Symmetric Electrolyte Nonrandom Two-Liquid Activity Coefficient Model

Jul 14, 2009 - Aspen Technology, Inc., 200 Wheeler Road, Burlington, Massachusetts 01803. The electrolyte nonrandom two-liquid (eNRTL) model is ...
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7788

Ind. Eng. Chem. Res. 2009, 48, 7788–7797

Symmetric Electrolyte Nonrandom Two-Liquid Activity Coefficient Model Yuhua Song and Chau-Chyun Chen* Aspen Technology, Inc., 200 Wheeler Road, Burlington, Massachusetts 01803

The electrolyte nonrandom two-liquid (eNRTL) model is reformulated as a symmetric activity coefficient model with the reference states chosen to be pure liquids for solvents and pure fused salts for electrolytes. These reference states are consistently used in the local interaction term, represented by a reformulated NRTL expression, and the long-range interaction term, represented by an extended symmetric Pitzer-Debye-Hu¨ckel expression. Retaining the local electroneutrality and like-ion repulsion hypotheses, the new symmetric electrolyte NRTL model yields simpler activity coefficient expressions for both molecular and ionic species. The utility of the model is demonstrated with vapor-liquid equilibrium, liquid-liquid equilibrium, and solid-liquid equilibria of several mixed solvent electrolyte systems. Introduction Modeling of properties and phase behaviors of electrolyte systems remains a major challenge.1 Some widely referenced activity coefficient models in the literature include the Pitzer model,2 the OLI MSE model of Wang et al.,3 the extended UNIQUAC model of Thomsen et al.,4 and the electrolyte nonrandom two-liquid (eNRTL) model.5,6 From a process modeling perspective, the use of the Pitzer model is cumbersome because of its limited applicability range for dilute aqueous electrolyte systems and its requirement for ternary interaction parameters. The OLI MSE model suffers from the inconsistent treatment of ion-ion and ion-molecule interactions with the virial expansion-type equation and molecule-molecule interaction with the UNIQUAC equation. Also questionable is its use of pure ions as reference state. The extended UNIQUAC model is faulted for its inconsistent treatments in both the reference state and the concentration scale. The pure ion reference state and the mole fraction concentration scale are used with the shortrange UNIQUAC term while the aqueous phase infinite dilution reference state and the molality concentration scale are used with the long-range Debye-Hückel expression. The electrolyte NRTL model was first proposed5,6 in the 1980s as an unsymmetric activity coefficient model for aqueous electrolytes with the reference state chosen to be aqueous phase infinite dilution for ions. This choice of reference state follows the convention for aqueous electrolyte thermodynamics. For example, the commonly available literature data for electrolyte systems such as thermophysical properties for ionic species,7 mean ionic activity coefficients, and osmotic coefficients for aqueous electrolytes8 all follow such a convention. Over the years the electrolyte NRTL model has evolved9-13 and it has become a widely used activity coefficient model for process modeling and simulation of aqueous and mixed-solvent electrolyte systems.14 While the formulation of the unsymmetric electrolyte NRTL model5,6 is general, the choice of the aqueous phase infinite dilution reference state for electrolytes has made the use of the unsymmetric model for nonaqueous electrolyte systems and mixed-solvent electrolyte systems rather cumbersome or even questionable. Not only does the choice of aqueous phase infinite dilution reference state become counterintuitive when one studies nonaqueous electrolyte systems, but also a small * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: 781-221-6420. Fax: 781-2216410.

concentration of water needs to be artificially introduced into the liquid composition in order for the unsymmetric model to compute properly the ionic activity coefficients with aqueous phase infinite dilution reference state. Moreover, the computation of the Born term to transfer the reference state to the aqueous phase infinite dilution solution for the long-range ion-ion interaction contribution is complicated by the uncertainties associated with the choices of the ionic radii and the mixing rules for the mixed-solvent dielectric constant and density. To highlight the need for a symmetric electrolyte NRTL model, a recent study15 presented a first formulation of the electrolyte NRTL model based on a symmetric reference state and it was successfully applied to the modeling of multicomponent liquid-liquid equilibrium for mixtures that involve ionic liquids. Specifically, a symmetrically referenced expression for the longrange ion-ion interaction term was obtained by renormalizing the unsymmetric expression of the Pitzer-Debye-Hu¨ckel model.16 In this study, we revisit the development of the electrolyte NRTL model and we present a new symmetric electrolyte NRTL model with the reference states consistently chosen to be pure liquids for nonelectrolytes and pure fused salts for electrolytes (called “hypothetical, homogeneously mixed, completely dissociated liquid electrolytes” in the 1986 model6) for both the local interaction contribution and the long-range interaction contribution. In addition, we present a new and general formulation for the local interaction contribution to remove the “ionic-charge fraction quantities”13 from the local composition excess Gibbs free energy expression and we extend the symmetric expression of the Pitzer-Debye-Hu¨ckel model for single electrolyte systems17 to multicomponent electrolyte systems for the long-range interaction contribution. We further examine the utility of the resulting symmetric electrolyte NRTL model to correlate phase behaviors of various mixed-solvent electrolyte systems. Symmetric Electrolyte NRTL Model The new electrolyte NRTL model contains two contributions: one from local interactions that exist at the immediate neighborhood of any species, and the other from the long-range ion-ion interactions that exist beyond the immediate neighborhood of an ionic species. To account for the local interactions, the model uses the local composition model known as the electrolyte NRTL expression.5,6 To account for the long-range interactions, the model extends the symmetric Pitzer-Debye-Hu¨ckel (PDH)

10.1021/ie9004578 CCC: $40.75  2009 American Chemical Society Published on Web 07/14/2009

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 16,17

formula to multicomponent electrolyte systems. The following equation is the basis of the electrolyte NRTL model for the excess Gibbs free energy of electrolyte systems Gex: Gex ) Gex,lc + Gex,PDH

(1)

where Gex,lc and Gex,PDH represent the contributions from the local and long-range interactions, respectively. The activity coefficient γi can be derived from eq 1 as follows: ln γi )

( )

1 ∂Gex RT ∂ni

) T,P,nj*i

(

( ) )

