Model for the Gibbs Excess Energy of Mixed-Solvent (Chemical

Ind. Eng. Chem. Res. 2005, 44, 201-225

201

Model for the Gibbs Excess Energy of Mixed-Solvent (Chemical-Reacting and Gas-Containing) Electrolyte Systems A Ä lvaro Pe´ rez-Salado Kamps† Lehrstuhl fu¨ r Technische Thermodynamik, University of Kaiserslautern, D-67653 Kaiserslautern, Germany

A model for the Gibbs excess energy of mixed-solvent (chemical-reacting) electrolyte systems is presented. The activities of solvent and solute species are calculated by applying a new extension of Pitzer’s equation for the excess Gibbs energy of aqueous electrolyte solutions, which allows for solvent mixtures. The capability of the new model to simultaneously describe such properties as, e.g., the mean ionic activity coefficient and the solubility of a salt (for example, sodium chloride) in a binary solvent mixture (for example, methanol + water), as well as the influence of that salt on the vapor-liquid equilibrium of that binary solvent mixture, is tested. The new model gives an explanation for the “salting-out” and “salting-in” effects resulting, e.g., from the addition of a salt to a liquid mixture of volatile solvents. The new model can also be applied to describe the gas solubility in mixed solvents in the presence of electrolytes. The capability of the new model to describe the solubility of a single gas in a binary solvent mixture is tested by dealing with the solubility of carbon dioxide in aqueous solutions of methanol over the whole range of solvent composition, i.e., from pure water to pure methanol. Introduction The present work was performed following a major project (in cooperation with the chemical industry) dealing with experimental and theoretical investigations on the solubility of single gases (as well as on the simultaneous solubility of chemical-reacting gases) in aqueous solutions of organic compounds and electrolytes. The prime aim of that project was to develop a practical, if necessary semiempirical but predictive, engineering-oriented, reliable, and comprehensive method, which, in principle, is able to thermodynamically consistently describe not only such gas solubilities but also, e.g., speciation, phase equilibria [osmotic coefficients, solid-liquid (SLE), vapor-liquid (VLE), vaporsolid-liquid (VSLE), and liquid-liquid (LLE) equilibria, etc.], mean ionic activity coefficients, enthalpy changes, etc., and which is suitable for chemical process simulation involving complex, chemical-reacting multiphase, multicomponent systems as encountered in industrial practice. Because of the lack of reliable experimental information, none of the existing models for describing the thermodynamic properties of mixed-solvent electrolyte systems (e.g., a review has been given by Anderko et al.;1 cf. also, e.g., ref 2) has yet been sufficiently tested for describing gas solubilities in mixed-solvent (chemicalreacting) electrolyte systems. Therefore, the goals of the project were, at first, to provide an extensive database on (1) the solubility of single gases in two-component solvent mixtures (water + an organic compound), (2) the influence of salts on the VLE of those two-component solvent mixtures, (3) the influence of salts on the gas solubility in such aqueous organic solutions, (4) the simultaneous solubilities of chemical-reacting gases in aqueous solvent mixtures, and (5) the influence of salts on the simultaneous solubilities of chemical-reacting gases in such solvent mixtures. Especially, the whole † Tel.: +49 631 205 2761. Fax: +49 631 205 3835. E-mail: [email protected].

solvent composition range was to be covered, from pure water to the pure organic component. Methanol was selected as the (first) organic solvent component and carbon dioxide as the (first) gas. Ammonia and carbon dioxide were chosen as the pair of chemical reactive gases and sodium chloride and sodium sulfate as the (first) strong electrolytes. The selection was mainly influenced by previous work (by Maurer and co-workers; see, e.g., refs 3-14) on the simultaneous solubilities of ammonia and carbon dioxide in water and aqueous solutions of these salts. In the present work, a model for the Gibbs excess energy of mixed-solvent (chemical-reacting and gascontaining) electrolyte systems is presented. Activities are calculated by applying a new extension of Pitzer’s equation for the excess Gibbs energy of aqueous solutions of strong electrolytes.15-17 Pitzer’s equation belongs to the group of osmotic virial equations, and therefore, in principle, it does not allow for a change in the solvent. However, the extension presented here thermodynamically consistently overcomes that restriction, allowing for solvent mixtures. [Pitzer’s equation has often been applied for describing mean ionic activity coefficients of salts in solvent mixtures (cf., e.g., refs 18 and 19), but general equations for calculating the activities of all species (ionic and molecular solute species, as well as solvent components) in such mixed solvents have not been presented before.] The capability of the new model to simultaneously describe mean ionic activity coefficients and the solubility of a salt (sodium chloride) in a binary solvent mixture (methanol + water), as well as the influence of that salt on the VLE of that binary solvent mixture (up to the solubility limit of the salt) is tested. The new model gives an explanation of the “salting-out” and “salting-in” effects resulting, e.g., from the addition of a salt to a liquid mixture of volatile solvents. For example, increasing the concentration of sodium chloride in the liquid (at constant temperature and salt-free concentration of methanol and water in that liquid) results in an increase

10.1021/ie049543y CCC: $30.25 © 2005 American Chemical Society Published on Web 12/03/2004

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Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005

of the concentration of methanol in the gaseous phase (methanol is “salted-out”, and water is “salted-in”; cf., e.g., refs 20-22). The new model can also be applied to describe the solubility of a single gas as well as the simultaneous solubilities of two (or more) gases in mixed-solvent chemical-reacting electrolyte solutions. In the present work, the new model is applied to describe the solubility of carbon dioxide in aqueous solutions of methanol over the whole range of solvent composition, i.e., from pure water to pure methanol.

The chemical potential of species i in the ideal solution results from

Thermodynamic Framework

solution ) µpure µideal W W (T,p) +

The Gibbs energy of a liquid solution of mixed solvents containing volatile and nonvolatile, neutral and ionic solutes may be, as usual, split into an ideal solution contribution and an excess contribution:

G ) Gideal solution + GE

(1)

Before equations are given for those two contributions, a concentration scale (e.g., the mole fraction scale, the molality scale, the molarity scale) must be chosen. The mole fraction scale was chosen here. For simplicity reasons, only a two-component solvent mixture [i ) W (water) and M (methanol)] is considered, but all equations can straightforwardly be extended to multicomponent solvent mixtures. The Ideal Solution The properties of the solvent components (i ) W and M) are normalized according to Raoult’s law; i.e., the reference state for the chemical potential of a solvent component i is the pure liquid i (at equilibrium tem(T,p). The properties of perature and pressure): µpure i the solute species k are normalized according to Henry’s law; i.e., the reference state for the chemical potential of a solute species k is the pure solute species k experiencing interactions as it is infinitely diluted in the solvent mixture (at equilibrium temperature and ˜ M), where pressure, cf., e.g., Sander et al.20): µ∞,x k (T,p,x x˜ M is the mole fraction of methanol in the solute-free solvent mixture:

x˜ i )

ni nW + nM

(i ) W and M)

(2)

The Gibbs energy of the ideal solution is therefore ideal solution

G

)

pure nWµpure W (T,p) + nMµM (T,p) nkµ∞,x ˜ M) + k (T,p,x k*W,M

∑

+

nk ln xk] (3)

k*W,M

xi is the true mole fraction of species i (i ) W, M, and all solute species):

xi ) ni/nT

i ) W, M, and all solute species

(

)

∂Gideal solution ∂ni

For a solute species k, this equation gives solution ) µ∞,x ˜ M) + RT ln xk µideal k k (T,p,x

(4)

where ni is the amount of substance of species i and nT is the total amount of substance in the solution (including solvent and solute species):

∑i ni

(7)

and for the solvent components water and methanol

RT ln xW - x˜ MM*

∑

( )( )

∑

( )( )

k*W,M solution ) µpure µideal M M (T,p) +

RT ln xM + x˜ WM*

k*W,M

mk ∂µ∞,x k

m°

∂x˜ M

i ) W, M, and all solute species (5)

(8)

T,p

mk ∂µ∞,x k

m°

∂x˜ M

(9)

T,p

respectively. M* is the (mean) relative molar mass of the solute-free solvent mixture divided by 1000. For the binary solvent mixture under consideration in the present work, it is

M* ) x˜ WM/W + x˜ MM/M

(10)

where M/W ) 0.018 015 28 (for water) and M/M ) 0.032 042 16 (for methanol). mk is the molality of solute species k (m° ) 1 mol/kg), i.e., the amount of substance nk of this species k per ˜M kilogram of a solute-free solvent mixture [m ˜ W and m are the masses of pure water and pure methanol, respectively (in kg)]:

mi )

ni m ˜W+m ˜M

(11)

xi mi 1 ) m° (xW + xM) M*

(12)

The molar Gibbs energy of transfer of solute species k from pure water to an aqueous solution of, e.g., methanol is defined (on the mole fraction scale) as

(13)

It is the change in the reference state of the chemical potential of solute species k encountered in the isothermal and isobaric transfer from pure water to the solvent mixture. Equations 8 and 9 may therefore also be written as solution µideal ) µpure W W (T,p) +

RT ln xW - x˜ MM*

∑

( )( )

∑

( )( )

k*W,M

nT )

(6)

T,p,nj*i

∞,x ˜ M) - µk,W (T,p) ∆tGxk(T,p,x˜ M) ) µ∞,x k (T,p,x

∑

RT[nW ln xW + nM ln xM +

solution µideal ) i

solution µideal ) µpure M M (T,p) +

RT ln xM + x˜ WM*

k*W,M

mk ∂∆tGxk

m°

∂x˜ M

mk ∂∆tGxk

m°

∂x˜ M

(14)

T,p

T,p

(15)

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 203

The last contribution in eqs 14 and 15 describes the “solute effect” on the chemical potentials of the solvent components in the ideal solution. As will be shown in the modeling section, this contribution is responsible for the so-called “salting-out” and “salting-in” effects resulting, for example, from the addition of a salt to a liquid mixture of volatile solvents. The (molar) Gibbs energy of transfer of an ionic solute (from a pure solvent to a solvent mixture) is usually determined from electromotive force (emf) measurements, whereas for molecular dissolved gases, it is determined from gas solubility measurements. Unfortunately, (molar) Gibbs energies of transfer are still only known for very few solutes and solvent mixtures in very limited temperature (and pressure) ranges (for ionic solutes, see, e.g., refs 23 and 24). If no experimental information is available, at least for the ionic solute species, those (molar) Gibbs energies of transfer (e.g., from pure water to a solvent mixture) may be estimated from the Born equation25,26 (which is assumed to be based on the molarity scale):

∆tGck

[

]

2 NAe2 1 1 zk ) 8π0 W rk

(16)

Equation 16 gives an approximation for ∆tGck resulting only from differences in the relative dielectric constant of pure water (W) and the solvent mixture () and can easily be converted to the molality or mole fraction scales. NA is Avogadro’s number, e the electron charge, 0 the permittivity of vacuum, zk the number of charges on the solute species k, and rk the ionic radius of solute species k (see, e.g., ref 27). The expression for the Gibbs excess energy used in the present work, which is presented in the next section, does not require a Born term (see, e.g., ref 28) because the reference state for the chemical potential of any (ionic or molecular) solute species k is the pure solute species k experiencing interactions as it is infinitely diluted in the solvent mixture (at equilibrium temper˜ M)]. However, informaature and pressure) [µ∞,x k (T,p,x tion on the transfer properties of solutes is required if, e.g., (i) the influence of salts on the partial pressures of volatile solvent components, (ii) chemical reactions (involving those ionic and molecular solute species), (iii) the solubility of a gas in the solvent mixture, (iv) LLE involving solute (ionic and/or neutral) species, etc., are to be described. Gibbs Excess Energy Model The Gibbs excess energy is assumed to result from two contributions:

GE ) GEI + GEII

The second term (GEII) describes any change in the Gibbs excess energy resulting from the addition of neutral or ionic solutes to the solvent mixture. It is expressed as an osmotic virial equation containing parameters only for interactions between solutes within the solvent mixture under consideration; i.e., those interaction parameters depend on the composition of the solvent mixture (and on the temperature). Because of its accuracy and relative simplicity, Pitzer’s GE equation15-17 is a commonly used tool for the description of thermodynamic properties of aqueous electrolyte systems, and it has turned out to be one of the most powerful equations for calculating activity coefficients in aqueous electrolyte solutions containing also volatile neutral species (i.e., gases) (see, e.g., refs 3-14 and 3051). Therefore, it was chosen here for this second term (GEII), but it was first converted from the molality scale to the mole fraction scale (in order to enable a combination with the UNIQUAC equation), and second it was extended, allowing for a solvent mixture of varying composition. The Gibbs excess energy of a solvent mixture solution, which does not contain any solutes, solely results from the first term in eq 17: GE ) GEI (GEII ) 0), and model parameters (e.g., UNIQUAC interaction parameters) may be taken from the literature. Furthermore, the Gibbs excess energy of a solution containing any number of solutes but only one solvent component solely results from the second term in eq 17: GE ) GEII (GEI ) 0). If water is the only solvent component, this second term exactly matches Pitzer’s GE equation for aqueous electrolyte solutions. For Pitzer’s model, a lot of reliable information (Pitzer’s parameters for interactions between ionic and/or neutral solutes in aqueous solutions) is available (see, e.g., refs 3-14, 16, and 30-51). Without any changes, the numerical values of these parameters can be adopted here. The Gibbs excess energy resulting from the UNIQUAC equation (GEI ) is calculated as the sum of combinatorial and residual terms: E E E GEI ) GUNIQUAC ) Gcombinatorial + Gresidual

For the binary solvent mixture where x˜ i is the mole

[

]

[

]

GE (nW + nM)RT

GE (nW + nM)RT

[

( )

φ˜ W z qWx˜ W ln 2 θ˜ W

( )

( ) ( )]

φ˜ W φ˜ M + x˜ M ln x˜ W x˜ M φ˜ M + qMx˜ M ln (19) θ˜ M

) x˜ W ln

combinatorial

(17)

The first term (GEI ) describes the Gibbs excess energy of the solute-free solvent mixture; i.e., it does only consider (interactions between) solvent components. Depending on that mixture and on its properties (e.g., complete miscibility or not, etc.), any common expression for the Gibbs excess energy may be chosen (e.g., NRTL, UNIQUAC, UNIFAC, etc.). The UNIQUAC equation29 (with temperature-dependent interaction parameters) is used in the present work to describe the properties of the water + methanol solvent mixture.

(18)

) -qWx˜ W ln(θ˜ W + θ˜ MΨMW) -

residual

qMx˜ M ln(θ˜ WΨWM + θ˜ M) (20)

fraction of component i in the solute-free solvent mixture (cf. eq 2). The coordination number z is set equal to 10. Segment fraction φ˜ i and area fraction θ˜ i are given by

φ˜ i )

x˜ iri x˜ WrW + x˜ MrM

(i ) W, M)

(21)

θ˜ i )

x˜ iqi x˜ WqW + x˜ MqM

(i ) W, M)

(22)

204


Pure-component UNIQUAC constants ri and qi may be adopted, e.g., from the Dortmund Data Bank.52 The two adjustable UNIQUAC interaction parameters appearing for each binary mixture (here ΨMW and ΨWM) are assumed to depend on the temperature according to

(

Ψij ) exp aij +

)

bij T

(ij ) MW, WM)

(23)

The Gibbs excess energy resulting from Pitzer’s equation (GEII) is calculated from

GEII ) GEPitzer + GEconv

stands for the extension of Pitzer’s Gibbs excess energy equation, allowing for any solvent (not only water) or any solvent mixture (based on the molality scale). It consists of Pitzer’s modified Debye-Hu¨ckel term and a virial expansion in the molalities of all solute species (i, j, and k), where the Debye-Hu¨ckel term and the osmotic virial coefficients depend on the temperature, ionic strength, and composition of the solvent mixture.

M*(nW + nM)RT

) f(I,x˜ M) +

m i mj

λij(I,x˜ M) + ∑∑ i,j*W,M m° m° mi mj mk

µijk(x˜ M) ∑∑∑ i,j,k*W,M m° m° m°

(25)

The ionic strength is given by

I)

1

(31)

(1) (2) All binary parameters β(0) ij , βij , and βij (and, in princi(1) (2) ple, also Rij and Rij ) depend (on the temperature and) ˜ M), on the composition of the solvent mixture [β(0) ij (x (2) (1) (2) β(1) (x ˜ ), β (x ˜ ), R (x ˜ ), and R (x ˜ )]. g(x) is defined as M M M M ij ij ij ij

g(x) )

2 [1 - (1 + x) exp(-x)] x2

(32)

In principle, “symmetrical and unsymmetrical mixing terms” (see ref 17) may be considered too. For simplicity reasons, they are not included here. Furthermore, it is the usual practice to even neglect all parameters describing interactions between ionic species carrying charges of the same sign. As mentioned before, the equation for GEPitzer given above (eq 25) is based on the molality scale, in which the reference state for the chemical potential of a solvent component i is still the pure liquid i (at equilibrium (T,p), whereas in the temperature and pressure): µpure i reference state for the chemical potential of a solute species k, that species has a concentration of 1 mol/kg of the solvent mixture and interacts as it is infinitely diluted in the solvent mixture (at equilibrium temper˜ i). Therefore, GEPitzer has ature and pressure): µ∞,m k (T,p,x to be converted to the mole fraction scale if it is to be combined with the UNIQUAC term (GEI ). GEconv accounts for this conversion:

GEconv/(nTRT) ) -[1 - (xW + xM) + ln(xW + xM)] (33)

mi

zi2 ∑ 2 i m°

(26) Activity Coefficients

where zi is the number of charges on the solute species i. The Debye-Hu¨ckel term, as modified by Pitzer, is

f(I,x˜ M) ) -Aφ

(1) (1) (2) (2) λij ) β(0) ij + βij g(Rij xI) + βij g(Rij xI)

(24)

GEPitzer

GEPitzer

According to Pitzer’s equation, λij(I,x˜ M) is expressed as

4I ln(1 + bxI) b

(27)

Because the GE equation presented here is based on the mole fraction scale, the excess chemical potential (and the activity coefficient) of both solute and solvent species result from

µEi ) (∂GE/∂ni)T,p,nj*i ) RT ln γxi i, j ) M, W, and all solute species (34)

Aφ is the Debye-Hu¨ckel parameter:

(

e2 1 Aφ ) (2πNAdm°)1/2 3 4π0kBT

)

The activity coefficient is therefore given by

3/2

(28)

kB is the Boltzmann constant and T is the absolute temperature. Because the specific density (d) and the relative dielectric constant () of the (solute-free) solvent mixture, as well as (in principle also) the parameter b (which was set to 1.2 by Pitzer for aqueous solutions), depend on the composition of the solvent mixture [d(x˜ M), (x˜ M), b(x˜ M)], f(I,x˜ M) is a function of that composition too. λij(I,x˜ M) and µijk(x˜ M) are osmotic virial coefficients for interactions between solute species. They are symmetric:

λij ) λji

(29)

µijk ) µikj ) µjik ) µjki ) µkij ) µkji

(30)

ln γxi ) ln γi,I + ln γi,II ) ln γi,UNIQUAC + ln γi,Pitzer + ln γi,conv (35) For a solute species i, the following expressions are obtained:

ln γi,UNIQUAC ) 0

(36)

ln γi,conv ) -ln(xW + xM)

(37)

mj zi2 ln γi,Pitzer ) f ′ + 2 λij + 2 j*W,M m° m j mk m j mk zi2 λ′jk + 3 µijk (38) 2 j,k*W,M m°m° j,k*W,M m°m°

∑

∑∑

∑∑


where

For methanol

f′)

() ∂f ∂I

λ′ij )

) -2Aφ

x˜ M

( ) ∂λij ∂I

x˜ M

[

]

xI 2 + ln(1 + bxI) b 1 + bxI

(39)

ln γM,Pitzer )

[

(1) (2) (2) ) β(1) ij g′(Rij xI) + βij g′(Rij xI) (40)

2

and

[ (

( )

∂g(x) g′(x) ) ∂I

]

)

mi mj mk

[( ) ∂f

∂x˜ M

()

[

θ˜ W θ˜ W + θ˜ MΨMW

-

θ˜ WΨWM + θ˜ M

ln

[( )

]

( ) [

θ˜ W + θ˜ MΨMW

-

(43)

( )

∂β(0) ∂β(1) ∂β(2) ∂λij ij ij ij (1) x ) + g(Rij I) + g(R(2) ij xI) + ∂x˜ M ∂x˜ M ∂x˜ M ∂x˜ M

θ˜ WΨWM + θ˜ M

[

xI

]

θ˜ M

]

]

(49)

β(1) ij

]

(1) (2) ∂g(R(1) ∂g(R(2) ij xI) ∂Rij ij xI) ∂Rij (2) + βij (50) ∂(R(1)xI) ∂x˜ M ∂(R(2)xI) ∂x˜ M ij

ij

with

[ (

)

∂g(x) x2 4 exp(-x) )- 3 1- 1+x+ ∂x 2 x

qM 1 - ln(θ˜ WΨWM + θ˜ M) θ˜ WΨMW

(47)

f 1 ∂d 3 ∂ 1 ∂b ∂f ) + [2(If ′ - f) - f] ∂x˜ M I 2 d ∂x˜ M ∂x˜ M b ∂x˜ M

φ˜ M z φ˜ M φ˜ M φ˜ M +1- q ln +1+ x˜ M x˜ M 2 M θ˜ M θ˜ M

[

(46)

(48)

I

()

∂x˜ M

(1) (1) (2) (2) λij + Iλ′ij ) β(0) ij + βij exp(-Rij xI) + βij exp(-Rij xI)

For methanol

ln γM,UNIQUAC )

+

I

I3/2 If ′ - f ) -2Aφ 1 + bxI

]

θ˜ MΨWM

∂x˜ M

(42)

φ˜ W z φ˜ W φ˜ W φ˜ W +1- q ln +1+ x˜ W x˜ W 2 W θ˜ W θ˜ W qW 1 - ln(θ˜ W + θ˜ MΨMW) -

( ) ( )]

where

1 - ln(xW + xM) xW + xM

[( )

I

mi mj ∂λij

∑∑ i,j*W,M m° m°

mi mj mk ∂µijk

For water

ln

+

+

∑∑∑ i,j,k*W,M m° m° m°

For a solvent component i (i ) W and M)

ln γW,UNIQUAC )

]

µijk ∑∑∑ i,j,k*W,M m° m° m°

M*(1 - x˜ M)

2 x2 )- 2 1- 1+x+ exp(-x) x˜ M 2 Ix (41)

ln γi,conv ) 1 -

m i mj

(λij + Iλ′ij) + ∑∑ i,j*W,M m° m°

-M/M (If ′ - f) +

(44)

]

(51)

Activities The chemical potential of all solute and solvent species results from

For water

ln γW,Pitzer )

[

-M/W (If ′ - f) +

M*x˜ M

∑∑ m° m°(λij + Iλ′ij) +

i,j*W,M

2

mi mj mk

µijk ∑∑∑ i,j,k*W,M m° m° m°

[( ) ∂f

∂x˜ M

I

solution + µEi µi ) µideal i i ) M, W, and all solute species (52)

mi mj

+

] ( ) ( )] -

mi mj ∂λij

∑∑ i,j*W,M m° m°

Furthermore, by definition

∂x˜ M

I

∂x˜ M

k ) solute species (53)

+ RT ln aj µj ) µpure j

+

mi mj mk ∂µijk

∑∑∑ i,j,k*W,M m° m° m°

x µk ) µ∞,x k + RT ln ak

j ) M, W

(54)

Therefore, the activity of a solute species k follows from

(45)

axk ) xkγxk

(55)

