extensive set of solubility data for NaBr in nonaqueous solvents and mixed solvents makes it a good candidate to study the performance of the symmetric model in correlating and predicting solubilities in solvents and solvent mixtures. Conceptual segment parameters and dielectric constants for water and various alcohols have been reported earlier.2,6 We identify conceptual segment parameters and the symmetric solubility product constant for NaBr from the three solubility data of NaBr in water, methanol, and ethanol at 298.15 K. Here the symmetric solubility product constant is defined in eq 77 sat sat Ksp ) xNa +γNa+ xCl-γCl-

initial results are very encouraging as the model predictions are qualitatively correct, although they are based solely on two molecule-specific parameters, z and e, and one solid phasespecific parameter Ksp. We further use the same conceptual segment parameters to predict the NaBr solubilities in various pure alcohols and compare the predictions against the 1939 data reported in the Linke compilation.13 Figure 2 gives the parity plot for the comparison between the model predictions and the literature data. While the parameters for NaBr are only fitted against the solubilities in water, methanol, and ethanol, the model yields robust predictions for NaBr

Figure 1. Model predictions vs data12 for sodium bromide in water-methanol, water-ethanol, and methanol-ethanol mixed solvents at 298.15 K.

(77)

is the concentration of ion i at saturation and γi is the xsat i symmetric activity coefficient of ion i at the system concentration. The resulting conceptual segment parameters are summarized in Table 2. The three solubility data points allow identification of the solubility product constant, the z parameter, and the e parameter. The relatively high value for the z parameter (2.140) suggests NaBr behaves like a hydrated electrolyte. Given the conceptual segment parameters identified for NaBr from the solubilities in water, methanol, and ethanol at 298.15 K, these parameters are then used to predict solubilities of NaBr in the mixed solvents. Figure 1 shows the predicted NaBr solubilities in the water-methanol binary solvent, the waterethanol binary solvent, and the methanol-ethanol binary solvent. The model qualitatively predicts correct trends for the highly nonideal NaBr solubilities in the mixed solvents. The solubility trends for NaBr in the water-ethanol binary and the watermethanol binary are well predicted, while the solubility trend for NaBr in the methanol-ethanol binary is off significantly. While the predictions should be improved upon further perhaps by properly optimizing the NRTL binary interaction parameters for the various molecule-electrolyte segment binary pairs, the

Figure 2. Model predictions vs data13 for sodium bromide solubilities in various pure solvents at 298.15 K.

5528


Figure 3. Model predictions vs data13 for sodium acetate solubilities in various pure solvents at 298.15 K.

solubilities in higher alcohols. It suggests that the model formulation adequatelyaccountsforthevariousmolecule-molecule,molecule-ion, and ion-ion interactions and the dielectric constant effect that determine the solubilities of NaBr. Sodium Acetate Solubility in Water-Ethanol Mixed Solvents. Solubility data of sodium acetate in five nonaqueous solvents (methanol, ethanol, 1-propanol, ethylene glycol, and acetone) are available14 and have been investigated with eNRTLSAC before.2 Among these data, the data for methanol and acetone were taken at 288.15 K, while the data for ethylene glycol and ethanol were taken at 298.15 K. The temperature for the 1-propanol data is not known. We treat all these data as if they were 298.15 K data. Also available are two solubility data of sodium acetate in water at 298.15 K. One of the two aqueous solubility data points has sodium acetate trihydrate solid phase. All other data have sodium acetate solid phase. In this study, these seven data points were fitted to obtain the conceptual segment parameters for sodium acetate and the solubility product constants for sodium acetate and sodium acetate trihydrate. The parity plot is given in Figure 3. Given that the data span over 4 orders of magnitude, we consider the fit excellent. Table 2 summarizes the regressed parameters. Like sodium bromide, sodium acetate is also “hydrated” with significant hydrophilicity. Also, the electrolyte parameter e for sodium acetate is found to be significantly less than unity, reflecting the nature of organic electrolytes. Solubility data of sodium acetate in aqueous ethanol solutions are available at 298.15 K.13 These data and model predictions are given in Figure 4. It is reported that the solid phase in contact with the solutions was sodium acetate trihydrate in all cases.13 However, this cannot be true because there would be no water present to form trihydrate solid phase if water content becomes too low in the aqueous ethanol solutions. The extreme case of that would be the nonaqueous ethanol solvent. The conceptual segment parameters in Table 2 were used to predict the sodium acetate solubility in the aqueous ethanol solutions. As shown in Figure 4, the model qualitatively predicts correct trends as the ethanol content varies from all water to all ethanol. The solubility remains relatively constant in water-rich solutions, but it drops off sharply in ethanol-rich solutions. Another interesting finding is that the model predicts the