1 ∂Gex,lc RT ∂ni

1 ∂Gex,PDH RT ∂ni

where υc is the cationic stoichiometric coefficient, υa is the anionic stoichiometric coefficient, and υ ) υc + υa (one mole of electrolyte releases υ moles of ions in solution). They are given by the chemical equation describing the dissociation of the electrolyte:

i, j ) m, c, a

(2)

ca f υcc+zc + υaa-za

(8)

υczc ) υaza

(9)

with

+

T,P,nj*i

where zc and za are charge numbers of cation and anion, respectively. Equation 7 can be rewritten in terms of zc and za:

T,P,nj*i

ln γca ) ln γ( )

or ln γi ) ln γlci + ln γPDH , i

i ) m, c, a

(3)

Each of the terms will be discussed separately in the following sections. In eq 2, R is the gas constant, T is the temperature, P is the pressure, and ni and nj are the mole numbers of components i and j, respectively; both i and j are used as component index for all components including molecular component m, cationic species c, and anionic species a. Reference States in Electrolyte Systems. The activity coefficient by eq 3 needs to be normalized by choosing reference states for molecular components and ionic species. Reference State for Molecular Components. The reference state for a molecular component m is defined as follows: γm(xm f 1) ) 1

(4)

This definition is the so-called standard state of pure liquids for molecular components, and it is also called the symmetric reference state for molecular components. Reference State for Ionic Species. The standard state of pure liquids is hypothetical for ionic species in electrolyte systems. The symmetric reference state for ionic species is defined as the pure fused salt state of each electrolyte component in the system. The conventional reference state for ionic species is the infinite dilution state in solutions. It is also called the unsymmetric reference state for ionic species. The definition of an unsymmetric reference state further depends on the composition of electrolyte-free solvent mixtures; that is, infinite dilution aqueous solutions or infinite dilution mixed-solvent solutions. In this work, we consider all three reference states for ionic species. Pure Fused Salt State of an Electrolyte Component. For an electrolyte component ca, the pure fused salt state can be defined as follows: γca(xca f 1) ) γ((xca f 1) ) 1

(5)

xca ) xc + xa

(6)

with

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: ln γca ) ln γ( )

1 (υ ln γc + υa ln γa) υ c

(7)

7789

za ln γc + zc ln γa zc + za

(10)

At the pure fused salt state, the total moles of ionic species (assume one mole of electrolyte) are υ ) υc + υa

(11)

Therefore, xc )

za υc ) υ zc + za

(12)

xa )

zc υa ) υ zc + za

(13)

xca ) xc + xa ) 1

(14)

The symmetric reference state defined by eq 5 is restricted to systems containing a single electrolyte component. For multicomponent electrolyte systems, the symmetric reference state can be generalized from eq 5 as follows: γca(xm f 0) ) γ((xm f 0) ) 1

(15)

where m applies to all molecular components in the system. The hypothetical symmetric reference state is a molecularcomponent-free media in which the multicomponent electrolytes form “ideal solution.” Infinite Dilution Solution. The condition of infinite dilution solution for ionic species can be written as follows: x c ) xa ) 0

(16)

This condition applies to all ionic species in the solution. However, there can be two different unsymmetric reference states, infinite dilution aqueous solutions and infinite dilution mixed-solvent solutions. Infinite Dilution Aqueous Solution. Water must be present in the solution when infinite dilution aqueous solution reference state is used. Therefore, the condition for the aqueous solutions (i.e., water (w)) as the unsymmetric reference state can be written as follows: γc(xw f 1) ) γa(xw f 1) ) 1

(17)

This equation applies to all ionic species in the solution. Infinite Dilution Mixed Solvent Solution. The condition for infinite dilution mixed solvent solution as the unsymmetric reference state can be written as follows:

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Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009

γc(xc f 0 and xa f 0) ) γa(xc f 0 and xa f 0) ) 1 (18) This equation applies to all ionic species in the solution. NRTL Term for Local Interaction Contribution. In an electrolyte system, all species can be categorized as one of three types: molecular component (solvents) m; cationic species (cations) c; and anionic species (anions) a. The model assumes that there are three types of local composition interactions. The first type consists of a central molecular component with other molecular components, cationic species, and anionic species in the immediate neighborhood. Here, local electroneutrality is maintained. The other two types are based on the like-ion repulsion assumption and have either a cationic or anionic species as the central species. They also have an immediate neighborhood consisting of molecular components and oppositely charged ionic species. In the case of the fused salt reference state, central ionic species will be surrounded by oppositely charged ions only. Accordingly, the excess Gibbs free energy from local interactions for an electrolyte system can be written as follows:

Gex,lc ) RT

( ) ( ) ( ) ( ) ( ) ( ) ∑XG

imτim

i

∑n

i

m

m

i

i

∑z n

+

∑XG

∑XG τ i*c

c

im

i

i

∑XG i

∑XG i

m′

i

∑ c

XcGmc

∑XG i

ex,lc

G ) nRT

∑X

∑XG i

m

m

imτim

i

+

∑XG i

∑X

i

XmGcm

∑

i*c

∑XG

c

i

im

ic ic

∑XG

a

ic

c

im

i

i

i

∑XG i

∑XG

a

i*a

ia ia

i*a

i

(20)

ia

i*a

with

Xi ) Cixi ) Ci n)

∑n

i

i

)

()

∑n m

ni , n

m

+

i ) m, c, a

∑n

c

c

+

(21)

∑n

a

(22)

a

where the first term in eq 19 is the contribution when a molecular component is at the center, the second is the contribution when a cationic species is at the center, and the third term is the contribution when an anionic species is at the center. In eq 21, Ci ) zi (charge number) for ionic species and Ci ) 1 for molecular components. Finally, in eqs 19 and 20, G and τ are local binary quantities related to each other by the NRTL nonrandomness factor parameter R: G ) exp(-Rτ)

ia ia

i*a

∑XG τ

a

i

∑XG τ

i*c

∑X

∑XG

m

i

ic

i

XmGam

∑

(23)