206


and the activities of the solvent components water and methanol result from

ln aW ) ln xW -

ln aM ) ln xM +

x˜ MM* RT

x˜ WM* RT

( )( ) ∑ ( )( ) ∑

k*W,M

mk

∂∆tGxk

m°

∂x˜ M

m°

∂x˜ M

(56) + ln

γxM

T,p

(57)

Conversion to the Molality Scale As mentioned before, Pitzer’s equation for the Gibbs excess energy is based on the molality scale. The conversion of this equation to the mole fraction scale was necessary in order to be able to combine that equation with the UNIQUAC equation and in order to easily derive the expressions for the activity coefficients. However, it is desirable to reconvert all expressions to the molality scale. In that case, the chemical potential of a solute species k is expressed as

µk ) µ∞,m + RT ln am k k

( ) ( ) ∂∆tGxk ∂x˜ M

γxW

T,p

mk ∂∆tGxk

k*W,M

+ ln

and

∂∆tGm k ∂x˜ M

)

T,p

(

T,p

)

M/M - M/W M*

- RT

(66)

(cf. eqs 56 and 57). Properties and Parameters Dependent on the Solvent Composition The specific density d and the relative dielectric constant of the solvent mixture, which are required for the Debye-Hu¨ckel parameter (cf. eq 28), are calculated from empirical correlations; e.g., the specific density of the water + methanol solvent mixture is correlated using a Redlich-Kister type of expansion for the excess volume (cf., e.g., ref 22):

d)

(1 - x˜ M)MW + x˜ MMM MW

(1 - x˜ M)

dW

+ x˜ M

MM dM

∞

+ (1 - x˜ M)x˜ M

∑ Av,n(1 - 2x˜ M)n

n)0

(67)

(58)

˜ M) ) µ∞,x ˜ M) + RT ln M* µ∞,m k (T,p,x k (T,p,x

(60)

dW and dM are the specific densities of pure components water and methanol, respectively. MW and MM are the molar masses of water and methanol, respectively. The Redlich-Kister parameters Av,n (n ) 0, 1, 2, ...) may depend on the temperature (and pressure). Oster’s rule53 (which follows from Kirkwood’s theory of polar liquids;54 after several assumptions are applied, cf., e.g., refs 55 and 56) is used for correlating the relative dielectric constant of the solvent mixture. If . (2)-1, that approximation can further be simplified. The resulting expression is corrected by also introducing a Redlich-Kister expansion; e.g., the relative dielectric constant of the water + methanol solvent mixture is correlated by means of

x am k ) ak/M*

(61)

)

and its activity is defined as

am k )

( )

mk m γ m° k

(59)

The reference state chemical potential, as well as the activity (and the activity coefficient) of a solute species k, can easily be converted from the mole fraction scale to the molality scale (and vice versa). If the solvent mixture is water + methanol, then

x γm k ) (xW + xM)γk

(62)

As expected, combining eqs 35-38 with eq 62 gives

γm k

) γk,Pitzer

(63)

Note that the activities of the solvent components are independent of the concentration scale chosen for the chemical potential of the solute species. Nevertheless, to maintain the molality-scale basis for all properties, in the forthcoming sections the molar Gibbs energies of transfer will also be given on that molality-scale basis. The molar Gibbs energy of transfer of solute species k from pure water to an aqueous solution of, e.g., methanol is defined (on the molality scale) as ∞,m ˜ M) ) µ∞,m ˜ M) - µk,W (T,p) ∆tGm k (T,p,x k (T,p,x

(64)

Therefore

˜ M) ) ∆tGxk(T,p,x˜ M) + RT ln(M*/M/W) (65) ∆tGm k (T,p,x

Wx˜ W

MW dW

MW

x˜ W

dW

+ Mx˜ M

+ x˜ M

MM dM

MM

∞

+ (1 - x˜ M)x˜ M

A,n(1 - 2x˜ M)n ∑ n)0

dM (68)

where A,n (n ) 0, 1, 2, ...) may depend on the temperature (and pressure). As mentioned above, the parameter b, as well as all (1) (2) (1) (2) of the binary parameters β(0) ij , βij , βij , Rij , and Rij , and the ternary parameters µijk depend, in principle, on the composition of the (solute-free) solvent mixture. All of those parameters are to be determined empirically. The following general expression may be chosen for describing the dependency of those parameters [F(x˜ M) ) (1) (2) (1) (2) b, β(0) ij , βij , βij , Rij , Rij , and µijk] on the (two-component) solvent mixture composition: ∞

F(x˜ M) ) [x˜ WFW + x˜ MFM] + x˜ Wx˜ M

AF,n(x˜ W - x˜ M)n ∑ n)0 (69)


The first term on the right-hand side of eq 69 represents a (mole fraction) average value for parameter F based on the values for the pure solvent components water and methanol (FW, FM), whereas the second term represents the deviation from that average value by means of a Redlich-Kister expansion. Because x˜ W + x˜ M ) 1, eq 69 can be written as

F(x˜ M) ) (1 - x˜ M)FW + x˜ MFM + ∞

(1 - x˜ M)x˜ M

AF,n(1 - 2x˜ M)n ∑ n)0

(70)

pressure is not too high, the influence of the pressure on the equilibrium constant Kr is usually neglected. The equilibrium condition given by eq 73 can also be applied for describing the solubility limit of a salt in a solvent mixture. The resulting chemical equilibrium constant, i.e., the solubility product Ksp, depends on the temperature (and pressure) and solvent mixture composition. Adopting eq 70 to describe the dependency of µ∞,m ˜ j) on the composition of the two-component k (T,p,x solvent mixture water + methanol results in ∞,m ∞,m µ∞,m ˜ M) ) (1 - x˜ M)µk,W + x˜ Mµk,M + k (x ∞

The partial derivative with respect to x˜ M is

( ) ∂F

∂x˜ M

(1 - x˜ M)x˜ M

∞

) FM - FW + (1 - 2x˜ M)

AF,n(1 - 2x˜ M)n ∑ n)0

∑ nAF,n(1 - 2x˜ M)n-1 n)1

k

(75)

Combining eqs 73-75 gives

∞

2(1 - x˜ M)x˜ M

∑ Aµ ,n(1 - 2x˜ M)n n)0

(71)

m ˜ M) ) (1 - x˜ M) ln Kr,W + ln Km r (T,p,x (1 - x˜ M)x˜ M ∞ m [ νk,rAµk,n](1 - 2x˜ M)n x˜ M ln Kr,M RT n)0 k

∑ ∑

For example, for the binary interaction parameter β(0) ij , eq 70 leads to

(solute species)

(76) ˜ M) β(0) ij (x

) (1 -

(0) x˜ M)βij,W

+

(0) x˜ Mβij,M ∞

(1 - x˜ M)x˜ M

+

Equation 76 may be written as

∑ Aβ n)0

(0),n

(1 - 2x˜ M)n (72)

(0) (0) , βij,M , and Aβ(0),n (n ) 0, 1, 2, ...) may depend where βij,W on the temperature. As already mentioned, a lot of information is available in the literature on Pitzer parameters for interactions between ionic and/or neutral solutes in aqueous solu(0) (1) (2) (1) (2) tions (bW ) 1.2, βij,W , βij,W , βij,W , Rij,W , Rij,W , and µijk,W), which can directly be adopted. Finally, to simplify the general equations given above, further assumptions may be introduced. It is assumed (2) that the parameters b, R(1) ij , and Rij are independent of the solvent mixture composition, and the parameter values for water are adopted for the solvent mixture (1) (2) (2) (b ) 1.2, R(1) ij ) Rij,W, and Rij ) Rij,W).

m ln Km ˜ M) ) (1 - x˜ M) ln Kr,W + r (T,p,x ∞ (1 - x˜ M)x˜ M m AKr,n(1 - 2x˜ M)n (77) x˜ M ln Kr,M RT n)0

∑

m Kr,W (T,p) and Km r,M(T,p) are the chemical equilibrium constants (on the molality scale) of reaction r in a pure water solvent and a pure methanol solvent, respectively. AKr,n (n ) 0, 1, 2, ...) are temperature-dependent (and m pressure-dependent) coefficients and, along with Kr,W m and Kr,M, are to be determined from experimental data. Combining eqs 73 and 74 with the definition of the molar Gibbs energy of transfer of solute species k from pure water to an aqueous solution of methanol (on the molality scale; cf. eq 64) results in

˜ M) ) ln Km r (T,p,x

Chemical Reactions The reference state for the chemical potential of a solute species k depends on the composition of the solvent mixture {µ∞,m ˜ j)}. Therefore, if solute spek (T,p,x cies are involved in a chemical reaction, the equilibrium constant of that reaction will also depend on that composition. The condition for chemical equilibrium yields the following equation for a chemical reaction r:

ln Kr(T,p,x˜ j) ) -

∆rG RT

m (T,p) ln Kr,W

∏i aiν

]

i,r

RT

∑k

Furthermore, from eqs 77 and 78

∑k

νk,r∆tGm ˜ M) ) k (T,p,x

x˜ M

(73)

∑k

νk,r∆tGm ˜ M)1) + k (T,p,x

(solute species)

where the molar Gibbs energy of reaction is defined as

∆rG(T,p,x˜ j) )

∑i νi,rµref i

νk,r∆tGm ˜ M) (78) k (T,p,x

(solute species)

(solute species)

) ln[

1

∞

(1 - x˜ M)x˜ M (74)

νi,r is the stoichiometric factor of reactant i in reaction r (νi,r > 0 for a product and νi,r < 0 for an educt). If the

AK ,n(1 - 2x˜ M)n ∑ n)0 r

(79)

VLE Because of the unsymmetrical convention chosen in the present work, the VLE condition results in the

208


extended Raoult’s law for all solvent components i (here i ) W and M)

[

vi(p psi φsi exp RT

]

psi )

ai ) yipφi

(80)

and in the extended Henry’s law for all volatile solute components j (i.e., gases)

kH,jaj ) yjpφj

(81)

psi , φsi , and vi are properties of the pure solvent component i (vapor pressure, fugacity coefficient of the saturated vapor, and molar liquid volume, respectively). p is the total pressure, R is the universal gas constant, and yi and φi are the mole fraction and the fugacity coefficient, respectively, of component i in the vapor phase. The nonideality of the vapor phase is described here by a truncated virial equation of state (for details, see the Appendix). kH,j(T,p,x˜ i) is Henry’s constant of component j in the solvent mixture under consideration, which depends on the temperature, pressure, and (solute-free) solvent mixture composition. It is defined by means of

ln

[

]

∆solGj kH,j(T,p,x˜ i) ) p° RT

ideal gas ˜ M) ) µ∞,m (T,p,x˜ M) - µpure (T,p°) ∆solGm j (T,p,x j j (83)

Combining eqs 82 and 83 gives

RT ln

[

]

m kH,j (T,p,x˜ M)

p°

(84)

Applying this equation to a solute j in the pure solvents water and methanol leads to

[

]

[

]

∞,m ideal gas (T,p) ) µpure (T,p°) + RT ln µj,W j

)

ideal gas µpure (T,p°) j

[

m kH,j,W (T,p) p° (85)

m kH,j,M (T,p) + RT ln p° (86)

Adopting eq 75 to describe the dependency of µ∞,m (T,p,x˜ M) on the composition of the binary solvent j mixture water + methanol and combining this equation with eqs 84-86 results in the following correlation equation for Henry’s constant for the solubility of gas j in the solvent mixture on the molality scale m (T,p,x˜ M): kH,j

[

]

]

∑

(m) (m) , kH,j,M , and Aµj,n (n ) As was already mentioned, kH,j,W 0, 1, 2, ...) depend on the temperature and pressure. However, eq 87 can be rearranged as follows. Neglecting the pressure dependency of v∞j (the partial molar volume of solute j in its reference state)