Figure 4. Model predictions vs data13 for sodium acetate solubilities in water-ethanol mixed solvent at 298.15 K.

trihydrate solid phase to be the less soluble solid phase in waterrich solutions until around 60 wt % ethanol. The model predicts anhydrous solid phase in ethanol-rich solutions or when the ethanol content exceeds 60 wt %. Sodium Salicylate Solubility in Water-Ethanol Mixed Solvents. Solubility data of sodium salicylate in 23 pure solvents are available,14 and 13 of these 23 data points have been investigated with eNRTL-SAC.2 In this study, we repeat the investigation on the same 13 solvents (water, methanol, ethanol, 1-pentanol, ethylene glycol, 1-octanol, dioxane, acetone, ethyl acetate, chloroform, chlorobenzene, formamide, N,N-dimethylformamide), and the regressed model parameters for sodium salicylate are summarized in Table 2. Like sodium bromide and sodium acetate, sodium salicylate also displays significant hydrophilicity, i.e., with a significant z value. As the size of the organic moiety increases from sodium acetate to sodium salicylate, the e value for sodium salicylate becomes smaller and the y+ value for polarity increases. The parity plot is shown in Figure 5. The overall goodness of the fit is similar to the 2005 study.2 Solubility data of sodium salicylate in aqueous mixtures of dioxane, acetone, methanol, ethanol, and 1-propanol have been reported in graphical form.15 However, while the data between these two references, one for pure solvents14 and the other for mixed solvents,15 are consistent for the sodium salicylate solubilities in pure water, methanol, ethanol, and acetone, the data deviate much from each other for the solubilities in pure dioxane, 1-propanol, and higher alcohols. For example, the pure solvent reference14 suggests sodium salicylate solubility of 0.0034 mol fraction in 1-pentanol, while the mixed solvent reference15 suggests 0.062 (∼98 mg/mL solvent), nearly 20 times higher. The model predictions for sodium salicylate solubilities in aqueous mixed solvents of dioxane, acetone, methanol, ethanol, and 1-propanol are given in Figure 6. Again, the model predicts qualitatively correct trends for the sodium salicylate solubilities even though the predicted solubility in pure water is about 27% too low. Of high significance are the abilities of the model to correctly predict the highly nonideal solubility trends of (1) the


5529

have made the development and validation of electrolyte thermodynamic models very challenging. The symmetric eNRTL-SAC model is presented here as a simple and practical thermodynamic framework to first correlate salt solubilities in a few representative solvents and then qualitatively predict salt solubilities in other solvents and solvent mixtures. The success in predicting solubilities qualitatively, though for a limited number of salts in nonaqueous solvents and mixed solvents, provides support that eNRTL-SAC properly accounts for the major thermodynamic factors controlling the liquid phase nonideality and that it can be a simple and practical thermodynamic formulation for correlating and predicting the phase behavior of electrolyte systems. Acknowledgment C.-C. Chen thanks G. M. Bollas for extensive discussions on the symmetric Pitzer-Debye-Hu¨ckel formula for the longrange interaction contribution. Literature Cited Figure 5. Model predictions vs data14 for sodium salicylate solubilities in various pure solvents at 298.15 K.