The activity coefficient of component i can be derived as follows:

ia

(

∑

+

i*c

1 ln γlca ) za

+

(

+

∑XG τ

(

∑ c

ic ic

i*c

∑XG i*c

XaGma

∑XG i

i

(

ic

ia

)

imτim

∑XG

i

i

(

im

ia

i*a

∑XG

∑XG i

ic

(

i

im

∑XG

)

ia

i*a

ia ia

i*a

∑XG

)

(25)

)

(26)

)

(27)

+

ia

+

∑XG τ i

τac -

+

)

ia ia

∑XG τ

imτim

i

XcGac

∑XG τ

i*a

i

)

+

i

∑XG i

im′

i

i

τca -

∑XG

i

i

i

XaGca

∑XG

i*a

∑XG i

im′τim′

i

i

i*c

(24)

i

τma -

i*a

τam -

im

i

i

τcm -

∑XG

m

i

ic ic

ic

(

∑XG

τmm′ -

im′

i

a

ia

∑XG τ

i

∑

i*a

or

∑XG τmc -

i*c

i

(19)

Xm′Gmm′

∑

+ im

i*c

ia ia

i*a

a a

a

)

∑XG τ

ic

∑XG τ

∑z n

ln

imτim

i

i

i*c

i

∑XG i

lc γm

+

∑XG

c c

i, j ) m, c, a

T,P,nj*i

The results are

1 ln γlcc ) zc

ic ic

( )

1 ∂Gex,lc RT ∂ni

ln γlci )

ic ic

i*c

∑XG i

i*c

ic

Binary Parameters. There are three types of model adjustable binary parameters: molecular-molecular binary parameters, molecule-electrolyte binary parameters, and electrolyte-electrolyte binary parameters. Here “electrolyte” is meant to represent a pair of cation c and anion a. The model adjustable binary parameters include the symmetric nonrandomness factor parameters, R, and the asymmetric binary interaction energy parameters, τ. In short, the adjustable binary parameters include Rmm′ ) Rm′m Rm,ca ) Rca,m Rca,ca′ ) Rca′,ca Rca,c′a ) Rc′a,ca (28) τmm′ ) τm′m τm,ca ) τca,m τca,ca′ ) τca′,ca τca,c′a ) τc′a,ca (29) As an example, for a system with two molecular components, m and w, and one electrolyte, ca, there are nine adjustable binary parameters: Rmw, Rm,ca, Rw,ca, τmw, τm,ca, τw,ca, τwm, τca,m, and τca,w. In practice, R’s are often fixed as constants, that is, 0.2, while τ’s are adjusted. However, from eqs 19 and 20, we also need the following binary parameters: Rcm Ram Rmc Rma Rca Rac

(30)

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009

τcm τam τmc τma τca τac

(31)

These nonadjustable binary parameters are calculated from the model adjustable binary parameters. A simple composition average mixing rule over cations or anions is adopted to calculate Rcm, Rmc, Ram, Rma, Rca, and Rac, respectively:

∑Y R

Rcm ) Rmc )

(32)

a ca,m

a

∑Y R

Ram ) Rma )

lc,I0x

x ln γlcc ) ln γlc,I - ln γc c

0

1 x ln γlc,I ) c zc

∑

∑

i

i

Yc′Rca,c′a

ic ic

i*c

(34)

∑

a′

(35)

where Yc is a cationic charge composition fraction and Ya is an anionic charge composition fraction; they are defined as follows: Xc

∑

1 x ln γlc,I ) a za

m

i

∑XG τ

(36)

i

ia ia

i*a

Xc′

∑XG

im

i

ia

+

c

(37)

i

i

(

im

ia

∑XG

imτim

∑XG

(

i

im

i

ic

)

i

ia

i*a

i

i

ia ia

i*a

i

∑XG

+

i

∑XG

XcGac

)

∑XG τ

τca -

i*a

i*c

0

∑XG

∑XG

(

∑

i*a

Xa

∑X

∑XG

imτim

i

XaGca

τam -

i

c′

Ya )

∑

XmGam

(47)

i

a

i*c

Ya′Rca,ca′

(

∑

ic

a′

Yc )

+

∑XG i

Ya′Rac,a′c )

im

∑XG i

τcm -

∑XG

∑XG τ

c′

Rac )

m

XmGcm i

c

Rca )

∑

(33)

c ca,m

(46)

x x - ln γlc,I ln γlca ) ln γlc,I a a

with

∑XG τ

ic ic

i*c

∑XG i

ic

i*c

x x ) ln γlc,I ln γlc,I c (xm f 0) c

a′

)

(48)

)

(49)

+

i

τac -

7791

(50)

a′

0

The same mixing rules are also applied to calculate Gcm, Gmc, Gam, Gma, Gca, and Gac, respectively: Gcm )

∑Y G

(38)

∑Y G

(39)

∑Y G

(40)

∑Y G

m,ca

(41)

∑Y G

ca,c′a

(42)

a

ca,m

a

Gam )

c

ca,m

c

Gmc )

a

m,ca

x x ) ln γlc,I ln γlc,I a a (xm f 0)

where Ix is defined as the ionic strength and I0x represents Ix at the reference state. In both eqs 50 and 51, xm f 0 applies to all molecular components in the solution. Ionic Species with Infinite Dilution Aqueous Solution Reference State. Applying eq 17, one can obtain the normalized unsymmetric activity coefficients for ionic species in aqueous solutions as follows:

a

Gma )

c

c

Gca )

c′

∑Y G a′

ac,a′c

)

a′

∑Y G a′

ca,ca′

(43)

a′

The binary parameters are then calculated from eq 23: τ)-

ln G R

(44)

Molecular Components. From eq 25, it is easy to show that activity coefficients for all molecular components are normalized; that is ln γlcm(xm f 1) ) 0

(52)

lc,Ix - ln γlc,∞ ln γ*lc a ) ln γa a

(53)

x ) ln γlc,I ln γlc,∞ c c (xw f 1) ) zc(Gcwτcw + τwc)

(54)

x ) ln γlc,I ln γlc,∞ a a (xw f 1) ) za(Gawτaw + τwa)