(T,p,x˜ M) ) µ∞,m (T,psWM,x˜ M) + v∞j (T,x˜ M)(p - psWM) µ∞,m j j (88) The influence of the pressure on Henry’s constant is so described by the Krichevsky-Kasarnovski equation (following from eqs 82, 83, and 88):

ln

[

]

m kH,j (T,p,x˜ M) ) p° m v∞j (T,x˜ M)(p - psWM) kH,j (T,psWM,x˜ M) + ln (89) p° RT

[

]

m (T,psWM,x˜ M) is usually referred to as the temperakH,j ture-dependent Henry’s constant (on the molality scale) of component j in the solvent mixture under consideration. From eqs 87 and 89

ln

[

]

m kH,j (T,psWM,x˜ M)

p°

[

(1 - x˜ M) ln

)

]

m kH,j,W (T,psW)

p°

∞ vj,W (psWM - psW)

(1 - x˜ M)

ideal gas µ∞,m (T,p,x˜ M) ) µpure (T,p°) + j j

∞,m µj,M (T,p)

]

m kH,j,W (T,p)

) (1 - x˜ M) ln + p° p° m (1 - x˜ M)x˜ M ∞ kH,j,M (T,p) + Aµj,n(T,p)(1 - 2x˜ M)n x˜ M ln p° RT n)0 (87)

ln

(82)

p° is a reference pressure, ∆solGj is the molar Gibbs energy of solution of component j in the solvent mixture, i.e., the change of the molar Gibbs energy of component j encountered in the isothermal transfer from the pure ideal gas at reference pressure into the reference state in the solvent mixture. If the solvent mixture consists, e.g., of water and methanol, and if ∆solGj is defined on the molality scale

[

m kH,j (T,p,x˜ M)

RT (1 - x˜ M)x˜ M RT

+ x˜ M ln + x˜ M

[

]

m kH,j,M (T,psM)

p° ∞ s vj,M(pWM - psM)

+

+

RT

∞

Aµ ,n(T,psWM)(1 - 2x˜ M)n ∑ n)0 j

(90)

Furthermore, from eq 75 v∞j (T,x˜ M) results in ∞ ∞ v∞j (T,x˜ M) ) (1 - x˜ M)vj,W + x˜ Mvj,M + ∞ ∂Aµ ,n j (1 - 2x˜ M)n (91) (1 - x˜ M)x˜ M ∂p T n)0

∑

( )

m m ∞ ∞ (T,psW), kH,j,M (T,psM), Aµj,n(T,psWM), vj,W kH,j,W , vj,M , and (∂Aµj,n/∂p)T (n ) 0, 1, 2, ...) only depend on the temperature and have to be determined from experimental data. Finally, following eqs 64, 82, and 83, (molar) Gibbs energies of transfer of (gaseous) solute i from pure water to an aqueous solution of, e.g., methanol may be calculated (on the molality scale) from

˜ M) ∆tGm j (T,p,x

[

) RT ln

]

m kH,j (T,p,x˜ M) m kH,j,W (T,p)

(92)


Modeling CH3OH + H2O. The VLE of the binary solvent mixture water + methanol is described by means of the extended Raoult’s law, applied to both (volatile) components

psWφsW exp

[ [

psMφsM exp

] ]

vW(p - psW) aW ) yWpφW RT

(93)

vM(p - psM) aM ) yMpφM RT

(94)

The vapor pressure of pure water (psW), as well as the specific density of pure liquid water (dW), which was approximated by the specific density of the saturated liquid, was calculated from correlation equations (valid for temperatures from 273 to 647 K) given by Saul and Wagner.57 The correlation equation for the vapor pressure of pure methanol (psM) given by Reid et al.58 (valid for temperatures from 288 to 512.6 K) was adopted. The specific density of pure liquid methanol (dM) was approximated by the specific density of the saturated liquid. It was calculated from the correlation given by Hales and Ellender59 (valid for temperatures from 273 to 440 K). Vapor-phase fugacity coefficients (φi) were calculated from a truncated virial equation of state (cf. the Appendix). The activities of i ) water and methanol result from

ai ) xiγ(x) i

(95)

Activity coefficients (γ(x) i ) were calculated from the UNIQUAC equation. Pure-component UNIQUAC constants were adopted from the Dortmund Data Bank52 (rW ) 0.92, qW ) 1.4, rM ) 1.431 11, and qM ) 1.432). Binary UNIQUAC interaction parameters were fitted to VLE data (for temperatures ranging from 298 to 423 K) from various authors60-74 {ΨMW ) exp[1.8134 728.69/T (K)] and ΨWM ) exp[-2.1338 + 790.08/T (K)]}. The specific density d of the (liquid) solvent mixture water + methanol was calculated by means of eq 67. Four Redlich-Kister parameters were fitted to experimental data for T ) 298.15 K taken from the literature: 75-104 A -3 L/mol, A -3 v,0 ) -4.07 × 10 v,1 ) -0.23 × 10 -3 L/mol, Av,2 ) 0.5 × 10 L/mol, and Av,3 ) 0.7 × 10-3 L/mol. The average relative (standard relative) deviation between experimental and calculated densities (at T ) 298.15 K) amounts to 0.034% (0.053%) and is mainly due to the scattering of the literature data. Those four parameters can be adopted at least for temperatures ranging from 263 to 353 K because the densities of the (liquid) solvent mixture reported in refs 75-104 in this temperature range (excluding the data at T ) 298.15 K) were predicted with an average relative (standard relative) deviation of 0.095% (0.15%). The relative dielectric constant of the solvent mixture water + methanol was calculated from eq 68. The relative dielectric constant of pure liquid water (W) was approximated by the relative dielectric constant of the saturated liquid and was calculated from the correlation equation given by Bradley and Pitzer105 (valid for temperatures from 273 to 623 K). The experimental data for the relative dielectric constant of pure liquid

methanol from refs 106-108 cover the temperature range from 228 to 511 K. The correlation equation

M ) -119.39 +

18517.3 + 18.4557 ln[T (K)] T (K) 0.0507761T (K) (96)

gives those data with an average relative (standard relative) deviation of 0.3% (0.35%). Four temperature-dependent Redlich-Kister parameters were fitted to the experimental data for the relative dielectric constant of liquid mixtures of water and methanol107-112 for temperatures ranging from 228 to 328 K: A,0 ) -25.2 + 11170/T (K), A,1 ) -5.4 + 3560/T (K), A,2 ) -27 + 10700/T (K), and A,3 ) -17 + 6250/T (K). The average relative (standard relative) deviation between experimental and calculated relative dielectric constants is 0.08% (0.12%). NaCl + H2O. Parameters describing interactions between sodium and chloride ions in a pure water (0) (1) φ solvent (βNa +,Cl-,W, βNa+,Cl-,W, and CNaCl,W) were taken 113 from Silvester and Pitzer. They are valid for temper(1) atures from 273 to 573 K. Note that RNa +,Cl-,W ) 2 and (2) (2) that RNa+,Cl-,W is not required because βNa +,Cl-,W ) 0. φ Furthermore, as usual, µNa+,Na+,Cl-,W ) 1/3CNaCl,W and µNa+,Cl-,Cl-,W ) 0. The equation for the (temperature-dependent) solubility product of sodium chloride in water given by Silvester and Pitzer113 was adopted here. It is based on m ). the molality scale (Ksp,NaCl,W NaCl + CH3OH. Barthel et al.114 reported experimental data for the vapor pressure depression (∆p ) psM - p) arising from the addition of sodium chloride to pure liquid methanol (up to about 0.22 mol/kg of methanol) at T ) 298.15 K. The activity of methanol in these mixtures is calculated from the extended Raoult’s law (again calculating the fugacity coefficient of pure methanol in the gaseous phase from the truncated virial equation of state):

[

]

∆p ∆p + ln 1 + s ln aM ) (vM - BM,M) RT p M

(97)

The osmotic coefficient follows from

φ)

ln aM ln aM )m j NaCl 1 12M/M xM m°

(98)

When Pitzer’s GE model is applied, the activity of methanol (in a methanolic solution containing only sodium chloride) is

{

( )

m j NaCl 2 (0) 1 [βNa+,Cl-,M + (If ′ - f) + 2 m° (1) (1) βNa+,Cl-,M exp(-RNa+,Cl-,MxI) +

ln aM ) -2M/M

(2) (2) βNa +,Cl-,M exp(-RNa+,Cl-,MxI)] + m j NaCl m j NaCl 3 φ CNaCl,M + m° m°

( )

}

(99)

where I ) m j NaCl/m°. Parameters describing interactions between sodium and chloride ions in a pure methanol solvent were fitted to the experimental data for the osmotic coefficient from Barthel et al.114

210


Figure 1. Osmotic coefficient of the system NaCl + CH3OH at 298.15 K: (b) Barthel et al.;114 (s) correlation, this work.

nicely represented (average relative deviation ) 2.4%), whereas the ones calculated from the data by Gladden and Fanning115 are not consistent with the data reported by Barthel et al.114 (The data of Gladden and Fanning, also at other temperatures, were therefore discarded in the present work.) Because of the lack of reliable experimental information on osmotic coefficients and/or on mean ionic activity coefficients in the system sodium chloride + methanol at temperatures other than 298.15 K, the set of interaction parameters determined for that temperature had to be adopted for other temperatures as well. Numerical values for the solubility product of sodium chloride in pure methanol (on the molality scale) were calculated from solubility data reported by Pinho and Macedo117 (at T ) 298.15 and 323.15 K) and Akerlof and Turck118 (at T ) 298.15 K) m Ksp,NaCl,M )

(

m j NaCl m γ m° (,NaCl,M

)

2

(102)

When eq 78 is applied, the Gibbs energy of transfer of NaCl from pure water to pure methanol (on the molality scale) is m (T) ∆tGNaCl,WfM

Figure 2. Mean ionic activity coefficient of NaCl in CH3OH at 298.15 K: (0) Gladden and Fanning;115 (b) Yan et al.;116 (s) prediction, this work. (1) As already mentioned, deliberately, RNa +,Cl-,M was set (1) equal to RNa+,Cl-,W ) 2. Just one interaction parameter (1) (βNa +,Cl-,M ) 2.4) was sufficient to describe the experimental osmotic coefficient data within experimental accuracy (see Figure 1). All other parameters were (0) (2) φ neglected (βNa +,Cl-,M ) βNa+,Cl-,M ) CNaCl,M ) 0, where µNa+,Na+,Cl-,M ) µNa+,Cl-,Cl-,M ) 0). The average relative (standard relative) deviation between experimental and correlated osmotic coefficients amounts to 0.36% (0.61%). The mean ionic activity coefficient of sodium chloride in methanol (on the molality scale) is

( )

m j NaCl f′ (0) + [2βNa +,Cl-,M + 2 m° (1) (1) (2) (2) βNa+,Cl-,M h(RNa+,Cl-,MxI) + βNa+,Cl-,Mh(RNa +,Cl-,MxI)] + 2 m j NaCl 3 φ C (100) m° 2 NaCl,M

m ln γ(,NaCl,M )

( )

) -RT ln

[

m Ksp,NaCl,M (T)

m Ksp,NaCl,W (T)

]

(103)

m m ∆tGNaCl,WfM (T) ) ∆tGNaCl (T,x˜ M)1) was calculated at T ) 298.15 and 323.15 K, and its temperature dependence was approximated by assuming that the entropy m of transfer ∆tSNaCl,WfM does not depend on the temperature:

m m ) ∆tGm, ° - ∆tSNaCl,WfM (T - T°) ∆tGNaCl,WfM NaCl,WfM (104)

where T° ) 298.15 K, resulting in ∆tGm, ° ) 20.46 NaCl,WfM m kJ/mol and ∆tSNaCl,WfM ) -121.6 J/(mol K). (Hefter et al.24 give 21.9 kJ/mol and -117 J/(mol K), respectively.) NaCl + H2O + CH3OH. Again when eq 78 is applied, the Gibbs energy of transfer of NaCl from pure water to the solvent mixture water + methanol (on the molality scale) results from

[

m (T,x˜ M) ) -RT ln ∆tGNaCl

]

m Ksp,NaCl (T,x˜ M) m Ksp,NaCl,W (T)