Figure 6. Model predictions for sodium salicylate solubilities in water-dioxane, water-acetone, water-ethanol, water-methanol, and water-1-propanol mixed solvents at 298.15 K.

initial rapid rise of the solubilities in aqueous dioxane solutions and the aqueous acetone solutions, (2) the initial slow rise of the solubilities in aqueous methanol solutions, (3) the leveling off of the solubilities after the initial rise of the solubilities, and (4) the existence of maximum peak solubilities in aqueous dioxane solutions. Conclusions The complexity of the molecule-molecule, molecule-ion, and ion-ion interactions in electrolyte solutions and the scarcity of reliable salt solubility data in pure solvents and mixed solvents

(1) Chen, C.-C.; Song, Y. Solubility Modeling with a Nonrandom TwoLiquid Segment Activity Coefficient Model. Ind. Eng. Chem. Res. 2004, 43, 8354–8362. (2) Chen, C.-C.; Song, Y. Extension of Non-Random Two-Liquid Segment Activity Coefficient Model for Electrolytes. Ind. Eng. Chem. Res. 2005, 44, 8909–8921. (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. Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086–1099. (5) Chen, C.-C. Correlation and Prediction of Phase Behavior of Organic Compounds in Ionic Liquids Using the Nonrandom Two-Liquid Segment Activity Coefficient Model. Ind. Eng. Chem. Res. 2008, 47, 7081–7093. (6) Chen, C.-C.; Crafts, P. A. Correlation and Prediction of Drug Molecule Solubility in Mixed Solvent Systems with the NonRandom TwoLiquid Segment Activity Coefficient (NRTL-SAC) Model. Ind. Eng. Chem. Res. 2006, 45, 4816–4824. (7) Tung, H.-H.; Tabora, J.; Variankaval, N.; Bakken, D.; Chen, C.-C. Prediction of Pharmaceutical Solubility via NRTL-SAC and COSMO-SAC. J. Pharm. Sci. 2008, 97, 1813–1820. (8) Kokitkar, P. B.; Plocharczyk, E.; Chen, C.-C. Modeling Drug Molecule Solubility to Identify Optimal Solvent Systems for Crystallization. Org. Process Res. DeV. 2008, 12, 249–256. (9) Chen, C.-C.; Song, Y. Generalized Electrolyte NRTL Model for Mixed-Solvent Electrolyte Systems. AIChE J. 2004, 50, 1928–1941. (10) Pitzer, K. S.; Simonson, J. M. Thermodynamics of Multicomponent, Miscible, Ionic Systems: Theory and Equations. J. Phys. Chem. 1986, 90, 3005– 3009. (11) Chen, C.-C.; Britt, H. I.; Boston, J. F.; Evans, L. B. Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. Part I: Single Solvent, Single Completely Dissociated Electrolyte Systems. AIChE J. 1982, 28, 588–596. (12) Pinho, S. P.; Macedo, E. A. Solubility of NaCl, NaBr, and KCl in Water, Methanol, Ethanol, and Their Mixed Solvents. J. Chem. Eng. Data 2005, 50, 29–32. (13) Linke, W. F. Solubilities: Inorganic and Metal-Organic Compounds, 4th ed.; D. Van Nostrand Company, Inc.: Princeton, NJ, 1958; Vol. I. (14) 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, 153–161. (15) Paruta, A. N.; Mauger, J. W. Solubility of Sodium Salicylate in Mixed Solvent Systems. J. Pharm. Sci. 1971, 60, 432–437.

ReceiVed for reView January 3, 2009 ReVised manuscript receiVed March 27, 2009 Accepted April 6, 2009 IE900006G

Symmetric Nonrandom Two-Liquid Segment Activity Coefficient Model

Recommend Documents