(55)

lc,∞ are activity coefficients at the infinite dilution where γlc,∞ c and γa aqueous solutions. Ionic Species with Infinite Dilution Mixed-Solvent Solution Reference State. Applying eq 18, one can obtain the normalized unsymmetric activity coefficients for ionic species in mixedsolvent solutions as follows:

(45)

Again m applies to all molecular components in the system. Ionic Species with Symmetric Reference State. Applying eq 15, one can obtain the normalized symmetric activity coefficients for ionic species in multicomponent electrolyte systems as follows:

x ) ln γlc,I - ln γlc,∞ ln γ*1c c c c

with

c′

Gac )

(51)

lc,Ix - ln γlc,∞ ln γ*lc c ) ln γc c

(56)

lc,Ix - ln γlc,∞ ln γ*lc a ) ln γa a

(57)

x ) ln γlc,I ln γlc,∞ c c (xc f 0 and xa f 0)

(58)

with

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Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 x ln γlc,∞ ) ln γlc,I a a (xc f 0 and xa f 0)

γlc,∞ c

(59)

∑x

c

γlc,∞ a

where and are activity coefficients at the infinite dilution mixed-solvent solutions. Extended PDH Model for Long-Range Interaction Contribution. To account for the long-range ion-ion interactions, we extend the symmetric Pitzer-Debye-Hu¨ckel (PDH) formula17 from single electrolyte systems to multicomponent electrolyte systems. Equation 60 shows the symmetric PDH equation for the excess Gibbs free energy, Gex,PDH, for single electrolyte systems:

[

1

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

]

( )

1 2πNA 3 V

Aφ )

∑ i

1 zi xi ) 2 2

∑ c

1/2

( ) Qe2 εkBT

1 zc xc + 2 2

∑

3/

za xa,

ln γPDH ) i

(

1 ∂Gex,PDH RT ∂ni

γPDH i

ln

(61) ln

γPDH i

) -Aφ

i ) m, c, a

∑x z

0 2 c c

+

c

)

1 2

∑x z

0 2 a a

)

(65)

a

c′

c′

+

(69)

{( ) [

]

1

1

3

-1/2 1

( )} n

∂Ix0 ∂ni

,

i ) c, a

(70)

For the unsymmetric reference state: n

∂Ix0 ) 0, ∂ni

i ) c, a

(71)

and for the symmetric reference state: n

∂Ix0 1 ) ∂ni 2

n

∑z

2 c′

c′

∂xc′0 ) ∂ni

( ) n

∂xc′0 1 + ∂ni 2

∂xa′0 ) ∂ni

2 a′

a′

δic′ - xc′0

∑x

c″

∑x

+

( ) n

∂xa′0 , ∂ni

i ) c, a (72)

,

i ) c, a

(73)

,

i ) c, a

(74)

i, j ) c, a

(75)

i, j ) c, a

(76)

a″

c″

n

∑z

a″

δia′ - xa′0

∑x

c″

∑x

+

a″

c″

a″

or n

n

∂Ix0 1 ) ∂ni 2

∂x0j ) ∂ni

∑

( )

z2j n

j

∂x0j , ∂ni

δij - x0j

∑x

c

+

c

,

∑x

a

a

For mixed solvents, the molar volume V and the dielectric constant ε for the single solvent need to be extended. A simple compositionaverage mixing rule is adequate to calculate them as follows:

∑x V

s s

xc

∑x

(68)

zi2Ix /2 - 2Ix /2 2zi2 1 + FIx /2 ln + 1 1 F 1 + F(Ix0) /2 1 + FIx /2

with x0c

i)s

,

1

1 + FIx /2

1 + F(Ix0) /2

(64)

where xm f 0 applies to all molecular components in the solution. This definition ensures that the excess Gibbs free energy from the long-range interactions will be zero at the symmetric reference state regardless of how many electrolytes are present in the solution. The expression for I0x can be easily carried out in general: 1 2

2AφIx /3

2Ix(Ix0)

(63)

Equation 63 reduces to the original value of 1/2 for 1-1 electrolyte systems.17 To extend the symmetric Pitzer-Debye-Hu¨ckel theory from single electrolyte systems to multicomponent electrolyte systems, we need to convert the symmetric reference state defined by eq 15 in terms of Ix0 as follows:

Ix0 )

i, j ) s, c, a

T,P,nj*i

For a cationic or anionic species, the activity coefficient can be carried out as follows:

where n is the total mole number of the solution,Aφ is the Debye-Hu¨ckel parameter, Ix is the ionic strength, F is the closest approach parameter, NA is Avogadro’s number, Qe is the electron charge, kB is the Boltzmann constant, and V and ε are the molar volume and the dielectric constant of the solvent, respectively. It is worth pointing out that eq 60 also applies to electrolyte systems with the unsymmetric reference state by setting Ix0 ) 0. Since the symmetric Pitzer-Debye-Hu¨ckel theory is originally based on a single electrolyte at its pure fused salt as the reference state, we can calculate I0x from eqs 12 and 13 for single electrolyte systems:

Ix(xm f 0) ) Ix0

)

For a solvent component s, the activity coefficient can be carried out as follows:

a

zcza 1 ) (xczc2 + xaza2) ) 2 2

a′

The activity coefficient of component i can be derived as follows:

(62)

I0x

a′

2

2

2

(67)

∑x

+

c

(60)

with

1 Ix ) 2

xa

x0a )

∑x a

(66) a

V)

s

∑x s

(77) s

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009

∑x M ε s

ε)

7793

s s

s

(78)

∑x M s

s

s

where s is a solvent component in the mixture and Ms is the solvent molecular weight. Each sum in eqs 77 and 78 is over all solvent components in the solution. Born Term Correction to Activity Coefficient. If 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 aqueous solution for the Debye-Hu¨ckel term. The Born term8,18 is used for this purpose:

(

Qe2 1 1 ∆GBorn ) nRT 2kBT ε εw

)∑ i

(

)

NAQe2 1 xizi2 -2 1 10 ) × ri 2RT ε εw

∑ i

2

xizi -2 10 ri

(79)

where ∆GBorn is the Born term correction to the Pitzer-DebyeHu¨ckel formula for infinite dilution aqueous solutions, εw is the dielectric constant of water, and ri is the Born radius of species i. The correction to the activity coefficient of component i can be derived as follows: ∆ln γBorn ) i

(

1 ∂∆GBorn RT ∂ni

)

i ) s, c, a

(80)

T,P,nj*i

For a cation or anion species, the correction to the activity coefficient can be carried out as follows: ∆ln

γBorn i

(

Figure 1. Prediction of NaCl molality scale mean ionic activity coefficient of a constant total molal NaCl-KCl-water system with various electrolyteelectrolyte binary interaction parameters. Solid lines, 1986 model results; dotted lines, current model results.