(105)

and according to eq 79 m m ∆tGNaCl (T,x˜ M) ) x˜ M∆tGNaCl,WfM (T) + ∞

where

h(x) ) g(x) + exp(-x)

(101)

Gladden and Fanning115 as well as Yan et al.116 reported some emf data for methanolic solutions containing sodium chloride at T ) 298.15 K. These data have been reevaluated; i.e., the standard electrode potentials were recalculated by means of the usual extrapolation proceduresusing Pitzer’s modified Debye-Hu¨ckel term as used heresand the mean ionic activity coefficients of sodium chloride were redetermined by using those (slightly different numbers for the) standard electrode potentials. As can be seen from Figure 2, the activity coefficients calculated from the data by Yan et al.116 are

(1 - x˜ M)x˜ M

∑ AK n)0

(1 - 2x˜ M)n (106)

sp,n

m ∆tGNaCl (T,x˜ M) results from experimentally determined standard electrode potentials of (specially arranged) electrochemical cells containing aqueous methanolic solutions with sodium chloride. Experimental emf data are available in the literature116,119-124 for temperatures from 288 to 318 K. Some of those data116,119-122,124 were used to redetermine the standard electrode potentials by means of the usual extrapolation proceduresusing Pitzer’s modified Debye-Hu¨ckel term (as given in the present work). Kozlowski et al.123 did not report the raw emf data. Their numbers for the standard electrode

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 211 Table 1. Comparison of Experimental Results with Model Correlations/Predictions for the System NaCl + CH3OH + H2O T/K m ∆tGNaCl

298.15

m γ(,NaCl

m j NaCl,sat.

m ∆tGNaCl

m γ(,NaCl

m j NaCl,sat. p, yM

a

308.15-318.15 288.15-318.15 298.15 308.15-318.15 298.15 298.15 298.15 323.15 273.15-298.15 339.25-372.75 323.15 273.15-298.15 339.25-372.75 339.25-372.75 342.05-372.65 342.7-362.15 298.15 318.15 313-397

average deviation

data points

kind of measurement

reference

0.131 kJ/mol 0.083 kJ/mol 0.078 kJ/mol 1.71% 1.83% 0.90% 0.80% 0.010 mol/kg 0.011 mol/kg

Correlated Data 5 11 5 116 66 7 9 7 9

Yan et al.116 Feakins and Voice120,121 Kozlowski et al.123 Yan et al.116 Feakins and Voice120,121 Pinho and Macedo117 Akerlof and Turck118 Pinho and Macedo117 Akerlof and Turck118

emf emf emf emf emf salt solubility salt solubility salt solubility salt solubility

0.186 kJ/mol 0.177 kJ/mol 0.376 kJ/mol 1.84% 0.77% 2.84% 6.17% 3.14% 0.96% 8.28% 0.047 mol/kg 0.027 mol/kg 0.112 mol/kg 3.29% 4.61% 6.88% 7.90% 7.62% 1.87% 1.50% 2.93% 1.99% 1.59% 2.53% 4.60%

Predicted Data 10 30a 9 206 35 35 47 7 7 12 7 7 12 12 17 25 7 30 48

Yao et al.122 Kozlowski et al.123 Akerlof119 Yao et al.122 Kozlowski et al.123 Akerlof119 Basili et al.124 Pinho and Macedo117 Armstrong and Vargas Eyre125 Johnson and Furter21 Pinho and Macedo117 Armstrong and Vargas Eyre125 Johnson and Furter21 Johnson and Furter21 Nishi et al.126,127 Morrison et al.128 Yang and Lee129 Yao et al.130 Jo¨decke et al.131

emf emf emf emf emf emf emf salt solubility salt solubility salt solubility salt solubility salt solubility salt solubility VLE VLE VLE VLE VLE VLE

Discarding data at T ) 298.15 K.

correlation results for T ) 298.15 K (see also Table 1). The standard electrode potentials calculated from the data reported by Akerlof119 considerably disagree with all other literature data and were therefore not considered in the correlation. (Basili et al.124 used a cell arrangement that did not allow for the calculation of the interesting standard electrode potential). Furthermore, the data compiled by Hefter et al.24, as well as the (poor) prediction resulting from Born’s equation, are shown in Figure 3 {cf. eq 16 with ionic radii from Marcus27 (rNa+ ) 102 pm and rCl- ) 181 pm) and after conversion to the molality scale}: Figure 3. Gibbs energy of transfer of NaCl from pure water to the solvent mixture H2O + CH3OH (on the molality scale) at 298.15 K: (2) Yan et al.;116 (O) Akerlof;119 ([) Feakins and Voice;120,121 (9) Kozlowski et al.;123 (0) compiled data from Hefter et al.;24 (s) correlation, this work, (- -) prediction from Born’s equation25,26 (cf. eq 107).

potentials were therefore adopted without reevaluation. Four Redlich-Kister parameters were fitted to the m ∆tGNaCl (T,x˜ M) data at T ) 298.15 K from refs 116, 120, 121, and 123 (AKsp,0 ) 7.055 kJ/mol, AKsp,1 ) 0.577 kJ/ mol, AKsp,2 ) 5.01 kJ/mol, and AKsp,3 ) -3.75 kJ/mol). The average absolute (standard absolute) deviation between experimental and correlated data at this temperature is 0.093 kJ/mol (0.112 kJ/mol) (one data point at x˜ M > 0.98 was not considered in the correlation). m Furthermore, the ∆tGNaCl (T,x˜ M) data from Yao et al.122 and Kozlowski et al.123 for temperatures ranging from 288 to 318 K are nicely predicted by that equation. The average absolute (standard absolute) deviation between experimental and calculated data is 0.180 kJ/mol (0.235 kJ/mol) (discarding the data at T ) 298.15 K). Figure 3 shows a comparison between experimental data and

m ∆tGNaCl,Born (T,x˜ M) )

[

][

]

( )

NAe2 1 1 d 1 1 + + 2RT ln (107) 8π0 W rNa+ rCldW When Pitzer’s GE model is applied, the mean ionic activity coefficient of sodium chloride (in an aqueous methanolic solution containing only sodium chloride) is (cf. eq 100)

( )

m j NaCl f′ (0) + [2βNa +,Cl- + 2 m° (1) (1) (2) (2) βNa+,Cl-h(RNa+,Cl-xI) + βNa+,Cl-h(RNa +,Cl-xI)] + m j NaCl 23 φ (108) C m° 2 NaCl

m ) ln γ(,NaCl

( )

Furthermore, the VLE of the ternary solvent mixture water + methanol + sodium chloride is described by means of the extended Raoult’s law, applied to both (volatile) components (see eqs 93 and 94). The activities of water and methanol result from a combination of eqs 35, 56, 57, and 66:

212


ln aW ) ln xW x˜ M

( )[( )(

m j NaCl M* - 2(M/M - M/W) + m° RT ∂x˜ M T,p ln γW,UNIQUAC + ln γW,Pitzer + ln γW,conv (109)

ln aM ) ln xM + x˜ W

]

)

m ∂∆tGNaCl

( )[( )(

]

)

m m j NaCl M* ∂∆tGNaCl - 2(M/M - M/W) + m° RT ∂x˜ M T,p ln γM,UNIQUAC + ln γM,Pitzer + ln γM,conv (110)

with eqs 42-46 and 106, where eqs 45 and 46 result in

[

ln γW,Pitzer ) -M/W (If ′ - f) +

( )

2

]

( )

m j NaCl 2 m j NaCl 3 φ (λNa+,Cl- + Iλ′Na+,Cl-) + 2 CNaCl m° m° M*x˜ M

[(

) ( )( ) ( )(

m j NaCl ∂f +2 ∂x˜ M I m°

2

∂λNa+,Cl+ ∂x˜ M I

m j NaCl m°

3

)]

φ ∂CNaCl ∂x˜ M

(111)

and

[

ln γM,Pitzer ) -M/M (If ′ - f) +

( ) ] [( ) ( ) ( ) ( ) ( )]

( )

2

m j NaCl 2 m j NaCl 3 φ (λNa+,Cl- + Iλ′Na+,Cl-) + 2 CNaCl + m° m° m j NaCl ∂f M*(1 - x˜ M) +2 ∂x˜ M I m°

2

∂λNa+,Cl+ ∂x˜ M I

m j NaCl m°

3

φ ∂CNaCl

(112)

∂x˜ M

As given by eqs 48 and 50 (0) λNa+,Cl- + Iλ′Na+,Cl- ) βNa +,Cl- + (1) (1) (2) (2) βNa +,Cl- exp(-RNa+,Cl-xI) + βNa+,Cl- exp(-RNa+,Cl-xI) (113)

and

( )

(0) (1) ∂βNa ∂βNa ∂λNa+,Cl+,Cl+,Cl(1) ) + g(RNa +,Cl-xI) + ∂x˜ M ∂x˜ M ∂x˜ M

φ is a linear combination of two ternary interacCNaCl tion parameters

φ CNaCl ) 3(µNa+,Na+,Cl- + µNa+,Cl-,Cl-)

(115)

µNa+,Cl-,Cl- may therefore, as usual, be set to zero. The influence of the solvent mixture composition φ (0) (1) (2) on βNa +,Cl-, βNa+,Cl-, βNa+,Cl-, and CNaCl is described by means of eq 70. The (temperature-dependent) RedlichKister parameters may be, in principle, fitted to experimental information on both the mean ionic activity coefficient of NaCl in mixtures of water and methanol and the influence of that salt on the VLE of that solvent mixture. From the emf data reported in refs 116, 119-122, and 124, the mean ionic activity coefficients of sodium chloride in mixtures of water + methanol were calculated by using the (slightly different numbers for the) standard electrode potentials redetermined in the present work. As already mentioned, Kozlowski et al.123 did not report the raw emf data, and only the numbers for the mean ionic activity coefficients given by those authors at T ) 298.15 K could be considered. Most of the experimental emf data found in the literature was taken at T ) 298.15 K. Only Yao et al.122 report emf data also at some higher temperatures (308.15 and 318.15 K). In addition, from experimental information on the solubility of sodium chloride in mixtures of water + methanol21,117,118,125 (for temperatures from 273.15 to 381.85 K), the mean ionic activity coefficient of sodium chloride in those saturated mixtures was calculated from m ) Ksp,NaCl

(

m j NaCl m γ m° (,NaCl

)

2

(116)

m (cf. eq 102), adopting Ksp,NaCl from eqs 105 and 106. Experimental information on the VLE of the system NaCl + water + methanol was found in refs 21 and 126-131 (for temperatures from 298.15 to 396.9 K). As was already mentioned, one may fit the paramφ (0) (1) (2) eters βNa +,Cl-, βNa+,Cl-, βNa+,Cl-, and CNaCl to experimenm tal information for both γ(,NaCl and the VLE of the ternary system. However, to test the predictive character of the model, the parameters mentioned above were m fitted only to some experimental information for γ(,NaCl at T ) 298.15 K (calculated from refs 116-118, 120, and 121) from emf and solubility measurements:

(0) (0) ˜ M) ) (1 - x˜ M)βNa βNa +,Cl-(T,x +,Cl-,W(T) + (0) ˜ M)x˜ M (117) x˜ MβNa +,Cl-,M(T) + 0.52970(1 - x

I

[

(2) ∂βNa +,Cl(2) g(RNa +,Cl-xI) + ∂x˜ M

(1) xI βNa +,Cl-

(1) (1) ∂g(RNa +,Cl-xI) ∂RNa+,Cl+ ∂x˜ M ∂(R(1) + -xI)

(2) βNa +,Cl-

Na ,Cl (2) ∂g(RNa +,Cl-xI)

(1) βNa ˜ M) ) +,Cl-(T,x (1) (1) ˜ MβNa (1 - x˜ M)βNa +,Cl-,W(T) + x +,Cl-,M(T) + (1 - x˜ M)x˜ M[-3.0315 - 1.0143(1 - 2x˜ M) -

]

(2) ∂RNa +,Cl(114) (2) ∂x ˜ M ∂(RNa+,Cl-xI)

(1) As was already mentioned, deliberately, RNa ˜ M) +,Cl-(T,x (1) was set equal to RNa+,Cl-,W ) 2. Therefore, the last contribution in eq 114 {+xI[...]} vanishes.

2.6378(1 - 2x˜ M)2 + 5.1646(1 - 2x˜ M)3] (118) φ φ CNaCl (T,x˜ M) ) (1 - x˜ M)CNaCl,W (T) + φ (T) - 0.029543(1 - x˜ M)x˜ M (119) x˜ MCNaCl,M

Note that six parameters were sufficient for a perfect correlation and that it was not necessary to consider (2) (2) βNa +,Cl- and RNa+,Cl-; i.e., these parameters were set


Figure 4. Mean ionic activity coefficient of NaCl in H2O + CH3OH (x˜ M ) 0.2726 and T ) 298.15 K): (0) Akerlof;119 (2) Feakins and Voice;120,121 (O) Kozlowski et al.;123 (b) Yan et al.;116 (4) Basili et al.;124 ([) Pinho and Macedo;117 (s) correlation, this work.

Figure 6. yM-x˜ M phase diagram for the system NaCl + H2O + CH3OH at p ) 0.10159 MPa (solutions are saturated with NaCl): (b) Johnson and Furter;21 (s) prediction, this work; (- -) no salt, calculation, this work.

Figure 5. Solubility of NaCl in H2O + CH3OH: (O) Pinho and Macedo117 (T ) 298.15 K); (0) Akerlof and Turck118 (T ) 298.15 K); (b) Pinho and Macedo117 (T ) 323.15 K); (s) correlation (at T ) 298.15 K), prediction (at T ) 323.15), this work.

to zero. Altogether 319 experimental data points for m (at T ) 298.15 K, from refs 116-121 and 123γ(,NaCl 125) were described by the model within experimental accuracy. A comparison between experimental and calculated results (by means of average deviations) is given in Table 1 (see also Figure 4). As expected, the model perfectly describes the solubility of sodium chloride in mixtures of methanol + water at T ) 298.15 K (see Table 1 and Figure 5). m at temperatures The experimental results for γ(,NaCl other than 298.15 K (from refs 21, 117, 122, and 125, which again result from both emf and solubility measurements; in total 228 data points) were predicted by the model almost within experimental accuracy with that set of parameters (see also Table 1). Therefore, the model also reliably predicts the experimental data (from refs 21, 117, and 125) for the concentration of sodium chloride in saturated liquid mixtures of methanol + water at temperatures other than 298.15 K (see Table 1 and Figure 5). The model was, furthermore, able to predict the experimental data for the VLE of that ternary system (from refs 21 and 126-131; see Table 1 and Figures 6 and 7). In particular, as can be seen from Figure 7, with increasing salt concentration in the liquid solvent mixture, generally the concentration of methanol in the gaseous phase increases; i.e., methanol is “salted-out”, whereas water is “salted-in”. This behavior is quantitatively predicted by the model. To better understand that effect, it might be useful to calculate the different contributions to the activities of the solvent components (cf. eqs 109 and 110). As an example, Figure 8 gives those contributions to the

Figure 7. Mole fraction of methanol in the vapor phase (upper diagram) and total pressure (lower diagram) versus the molality of NaCl in aqueous solutions of methanol at T ≈ 314.6 K (full symbols, experimental data by Jo¨decke et al.;131 blank symbols, prediction, this work): (b, O) x˜ M ≈ 0.0328, p ) 0.01 MPa; (2, 4) x˜ M ≈ 0.0789, p ) 0.013 MPa; (9, 0) x˜ M ≈ 0.216, p ) 0.018 MPa; (1, 3) x˜ M ≈ 0.466, p ) 0.025 MPa.

activities of methanol and water (at T ) 298.15 K, x˜ M ) 0.2) plotted versus the stoichiometric molality of sodium chloride in the liquid solvent mixture (up to the solubility limit of the salt). Apart from the contributions resulting from the dilution term (ln xi) and from the (constant) UNIQUAC term, the (linear) term involving the partial derivative of the (molar) Gibbs energy of transfer of that salt (from one pure solvent to the solvent mixture) with respect to x˜ M (called the “salt effect term” in Figure 8) has a tremendous influence on the activities of methanol and water. Furthermore, this contribution

214


Figure 8. Contributions to the activities of the solvent components methanol (left) and water (right) (cf. eqs 109 and 110) plotted versus the molality of NaCl in the liquid solvent mixture (up to the solubility limit of the salt) at T ) 298.15 K and x˜ M ) 0.2.

results in an increase of the activity of methanol with rising salt concentration (methanol is “salted-out”) and in a decrease of the activity of water (water is “saltedin”). Obviously, that term is responsible for that salt effect. The contribution resulting from the Pitzer term only slightly reduces that salt effect (cf., e.g., refs 20 and 132), and the contribution resulting from the conversion term is nearly negligible. Of course, the importance of any of those contributions may highly depend on the salt and on the solvent mixture under consideration (as well as on the concentration of all components, the temperature, and, in principle, also the pressure). It might be, for example, interesting to investigate the influence of different salts (or cations and anions) in different binary solvent mixtures on the “salt effect term” [by comparing the partial derivative of the molar Gibbs energy of transfer of those salts from one pure solvent to the binary solvent mixtures (with respect to the salt-free mole fraction of one of the solvent components) and establishing series, which may be compared to the Hofmeister series]. CO2 + H2O. The VLE is expressed by the extended Raoult’s law (cf. eq 93) for water and the extended Henry’s law for carbon dioxide: m m kH,CO (T,p) aCO ) yCO2pφCO2 2,W 2

(120)

Henry’s constant of CO2 in pure water (based on the molality scale) is

m (T,p) kH,CO 2,W

)

m kH,CO (T,psW) 2,W

[

exp

]

∞ vCO (p - psW) 2,W

RT

(121)

m kH,CO (T,psW) is Henry’s constant of CO2 in pure water 2,W at the vapor pressure of water. It is adopted from Rumpf and Maurer:5

9624.41 T (K) 28.7488 ln[T (K)] + 0.0144074T (K) (122)

m (T,psW) (MPa)] ) 192.876 ln[kH,CO 2,W

This correlation equation can be applied for tempera∞ is the partial molar tures from 273 to 473 K. vCO 2,W volume of CO2 infinitely diluted in water and is calculated as recommended by Brelvi and O’Connell133 (cf. the Appendix). For a solution containing any number of solutes but only water as a single solvent, the Gibbs excess energy equation presented in this work exactly matches Pitzer’s GE equation for aqueous electrolyte solutions. Because the amount of ionic species (e.g., bicarbonate, carbonate, ...), i.e., the ionic strength, is negligible at the carbon dioxide molality under consideration, the Debye-Hu¨ckel term vanishes from those equations. The activities of water and carbon dioxide (based on the molality scale), therefore, result from

1 xW

ln aW ) 1 M/W

[( ) mCO2

2 (0) βCO +2 2,CO2,W

m°

m aCO ) 2

( ) mCO2 m°

]

3

µCO2,CO2,CO2,W (123)

mCO2 γm m° CO2

(124)

where

ln

m γCO 2

( )

)2

mCO2 m°

(0) βCO 2,CO2,W

+3

( ) mCO2 m°

2

µCO2,CO2,CO2,W (125)

Because the molality of dissolved carbon dioxide mCO2 (i.e., the amount of substance of the gas per kilogram of pure solvent water) remains small even at high pressures, Rumpf and Maurer5 neglected Pitzer parameters for interactions between carbon dioxide molecules (0) in pure water: βCO ) µCO2,CO2,CO2,W ) 0. 2,CO2,W CO2 + CH3OH. With the same reasoning as that presented for the binary system CO2 + water, the equilibrium condition results for methanol in eq 94, and for carbon dioxide, it results in m m (T,p) aCO ) yCO2pφCO2 kH,CO 2,M 2

(126)


Figure 9. Total pressure above solutions of CO2 + CH3OH + H2O: [2 (x˜ M ) 0.05), O (x˜ M ) 0.1), 9 (x˜ M ≈ 0.25), ] (x˜ M ≈ 0.5), 1 (x˜ M ≈ 0.75), 4 (x˜ M ≈ 0.9), b (x˜ M ≈ 0.95), 0 (x˜ M ) 1)] experimental results, Xia et al.;134 (s) correlation, this work; (- -) solubility of CO2 in pure water, correlation, Rumpf and Maurer.5

Henry’s constant of CO2 in pure methanol (based on the molality scale) is m kH,CO (T,p) 2,M

)

m kH,CO (T,psM) 2,M

[

exp

∞ vCO (p 2,M

-

RT

]

(127)

where is Henry’s constant of CO2 in pure ∞ methanol at the vapor pressure of methanol and vCO 2,M is the partial molar volume of CO2 infinitely diluted in methanol. The activity of methanol is calculated from

M/M

[( ) mCO2 m°

1 xM 2 (0) βCO +2 2,CO2,M

( ) mCO2 m°

3

]

m aCO ) 2

psM)

m kH,CO (T,psM) 2,M

ln aM ) 1 -

The activity of carbon dioxide again results from

µCO2,CO2,CO2,M (128)

mCO2 γm m° CO2

(129)

where m ln γCO )2 2

( ) mCO2 m°

( )

(0) βCO +3 2,CO2,M

mCO2 m°

2

µCO2,CO2,CO2,M (130)

mCO2 is the molality of carbon dioxide (i.e., the amount of substance of the gas per kilogram of pure solm ∞ (0) vent methanol). kH,CO (T,psM), vCO , βCO , and 2,M 2,M 2,CO2,M µCO2,CO2,CO2,M were determined from the experimental pressures above the solution of carbon dioxide + methanol recently presented by Xia et al.134 (cf. Figure 9; x˜ M ) 1), which range from about 313 to 395 K.

216


Figure 10. Influence of p - psWM (p ) total pressure above CO2 + CH3OH + H2O; psWM ) vapor pressure above CH3OH + H2O) on the ratio of (approximated) CO2 fugacity (in gaseous phase) to CO2 (liquid phase) molality: [2 (x˜ M ) 0.05), O (x˜ M ) 0.1), 9 (x˜ M ≈ 0.25), ] (x˜ M ≈ 0.5), 1 (x˜ M ≈ 0.75), 4 (x˜ M ≈ 0.9), b (x˜ M ≈ 0.95), 0 (x˜ M ) 1)] calculated from experimental results, Xia et al.;134 (s) linear extrapolation. Table 2. Henry’s Constant of CO2 in CH3OH + H2O (on the Molality Scale) and Standard Deviations (∆s)a T ) 313.75 K

T ) 354.35 K

T ) 395.0 K

x˜ M

ln[km H (MPa)]

∆s ln[km H (MPa)]

ln[km H (MPa)]

∆s ln[km H (MPa)]

ln[km H (MPa)]

∆s ln[km H (MPa)]

0b 0.05 0.1 0.2515 0.4999 0.7508 0.9015 0.9527 1

1.456 1.359 1.327 1.106 0.494 0.084 -0.310 -0.410 -0.496

0.011 0.007 0.015 0.008 0.004 0.013 0.019 0.017

2.058 1.940 1.849 1.522 0.866 0.495 0.205 0.086 0.009

0.007 0.006 0.012 0.004 0.016 0.006 0.010 0.006

2.316 2.180 2.057 1.665 1.022 0.703 0.455 0.358 0.294

0.003 0.006 0.009 0.005 0.019 0.007 0.010 0.004

a

Based on experimental data from Xia et al.134

b

Adopted from Rumpf and Maurer.5

m It is common practice to first evaluate kH,CO (T,psM) 2,M from an extrapolation procedure (at constant temperature). For example, from eqs 126 and 127

m ln kH,CO (T,psM) ) lim ln 2,M pfpsM

[

]

fCO2(T,p,yCO2) mCO2/m°

(131)

where

fCO2(T,p,yCO2) ) yCO2pφCO2(T,p,yCO2)