)

(

)

2 Qe2 1 NAQe2 1 1 zi -2 1 ) 10 ) × 2kBT ε εw ri 2RT ε εw zi2 -2 10 i ) c, a (81) ri

The correction to the activity coefficient for a solvent component is zero: ) 0, ∆ln γBorn i

i)s

(82)

Difference Between 1986 Model and this Formulation. In addition to the choice of hypothetical pure fused salts as the reference state for ions and the use of an extended symmetric formulation for the Pitzer-Debye-Hu¨ckel term for the longrange ion-ion interaction contribution, it is important to highlight the difference in the NRTL term for local interaction contribution between the original electrolyte NRTL model of 19866 and the current formulation. In the 1986 model, the choice of the fused salt reference state for the local interaction contribution was considered and applied during the formulation of the excess Gibbs free energy expression. In the current formulation, the excess Gibbs free energy expression is developed initially as a generalized expression without consideration to the fused salt reference state. Rather, this fused salt reference state along with the like-ion repulsion and the local electroneutrality hypotheses are introduced in formulating the binary interaction parameters for the excess Gibbs free energy expression. This simplification results in the elimination of the “ionic charge fraction quantities” from the excess Gibbs free energy expression, eq 19, and the subsequent derivations for activity coefficients. For single electrolyte systems, both the 1986 model and the current formulation yield

identical NRTL expressions for both molecular and ionic activity coefficients and identical values for the NRTL moleculeelectrolyte binary parameters. For multicomponent electrolyte systems, both the 1986 model and the current formulation generally follow the Harned’s rule, that is, the logarithm of the mean ionic activity coefficient of one electrolyte in a mixture of constant molality is directly proportional to the molality of the other electrolyte.8 Figures 1 and 2 show the molality scale mean ionic activity coefficients of sodium chloride and potassium chloride, respectively, for a constant total molal sodium chloride-potassium chloride-water system (4m) as functions of the mole fraction of potassium chloride in the salt mixture. The solid lines represent the mean ionic activity coefficients as computed from the 1986 model with τNaCl,KCl ) -τKCl,NaCl ) -1, 0, and 1. The dotted lines are from the new model. Although the two models do give slightly different predictions for the mean ionic activity coefficients, the models behave rather similarly and all curves are nearly linear and follow the Harned’s rule. Note that, in generating the plots, the binary interaction parameters for the water-sodium chloride pair and the water-potassium chloride pair are obtained directly from the 1986 paper. Model Applications To illustrate the utility of the symmetric eNRTL model, we present modeling results with four mixed-solvent electrolyte systems, since this symmetric model should offer significant advantages with such systems over the unsymmetric one. We demonstrate the use of the model with systems that involve vapor-liquid equilibrium, liquid-liquid equilibrium, and solid-liquid equilibrium. In addition, we choose two different classes of electrolytes: elemental inorganic electrolytes such as NaCl and NaBr and complex organic electrolytes such as ionic liquids. Note that these four systems are single electrolyte systems. Therefore, both the 1986 model and the current model

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Figure 2. Prediction of KCl molality scale mean ionic activity coefficient of a constant total molal NaCl-KCl-water system with various electrolyteelectrolyte binary interaction parameters. Solid lines, 1986 model results; dotted lines, current model results.

Figure 3. Representation of the experimental equilibrium composition (mole fraction) data19 for the ternary system 2-propanol (1)-water (2)-potassium acetate (3) at 101.3 kPa. Note that z_salt stands for liquid phase mole fraction for potassium acetate.

reported in Table 1. Also included in Table 1 is the dielectric constant of 2-propanol.

Table 1. eNRTL Binary Interaction Parameters for the 2-Propanol-Water-Potassium Acetate Systema component (1)

water

water

2-propanol

component (2)

2-propanol

potassium acetate

potassium acetate

τ12 τ21 R21 ) R21

826.6/T (K) 154.4/T (K) 0.4915

8.576 -4.596 0.2

2.962 3.058 0.6

a

ε2-propanol ) 19.9.

should yield the same results except that the last two systems could not be conveniently studied with the 1986 unsymmetric model because they involve nonaqueous electrolyte systems. Vapor-Liquid Equilibrium of 2-Propanol-Water-Potassium Acetate System. Wu et al.19 reported vapor-liquid equilibrium data at atmospheric pressure for the systems 1-propanol-water-potassium acetate and 2-propanol-water-potassium acetate under fixed salt mole fractions. We model the vapor-liquid equilibrium data for the 2-propanol-water-potassium acetate system in this investigation. Modeling the ternary system of 2-propanol-water-potassium acetate requires binary interaction parameters for the 2-propanol-water pair, the water-potassium acetate pair, and the 2-propanol-potassium acetate pair. We obtain the binary parameters for the 2-propanol-water pair and the water-potassium acetate pair from the Aspen Plus databank. These values are given in Table 1. With the nonrandomness factor for the 2-propanol-potassium acetate pair fixed at 0.6, the binary interaction parameters for the 2-propanol-potassium acetate pair are fitted against the isobaric VLE data of Wu et al.19 for the ternary system. The standard deviations for temperature, pressure, vapor composition of 2-propanol, and liquid composition of 2-propanol are set at 0.1 K, 1%, 0.01, and 0.01, respectively. The vapor and liquid compositions of potassium acetate are set to be error-free. The residual root-mean-square error (rrmse) for the fit, as defined in eq 83, is 2.34. The final regressed values for the binary interaction parameters are

rrmse )



k

m

∑∑ i)1 j)1

(

Zij - ZMij σij

k-n

)