(132)

is the fugacity of carbon dioxide in the gaseous phase at equilibrium temperature, pressure, and gas-phase composition. The gas-phase composition is not experimentally known but can be estimated (in an iterative procedure) for each experimental point (at given temperature, liquid-phase composition, and solubility pressure) from eqs 94, 128, and 132, disregarding Pitzer interaction parameters and applying the virial equation of state for calculating fugacity coefficients in the gaseous phase (cf. the Appendix). In Figure 10, the calculated values for ln[fCO2/(mCO2/ m°)] (for a preset temperature) are plotted versus the difference between the total pressure above CO2 + CH3OH and the vapor pressure of pure methanol. Extrapolations were done by linear regression. Henry’s constants resulting from the extrapolations (as well as the standard deviations between the experimental values and the linear fit) are given in Table 2. They are plotted in Figure 11 versus the inverse temperature (full symbols). Furthermore, this extrapolation procedure was applied to solubility data found in the literature (refs 135154), which range from 213 to 477 K. As can be seen

Figure 11. Henry’s constant of CO2 in CH3OH (on the molality scale): (b) extrapolated experimental results, Xia et al.;134 (O) extrapolated experimental results, literature;135-154 (s) correlation, this work.

from Figure 11 (blank symbols), results from the literature largely scatter, and it is very difficult to determine which of those data are reliable. Nevertheless, to more or less account for the literature data (especially outside the temperature range investigated in the present work), the following four-parameter equation for Henry’s constant of CO2 in methanol (based on the molality scale) was adjusted to all literature data exactly matching the three numerical values determined in the present work:

1022.94 + T (K) 22.4222 ln[T (K)] - 0.0455865T (K) (133)

m ln[kH,CO (T,psM) (MPa)] ) -118.349 + 2,M

Figure 11 shows the resulting correlation curve.

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 217 ∞ (0) In a next step, vCO , βCO , and µCO2,CO2,CO2,M 2,M 2,CO2,M were simultaneously fitted to the new experimental pressures above solutions of carbon dioxide + methanol (from Xia et al.),134 resulting in

ln aM ) ln xM +

( )[( )(

∞ vCO (cm3/mol) ) -0.63 - 0.4555T (K) (134) 2,M (0) βCO 2,CO2,M

) 0.065421 - 25.828/T (K)

M/M

(135)

µCO2,CO2,CO2,M ) -1.3275 × 10-3 + 0.4333/T (K) (136) The numbers for the partial molar volume of carbon dioxide at infinite dilution in pure methanol as calculated from eq 134 are negative. The experimental data for the total pressure reported by Xia et al.134 (29 experimental data points) are correlated (with those six parameters) with an average relative (absolute) deviation of 0.52 % (0.024 MPa) (cf. Figure 9). CO2 + H2O + CH3OH. The phase equilibrium condition for the solubility of carbon dioxide in a mixed solvent of water and methanol results in the extended Raoult’s law (cf. eqs 93 and 94) for both solvent components and in the extended Henry’s law for carbon dioxide: m m (T,p,x˜ M) aCO ) yCO2pφCO2 kH,CO 2 2

m (T,p,x˜ M) ) kH,CO 2

[

m (T,psWM,x˜ M) exp kH,CO 2

]

∞ vCO (T,x˜ M)(p - psWM) 2

RT

(138)

x˜ M

( )[( )( m°

)

m M* ∂∆tGCO2 RT ∂x˜ M

ln γW,UNIQUAC + 1 M/W

[(

m°

( ) [( ) 2

M*x˜ M

mCO2 m°

)

mCO2

mCO2 m°

]

- (M/M - M/W) +

T,p

1 - ln(xW + xM) xW + xM

2 (0) βCO (T,x˜ M) + 2,CO2

]

3

µCO2,CO2,CO2(T,x˜ M) -

2 ∂β(0) CO2,CO2

∂x˜ M

+

( ) mCO2 m°

3

]

∂µCO2,CO2,CO2 ∂x˜ M

)

mCO2 m°

mCO2

(0) βCO (T,x˜ M) + 2,CO2

(139)

]

3

µCO2,CO2,CO2(T,x˜ M) +

2 ∂β(0) CO2,CO2

m°

]

( ) mCO2

+

∂x˜ M

m (T,p,x˜ M) ∆tGCO 2

[

) RT ln

3

∂µCO2,CO2,CO2

m°

∂x˜ M (140)

]

m kH,CO (T,p,x˜ M) 2 m kH,CO (T,p) 2,W

(141)

The partial derivative of the (molar) Gibbs energy of transfer of CO2 (from pure water to the solvent mixture) with respect to x˜ M, therefore, results in (cf. also eq 138)

(

∂x˜ M

)

(

) RT

T,p

)

m ∂ ln kH,CO (T,psWM,x˜ M) 2

∂x˜ M (p -

is Henry’s constant of CO2 in the solvent mixture of water and methanol at the vapor pressure of that solvent mixture (psWM), which is calculated as explained before (see the section on VLE in ∞ (T,x˜ M) is the partial molar water + methanol). vCO 2 volume of CO2 infinitely diluted in the solvent mixture. From the Gibbs energy equation given above, the following expressions result for the activities of water and methanol:

mCO2

m°

2

with ln γi,UNIQUAC (i ) W and M) according to eqs 43 and 44. mCO2 is the molality of carbon dioxide (i.e., the amount of substance of the gas per kilogram of solvent mixture water + methanol). The (molar) Gibbs energy of transfer of CO2 (from pure water to the solvent mixture) is directly related to Henry’s constant of CO2 in those mixtures (cf. eq 92) by

m ∂∆tGCO 2

m (T,psWM,x˜ M) kH,CO 2

ln aW ) ln xW -

M*(1 - x˜ M)

[(

mCO2

( ) [( )

2

(137)

The (molality scale based) Henry’s constant of CO2 in solvent mixtures of water and methanol is expressed as (cf. eq 89)

]

)

m M* ∂∆tGCO2 - (M/M - M/W) + x˜ W m° RT ∂x˜ M T,p 1 - ln(xW + xM) ln γM,UNIQUAC + 1 xW + xM

mCO2

psWM)

(

T

+

)

∞ ∂vCO (T,x˜ M) 2

∂x˜ M

(142)

T

The activity of carbon dioxide again results from m aCO ) 2

mCO2 γm m° CO2

(143)

where

( )

m ln γCO )2 2

mCO2 m°

(0) βCO (T,x˜ M) + 2,CO2

3

( ) mCO2 m°

2

µCO2,CO2,CO2(T,x˜ M) (144)

m ∞ (0) kH,CO (T,psWM,x˜ M), vCO (T,x˜ M), βCO (T,x˜ M), and 2 2 2,CO2 µCO2,CO2,CO2(T,x˜ M) were determined from the experimental pressures above solutions of carbon dioxide + water + methanol reported by Xia et al.134 (x˜ M * 1), which range from about 313 to 395 K. m (T,psWM,x˜ M) was determined by the exFirst kH,CO 2 trapolation procedure explained before (at constant temperature and solvent mixture composition):

m (T,psWM,x˜ M) ) lim ln ln kH,CO 2 pfpsWM

[

]

fCO2(T,p,yj) (mCO2/m°)

(145)

218

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 ∞ ∞ vCO (T,x˜ M) (cm3/mol) ) (1 - x˜ M)vCO (T) (cm3/mol) + 2 2,W ∞ (T) (cm3/mol) + x˜ MvCO 2,M

{[

(1 - x˜ M)x˜ M(103) 0.6585 -

[

]

1079 (1 - 2x˜ M) + T (K) 421.1 -0.9378 + (1 - 2x˜ M)2 T (K)

-3.153 +

[

]

243.2 + T (K)

]

}

(148)

(0) βCO (T,x˜ M) ) 2,CO2

Figure 12. Henry’s constant of CO2 in CH3OH + H2O (on the molality scale): [0 (313.75 K), 2 (354.35 K), O (395 K)] extrapolated experimental results, Xia et al.;134 (s) correlation, this work.

where

(0) (0) (T) + x˜ MβCO (T) + (1 - x˜ M)βCO 2,CO2,W 2,CO2,M

[

(1 - x˜ M)x˜ M -0.29602 +

]

71.949 (149) T (K)

µCO2,CO2,CO2(T,x˜ M) ) fCO2(T,p,yj) ) yCO2pφCO2(T,p,yj)

(146)

The gas-phase composition was estimated (in an iterative procedure) for each experimental point (at given temperature, liquid-phase composition, and solubility pressure) from eqs 93, 94, 139, 140, and 146, disregarding Pitzer’s interaction parameters as well as the “gas effect term” involving the partial derivative of the (molar) Gibbs energy of transfer of CO2 (from pure water to the solvent mixture) with respect to x˜ M and applying the virial equation of state for calculating fugacity coefficients in the gaseous phase (cf. the Appendix). The calculated values for ln[fCO2/(mCO2/m°)] (for preset temperature and solvent mixture composition) are plotted in Figure 10 versus the difference between the total pressure above CO2 + H2O + CH3OH and the vapor pressure of the methanol + water solvent mixture. Extrapolations were done by linear regression. Henry’s constants resulting from the extrapolations (as well as the standard deviations between the experimental values and the linear fit) are given in Table 2. Henry’s constant is plotted in Figure 12 versus the mole fraction of methanol in the (solute-free) solvent mixture (x˜ M) for each temperature investigated. Henry’s constant is correlated according to eq 90, resulting in m (T,psWM,x˜ M) ln[kH,CO 2

(1 -

m (T,psW) x˜ M) ln[kH,CO 2,W

[

[

The experimental data (in total 192 data points) for the total pressure are correlated (with 12 parameters) with an average relative (absolute) deviation of 1.96% (0.083 MPa; see Figure 9, where the dashed line gives the results of a former correlation of experimental data on the solubility of carbon dioxide in pure water).5 The largest absolute deviation between the experiment and correlation (up to 0.84 MPa) appears on the water-rich solvent mixture side (x˜ M ≈ 0.05, T ≈ 353 K, and mCO2 ≈ 0.963 mol/kg). However, this rather large deviation in the pressure corresponds to a deviation in the molality of carbon dioxide of only about 0.053 mol/kg. Thermodynamic properties of solution of CO2 in mixtures of H2O + CH3OH can be calculated from the correlation of Henry’s constant given above by applying the usual thermodynamic relations (cf. eqs 82 and 83), e.g. m m (T,p,x˜ M) ) RT ln[kH,CO (T,p,x˜ M)/p°] ∆solGCO 2 2

)R

(MPa)] +

(MPa)] +

(1 - x˜ M)x˜ M -5.807 +

]

0.037792 (150) T (K)

(1 - x˜ M)x˜ M 0.0002955 +

m (T,p,x˜ M) ∆solHCO 2

(MPa)] )

m (T,psM) x˜ M ln[kH,CO 2,M

(1 - x˜ M)µCO2,CO2,CO2,W(T) + x˜ MµCO2,CO2,CO2,M(T) +

(

(151)

)

m ∂ ln[kH,CO (T,p,x˜ M)/p°] 2

∂(1/T)

p,x˜ M

(152)

]

1907.7 (147) T (K)

The contributions in eq 90 involving the partial molar volumes of carbon dioxide in water and methanol ∞ ∞ (vCO and vCO ) are very small and were therefore 2,W 2,M discarded. Figure 12 shows a comparison between the experimental data and correlation results. ∞ (0) (T,x˜ M), βCO (T,x˜ M), and Subsequently, vCO 2 2,CO2 µCO2,CO2,CO2(T,x˜ M) were simultaneously fitted to the experimental pressures above solutions of carbon dioxide + water + methanol from Xia et al.,134 resulting in (cf. eqs 70, 72, and 91)

m m m ∆solSCO ) (∆solHCO - ∆solGCO )/T 2 2 2

m ∆solCp,CO 2

)