2

(83)

where ZM ) measured (experimental) value; Z ) calculated value; σ ) standard deviation; i ) data point number; k ) total number of data points; j ) measured variable for a data point (such as temperature, pressure, or mole fraction); m ) number of measured variables for a data point; and n ) total number of adjustable parameters. Figure 3 shows the isobaric VLE data of Wu et al.19 at varying potassium acetate concentration and the correlation results. Although the model correlations do not overlap exactly with the reported equilibrium compositions, the model correlations show proportional increases in equilibrium vapor phase 2-propanol composition (salting-out) as the potassium acetate salt is introduced to the 2-propanol-water mixed solvent and then gradually increased. The model representation of the isobaric VLE data is satisfactory. Liquid-Liquid Equilibrium of Water-Sodium Chloride-1-Propanol System. Gomis et al.20 reported liquid-liquid equilibrium data for the ternary systems water-sodium chloride-1-propanol, water-potassium chloride-1-propanol, and water-sodium chloride-2-propanol, all at 298.15 K. The liquid-liquid phase equilibrium behaviors of these three systems are quite similar. In this study, we focus on modeling liquid-liquid equilibria for the ternary system water-sodium chloride-1-propanol. Modeling the ternary system of water-sodium chloride-1propanol requires binary interaction parameters for the water-1propanol pair, the water-sodium chloride pair, and the 1-propanol-sodium chloride pair. The binary parameters for the water-1-propanol pair and for the water-sodium chloride pair are obtained from the Aspen Plus databank and the 1986 electrolyte NRTL paper, respectively. Their values are given

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 Table 2. eNRTL Binary Interaction Parameters for the Water-Sodium Chloride-1-Propanol Systema component (1)

water

water

1-propanol

component (2)

1-propanol

sodium chloride

sodium chloride

τ12 τ21 R21 ) R21

928.0/T (K) 181.2/T (K) 0.4687

9.023 -4.592 0.2

3.111 2.039 0.2

a

ε1-propanol ) 20.1.

Sodium Bromide Solubilities in Water-Methanol-Ethanol Mixed Solvents. Pinho and Macedo21 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. The extensive set of solubility data for NaBr in nonaqueous solvents and mixed solvents makes it a good candidate to study the usability of the symmetric model in correlating solubilities in solvents and solvent mixtures. Binary interaction parameters for various molecule-molecule pairs and water-sodium bromide pairs are readily available and they are retrieved from the Aspen Plus databank. Their values and the dielectric constants for various alcohols are summarized in Table 3. Binary interaction parameters for the methanol-sodium bromide pair and the ethanol-sodium bromide pairs are not available, and together with the symmetric solubility product constant for sodium bromide, they are used as adjustable parameters to fit against the solubility data of sodium bromide in water, methanol, ethanol, and their mixed solvents at 298.15 K. Here the symmetric solubility product constant, Ksp, is defined as sat sat Ksp ) xNa +γNa+xCl-γCl-

Figure 4. Representation of the experimental equilibrium composition (% mass) data20 for the ternary system water (1)-sodium chloride (2)-1propanol (3) at 298.15 K. Solid lines, experimental tie line; dotted lines, model prediction.

in Table 2. With the nonrandomness factor for the 1-propanolsodium chloride pair fixed at 0.2, the binary interaction parameters for the 1-propanol-sodium chloride pair are fitted against the LLE data of Gomis et al. for the ternary system. The standard deviations for the aqueous phase liquid compositions of water and sodium chloride are fixed at 0.01 while the standard deviations for the 1-propanol phase liquid compositions of water and sodium chloride are fixed at 0.01 and 0.001, respectively. The residual root-mean-square error (rrmse) for the fit, as defined in eq 83, is 1.79. The final regressed values for the binary interaction parameters are reported in Table 2. Also included in Table 2 is the dielectric constant of 1-propanol at 298.15 K. Figure 4 shows the experimental tie-lines and the calculated tie-lines from the model. The model provides excellent representation of the liquid-liquid equilibrium for the water-sodium chloride-1-propanol ternary system. Note the significant sodium chloride solubilities in the organic phase. These are results of relatively high dielectric constant for the organic solvent 1-propanol.

7795

(84)

where xsat i is the concentration of ion i at saturation and γi is the symmetric activity coefficient of ion i at the system concentration. In the regression, the nonrandomness factors for the methanol-sodium bromide pair and the ethanol-sodium bromide pair are fixed at 0.2 and 0.1, respectively. The standard deviations for sodium bromide solubilities in water, methanol, and ethanol are set to be 10%, 10%, and 20%, respectively. The standard deviations for the liquid compositions of the mixed solvents (i.e., water-methanol, water-ethanol, and methanolethanol) saturated with sodium bromide are set to be 1%. The residual rrmse for the fit, as defined in eq 83, is 6.35. The regressed binary interaction parameters and the symmetric solubility product constant are summarized in Table 3. Figure 5 shows the correlated sodium bromide solubilities in the water-methanol binary solvent, the water-ethanol binary solvent, and the methanol-ethanol binary solvent. The model provides adequate correlation for the highly nonideal sodium bromide solubilities in the mixed solvents. The solubility trends for sodium bromide in the water-methanol binary and the methanol-ethanol binary are well captured while the solubility trend for sodium bromide in the water-ethanol binary is off slightly. While the correlations could be improved upon further, perhaps by properly optimizing the nonrandomness factor in addition to the binary interaction parameters for the various molecule-electrolyte binary pairs, the results are very encouraging as the model predictions are qualitatively correct.

Table 3. eNRTL Binary Interaction Parameters for the Sodium Bromide-Water-Methanol-Ethanol Systema component (1)

water

water

methanol

water

methanol

ethanol

component (2)

methanol

ethanol

ethanol

sodium bromide

sodium bromide

sodium bromide

τ12 τ21 R21 ) R21

425.3/T (K) -127.8/T (K) 0.2994

670.4/T (K) -55.2/T (K) 0.3031

-13.1/T (K) 6.4/T (K) 0.3356

9.527 -4.790 0.2

5.910 -3.863 0.2

6.118 -4.450 0.1

a

1 1 ( T(K) 298.15 ) 1 1 ) 24.1113 + 12601.6( T(K) 298.15 )

εmethanol ) 32.6146 + 12805.8 εethanol

Ksp,NaBr(s) ) -7.157

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Figure 6. Representation of the experimental equilibrium composition (% mole) data23 for the ternary system [hmim][Tf2N]-water-ethanol at 295 K and 1 atm. Solid lines and solid squares, experimental tie line; dotted lines and empty squares, model prediction. Figure 5. Model predictions vs experimental solubility (% mass) data21 for sodium bromide in water-methanol, water-ethanol, and methanolethanol mixed solvents at 298.15 K. Table 4. eNRTL Binary Interaction Parameters for the [hmim][Tf2N]-Ethanol-Water Systema component (1)

water

water

ethanol

component (2)

ethanol

[hmim][Tf2N]

[hmim][Tf2N]

τ12 τ21 R21 ) R21

670.4/T (K) -55.2/T (K) 0.3031

7.123 -1.397 0.2

6.480 -3.274 0.2

a

εethanol ) 24.1113 + 12601.6

1 1 ( T(K) 298.15 )

Liquid-Liquid Equilibria of [hmim][Tf2N]-1-ButanolWater System and [hmim][Tf2N]-Ethanol-Water System. Ionic liquids behave like nonelectrolytes and their phase behaviors can be qualitatively modeled with nonelectrolyte models.15,22 However, partial dissociation of ionic liquids has been recognized and investigated even for pure ionic liquids.15 Chapeaux et al.23 recently reported ternary liquid-liquid equilibrium phase diagrams for the systems 1-hexyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl) imide ([hmim][Tf2N])ethanol-water and 1-hexyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl) imide-1-butanol-water at 295 K and 1 atm. In addition, they used four excess Gibbs free energy models (NRTL, eNRTL, UNIQUAC, and UNIFAC) to predict the behaviors of the ternary LLE systems. To examine the potential applicability of the symmetric model to ionic liquids, here we apply the model to the data of Chapeaux et al. and we treat ionic liquids (ILs) as completely dissociated electrolytes. Table 4 reports the values of the various binary interaction parameters for the system [hmim][Tf2N]-ethanol-water. The water-ethanol binary parameters are obtained directly from the Aspen Plus databank. The ethanol-[hmim][Tf2N] binary parameters are obtained from fitting VLE data for the ethanol-[hmim][Tf2N] binary at 353.16 K.24 The water-[hmim][Tf2N] binary parameters are obtained from fitting the liquid-liquid equilibrium experimental tie line data of Chapeaux et al. for the [hmim][Tf2N]-ethanol-water ternary system.

Table 5. eNRTL Binary Interaction Parameters for the [hmim][Tf2N]-1-Butanol-Water Systema component (1)

water

water

1-butanol

component (2)

1-butanol

[hmim][Tf2N]

[hmim][Tf2N]

τ12 τ21 R21 ) R21

4.103 -0.196 0.3

7.123 -1.397 0.2

4.950 -2.526 0.2

a

ε1-butanol ) 17.7.

In fitting the ternary LLE data of Chapeaux et al., the nonrandomness factor for the water-[hmim][Tf2N] pair is fixed at 0.2. The standard deviations for the aqueous phase compositions of [hmim][Tf2N] and water are fixed at 0.001 and 0.01, respectively. The standard deviations for the IL phase compositions of [hmim][Tf2N] and water are fixed at 0.01. The rrmse for the fit, as defined in eq 83, is 7.51. Figure 6 shows the model adequately represents the “Type 1 LLE phase diagram” reported by Chapeaux et al.23 for the ternary system [hmim][Tf2N]-ethanol-water at 295 K. The [hmim][Tf2N]-water binary is partially miscible while the ethanol-water binary and the ethanol-[hmim][Tf2N] binary are completely miscible. There is a significant departure between the model-predicted LLE tie-lines and the experimental LLE tie-lines as the ethanol content increases. However, given the “complete dissociation” simplifying assumption of the model treatment, the general scattering of the experimental LLE data, and the fact that the ethanol-[hmim][Tf2N] binary parameters are obtained from fitting VLE data at 353.16 K, the model representation for the ternary system can be considered satisfactory. Table 5 shows the values of the various binary interaction parameters for the system [hmim][Tf2N]-1-butanol-water. Here the binary parameters for the water-1-butanol binary are obtained from fitting miscibility data of the water-1-butanol binary at 303.15 K25. The water-[hmim][Tf2N] binary parameters are taken directly from Table 4. The 1-butanol-[hmim][Tf2N] binary parameters are obtained from fitting the liquid-liquid equilibrium experimental tie line data of Chapeaux et al. for the [hmim][Tf2N]-1-butanol-water ternary system. In fitting the ternary LLE data of Chapeaux et al., the nonrandomness factor for the 1-butanol-[hmim][Tf2N] pair is fixed at 0.2. The standard deviations for the aqueous phase compositions of [hmim][Tf2N] and water are fixed at 0.0001

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009

Figure 7. Representation of the experimental equilibrium composition (% mole) data23 for the ternary system [hmim][Tf2N]-water-1-butanol at 295 K and 1 atm. Solid lines and solid squares, experimental tie line; dotted lines and empty squares, model prediction.

and 0.01, respectively. The standard deviations for the second liquid phase compositions of [hmim][Tf2N] and water are fixed at 0.01. The rrmse for the fit, as defined in eq 83, is 1.63. Figure 7 shows the model gives excellent representation of the “Type 2 LLE phase diagram” reported by Chapeaux et al.23 for the ternary system [hmim][Tf2N]-1-butanol-water. The [hmim][Tf2N]-water binary and the water-1-butanol are partially miscible while the 1-butanol-[hmim][Tf2N] binary is completely miscible. There is an almost perfect match between the model-predicted LLE tie-lines and the experimental LLE tie-lines throughout the phase diagram. The model clearly captures this phase diagram very well. Conclusions In this paper we present a symmetric electrolyte NRTL activity coefficient model with the reference states chosen to be pure liquids for solvents and pure fused salts for electrolytes. The model includes a reformulated NRTL local interaction contribution expression and an extended Pitzer-Debye-Hu¨ckel long-range interaction contribution expression for multicomponent electrolytes. Retaining the basic hypotheses of the original electrolyte NRTL model, the model reduces to the original 1982 model for aqueous single electrolytes and closely resembles the 1986 model for aqueous multicomponent electrolytes when the model is transformed back to the aqueous phase infinite dilution reference state for ions. While the 1986 unsymmetric model provides a useful engineering model for correlating phase behavior of aqueous electrolytes, the symmetric electrolyte activity coefficient model provides a comprehensive thermodynamic framework for correlating phase behaviors of aqueous electrolytes, nonaqueous electrolytes, and mixed solvent electrolytes. Acknowledgment C.-C. Chen thanks G. M. Bollas (M.I.T.) and L.D. Simoni (University of Notre Dame) for their discussions and assistance during the development of this symmetric electrolyte NRTL model. Literature Cited (1) Chen, C.-C. Toward Development of Activity Coefficient Models for Process and Product Design of Complex Chemical Systems. Fluid Phase Equilib. 2006, 241, 103–112.

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(2) Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268–277. (3) Wang, P.; Anderko, A.; Young, R. D. A Speciation-Based Model for Mixed-Solvent Electrolyte Systems. Fluid Phase Equilib. 2002, 203, 141–176. (4) Thomsen, K.; Rasmussen, P.; Gani, R. Simulation and Optimization of Fractional Crystallization Processes. Chem. Eng. Sci. 1998, 53, 1551–1564. (5) 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. (6) Chen, C.-C.; Evans, L. B. A Local Composition Model for the Excess Gibbs Energy of Aqueous Electrolyte Systems. AIChE J. 1986, 32, 444–454. (7) Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. The NBS Tables of Chemical Thermodynamic Properties: Selected Values for Inorganic and C1 and C2 Organic Substances in SI Units. J. Phys. Chem. Ref. Data. 1982, 11, Suppl. No. 2. (8) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths, ON, Canada, 1970. (9) Mock, B.; Evans, L. B.; Chen, C.-C. Thermodynamic Representation of Phase Equilibria of Mixed-Solvent Electrolyte Systems. AIChE J. 1986, 32, 1655–1664. (10) Chen, C.-C.; Mathias, P. M.; Orbey, H. Use of Hydration and Dissociation Chemistries with the Electrolyte-NRTL Model. AIChE J. 1999, 45, 1576–1586. (11) Chen, C.-C.; Bokis, C. P.; Mathias, P. M. Segment-Based Excess Gibbs Energy Model for Aqueous Organic Electrolytes. AIChE J. 2001, 47, 2593–2602. (12) Chen, C.-C.; Song, Y. Generalized Electrolyte-NRTL Model for Mixed-Solvent Electrolyte Systems. AIChE J. 2004, 50, 1928–1941. (13) Bollas, G. M.; Chen, C.-C.; Barton, P. I. Refined Electrolyte-NRTL Model: Activity Coefficient Expressions for Application to Multi-electrolyte Systems. AIChE J. 2008, 54, 1608–1624. (14) Chen, C.-C.; Mathias, P. M. Applied Thermodynamics for Process Modeling. AIChE J. 2002, 48, 194–200. (15) Simoni, L. D.; Lin, Y.-D.; Brennecke, J. F.; Stadtherr, M. A. Modeling Liquid-Liquid Equilibrium of Ionic Liquid Systems with NRTL, ElectrolyteNRTL, and UNIQUAC. Ind. Eng. Chem. Res. 2008, 47, 256–272. (16) Pitzer, K. S. Electrolytes. From Dilute Solutions to Fused Salts. J. Am. Chem. Soc. 1980, 102, 2902–2906. (17) Pitzer, K. S.; Simonson, J. M. Thermodynamics of Multicomponent, Miscible, Ionic Systems: Theory and Equations. J. Phys. Chem. 1986, 90, 3005–3009. (18) Rashin, A. A.; Honig, B. Reevaluation of the Born Model of Ion Hydration. J. Phys. Chem. 1985, 89, 5588–5593. (19) Wu, W. L.; Zhang, Y. M.; Lu, X. H.; Wang, Y. R.; Shi, J.; Lu, B. C.-Y. Modification of the Furter Equation and Correlation of the VaporLiquid Equilibrium for Mixed-Solvent Electrolyte Systems. Fluid Phase Equilib. 1999, 154, 301–310. (20) Gomis, V.; Ruiz, F.; De Vera, G.; Lopez, E.; Saquete, M. D. Liquid-Liquid-Solid Equilibria for the Ternary Systems Water-Sodium Chloride or Potassium Chloride-1-Propanol or 2-Propanol. Fluid Phase Equilib. 1994, 98, 141–147. (21) 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. (22) Chen, C.-C.; Simoni, L. D.; Brennecke, J. F.; Stadtherr, M. A. 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. (23) Chapeaux, A.; Simoni, L. D.; Ronan, T. S.; Stadtherr, M. A.; Brennecke, J. F. Extraction of Alcohols from Water with 1-n-Hexyl-3methylimidazolium Bis(trifluoromethylsulfonyl) Imide. Green Chem. 2008, 10, 1301–1306. (24) Kato, R.; Gmehling, J. Systems with Ionic Liquids: Measurement of VLE and γ∞ Data and Prediction of Their Thermodynamic Behavior Using Original UNIFAC, Mod. UNIFAC(Do), and COSMO-RS(O1). J. Chem. Thermo. 2005, 37, 603–619. (25) Marigliano, A. C. G.; de Doz, M. B. G.; So´limo, H. N. Influence of Temperature on the Liquid-Liquid Equilibria Containing Two Pairs of Partially Miscible Liquids. Water + Nitromethane + 1-Butanol Ternary System. Fluid Phase Equilib. 1998, 149, 309–322.

ReceiVed for reView March 20, 2009 ReVised manuscript receiVed June 26, 2009 Accepted June 30, 2009 IE9004578