(

)

m ∂∆solHCO 2

∂T

p,x˜ M

(153)

(154)

At standard temperature and pressure (T° ) 298.15 K and p° ) 0.1 MPa) and at several solvent mixture compositions (x˜ M), numerical values for ∆solGm, CO°2, m, m, ° , ∆ S ° , and ∆ C ° resulting from those ∆solHm, sol CO2 sol p,CO2 CO2 equations (on the molality scale) are given in Table 3. Thermodynamic properties of transfer (e.g., of CO2 from pure water to mixtures of H2O + CH3OH) are calculated from the correlation of Henry’s constant given

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 219 Table 3. Standard State (T° ) 298.15 K and p° ) 0.1 MPa) Thermodynamic Properties of Solution of CO2 in Mixtures of H2O + CH3OH (on the Molality Scale)a x˜ M

∆solGm,°/ (kJ/mol)

∆solHm,°/ (kJ/mol)

∆solSm,°/ [J/(mol K)]

∆solCpm,°/ [J/(mol K)]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

8.42 8.10 7.74 7.35 6.93 6.48 6.01 5.50 4.97 4.41 3.82

-19.40 -17.34 -15.62 -14.25 -13.20 -12.47 -12.05 -11.93 -12.12 -12.62 -13.42

-93.3 -85.3 -78.4 -72.4 -67.5 -63.6 -60.5 -58.5 -57.3 -57.1 -57.8

167.7 154.8 141.8 128.9 115.9 103.0 90.0 77.1 64.1 51.0 37.5

a


Table 4. Standard State (T° ) 298.15 K and p° ) 0.1 MPa) Thermodynamic Properties of Transfer of CO2 from Pure Water to Mixtures of H2O + CH3OH (on the Molality Scale)a x˜ M

∆tGm,°/ (kJ/mol)

∆tHm,°/ (kJ/mol)

∆tSm,°/ [J/(mol K)]

∆tCpm,°/ [J/(mol K)]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.32 -0.68 -1.07 -1.49 -1.94 -2.41 -2.92 -3.45 -4.01 -4.60

2.06 3.78 5.15 6.20 6.93 7.35 7.47 7.28 6.78 5.98

8.0 15.0 20.9 25.8 29.8 32.8 34.8 36.0 36.2 35.5

-12.9 -25.8 -38.8 -51.7 -64.7 -77.6 -90.6 -103.6 -116.7 -130.2

a


m above (cf. eq 92). ∆tGCO (T,p,x˜ M) results from eq 141; 2 furthermore

([ ∂ ln

m ∆tHCO (T,p,x˜ M) 2

)R

])

m kH,CO (T,p,x˜ M) 2 m kH,CO (T,p) 2,W

∂(1/T)

(155)

p,x˜ M

m m m ∆tSCO ) (∆tHCO - ∆tGCO )/T 2 2 2

(156)

m m ) (∂∆tHCO /∂T)p,x˜ M ∆tCp,CO 2 2

(157)

At standard temperature and pressure and at several solvent mixture compositions (x˜ M), numerical values for m, m, m, ° 2 (on the molality ∆tGm, CO°2, ∆tH CO°2, ∆tS CO°2, and ∆tC p,CO scale) resulting from those equations (or directly from the data given in Table 3) are given in Table 4. Apart from the data reported by Xia et al.,134 little experimental information can be found in the literature on the solubility of carbon dioxide in mixtures of water and methanol. Chang and Rousseau144 give some experimental results (35 data points) for x˜ M ≈ 0.837 at 243, 258, 273, and 298 K, i.e., at temperatures below the range investigated by Xia et al.134 and at (total) pressures up to about 5.3 MPa. Carbon dioxide molalities range up to about 60 mol/kg of solvent mixture. Model predictions agree well with that literature data as long as the molality of carbon dioxide in the solvent mixture is below about 25 mol/kg. The average relative and absolute deviations between experimental and predicted results then amount to 21.8, 17.3, 10.7, and

Figure 13. Comparison of experimental (total) pressures above solutions of CO2 + CH3OH + H2O from Chang and Rousseau144 [x˜ M ≈ 0.837: 2 (243 K), O (258 K), 4 (273 K), b (298 K)] with model predictions (s), this work.

13.1% and 0.13, 0.15, 0.16, and 0.27 MPa at 243, 258, 273, and 298 K, respectively. As can be seen from Figure 13, those deviations are systematic because predicted pressures are always larger than experimental pressures. (The data by Chang and Rousseau were not taken into account in the model parameter estimation because they are limited to one single solvent mixture composition.) Furthermore, Yoon et al.155 reported carbon dioxide solubility pressures above aqueous solutions of methanol (14 data points) at 313 K [at constant (total) pressures of 7, 10, and 12 MPa] for (gas-free) solvent mixture methanol mole fractions from about 0.2 to about 0.85 and for carbon dioxide molalities ranging up to about 25 mol/kg. The average relative and absolute deviations between experimental and predicted results amount to 4.7, 12.2, and 21.1% and 0.33, 1.2, and 2.5 MPa at 7, 10, and 12 MPa, respectively. Although, at least at 7 and 10 MPa, the data by Yoon et al. lie within the experimental ranges investigated by Xia et al.,134 a rather large disagreement is observed. Therefore, these data were also not considered in the model parameter estimation. Conclusions In previous work (by Maurer and co-workers)3-14,30-51 a thermodynamic model for describing single-gas and simultaneous gas solubilities in chemical-reacting electrolyte solutions of single-solvent water was developed. Activity coefficients were calculated from Pitzer’s GE equation for aqueous electrolyte solutions.15-17 Fugacity coefficients (in the gaseous phase) were calculated from a truncated virial equation of state. Model parameters were systematically determined from experimental information on the properties of the pure components, binary systems, ternary systems, etc., up to multicomponent systems. In a continuation of this work, that model is now extended, allowing for mixed solvents. A model for the Gibbs excess energy of mixed-solvent (chemical-reacting and gas-containing) electrolyte systems is presented. Choosing the pure liquid state (at equilibrium temperature and pressure) as the reference state for the chemical potential of the solvent components and the “pseudopure” state (at equilibrium temperature and pressure) as the reference state for the chemical potential of the solute species (i.e., the pure solute species,

220


experiencing interactions as infinitely diluted in the solvent mixture) results in a contribution to the activities of the solvent components in the ideal solution, which depends on the composition of the solute-free solvent mixture, the concentration of the solutes, and the partial derivative of the (molar) Gibbs energy of transfer of those solutes (from one pure solvent to the solvent mixture) with respect to the concentration of the solvent components. That contribution explains the salt effect (“salting-out” and “salting-in”) observed when salts are added to liquid mixtures of solvents, leading (in equilibrium) to an increase (or a decrease) of the concentrations of those solvent components in a second (vapor or liquid) phase. The activity coefficients of solvent and solute species are calculated by applying a new extension of Pitzer’s equation for the excess Gibbs energy of aqueous electrolyte solutions, which allows for solvent mixtures. The equation for the Gibbs excess energy results from the addition of two terms. The first term describes the Gibbs excess energy of the solute-free solvent mixture. Arbitrarily, the UNIQUAC equation29 was chosen. The second term describes any changes in the Gibbs excess energy resulting from the addition of neutral or ionic solutes to the solvent mixture. To accomplish that, Pitzer’s GE equation for aqueous electrolyte solutions was extended, allowing for any solvent (not only water) or any solvent mixture. It is an osmotic equation containing parameters only for interactions between solute species in the solvent. Those parameters depend on the solvent mixture composition. A main advantage of the new GE equation is that if no solutes are present, it results in the ordinary UNIQUAC equation, whereas if water is the only solvent present, it results in the ordinary Pitzer’s GE equation for aqueous electrolyte solutions; i.e., parameters describing molecular interactions in those subsystems can directly be adopted from the literature. The capability of the new model to simultaneously describe mean ionic activity coefficients and the solubility of a salt (sodium chloride) in a binary solvent mixture (methanol + water) has been successfully tested. Experimental information on the (molar) Gibbs energy of transfer of that salt from one pure solvent to the solvent mixture is taken into account by the model. The model is able to predict the influence of that salt on the VLE of that binary solvent mixture (up to the solubility limit of the salt), if additionally experimental information is available on the VLE of the salt-free binary solvent mixture. Furthermore, the capability of the model to describe the solubility of a single gas (carbon dioxide) in a binary solvent mixture (methanol + water) has been successfully tested. In forthcoming publications, the new model will be applied, for example, to describe (1) single-gas solubilities in other two-component solvent mixtures, (2) VLE and SLE for salt-containing two-component solvent mixtures (including mean ionic activity coefficients) (for other salts and other solvent components), (3) the influence of salts on the solubility of a single gas in twocomponent solvent mixtures, (4) the simultaneous solubility of chemical-reacting gases in two-component solvent mixtures, and (5) the influence of salts on the simultaneous solubilities of chemical-reacting gases in two-component solvent mixtures.

Appendix Fugacity Coefficients (in Gaseous Phases). These are calculated from a truncated virial equation of state

z)

pv p )1+B RT RT

(A1)

∑i ∑j yiyjBij

(A2)

where

B)

Fugacity coefficients result from

ln φi ) [2

p

∑j yjBij - B]RT

(A3)

Pure-component second virial coefficients Bii are calculated from a correlation based on experimental data recommended by Hayden and O’Connell.156 i

ai

bi

ci

di

T/K

CO2 H2O CH3OH

65.703 -53.527 -59.649

-184.854 -39.287 -103.781

304.16 647.3 513.2

1.36 4.277 5.7

273-573 373-577 335-573

Bii (cm3/mol) ) ai + bi

[ ] ci

di

(A4)

T (K)

Mixed second virial coefficients Bij are calculated as proposed by Hayden and O’Connell.156 Critical temperatures and pressures (Tc,i and pc,i), molecular dipole moments (µi), and mean radii of gyration (RD,i) of the pure components, as well as association parameters (ηij), were taken from refs 5 and/or 156 and are listed below. i

Tc,i/K

pc,i/MPa

µi/10-30 C m

RD,i/10-10 m

CO2 H2O CH3OH

241.0 647.3 513.2

5.38 22.13 7.95

0 6.10 5.54

0.9918 0.615 1.536

ηij

CO2

H2O

CH3OH

CO2 H2O CH3OH

0.16 0.3 0.32

0.3 1.7 0

0.32 0 1.63

The following numerical values result for Bij (at several temperatures): Bij/(cm3/mol) T/K

CO2, H2O

CO2, CH3OH

H2O, CH3OH

313.75 354.35 395.0

-162.5 -128.1 -103.3

-256.6 -201.5 -162.0

-258.9 -188.0 -142.6

∞ . According to Brelvi Partial Molar Volume vCO 2,W 133 and O’Connell, the partial molar volume of a solute i ∞ at infinite dilution in a solvent s (vi,s ) is estimated from

∞ + 0.62 vi,s ) as[1 + (v+ exp(bs)] i /vs )

(A5)

+ v+ i and vs are characteristic parameters for each substance.

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 221 Table 5. Overview of All Correlation Equations and Parameters Determined in the Present Work as Well as Sources for Correlation Equations and Parameters Adopted from the Literature psW dW W rW, qW (UNIQUAC) psM dM M rM, qM (UNIQUAC)

H2O Saul and Wagner57 Saul and Wagner57 Bradley and Pitzer105 Dortmund Data Bank52 (rW ) 0.92 and qW ) 1.4) CH3OH Reid et al.58 Hales and Ellender59 M ) -119.39 + 18517.3/T (K) + 18.4557 ln[T (K)] - 0.0507761T (K) (based on experimental data from refs 106-108) Dortmund Data Bank52 (rM ) 1.43111 and qM ) 1.432) CH3OH + H2O

(1 - x˜ M)MW + x˜ MMM

d

d)

MW

(1 - x˜ M)

dW

+ x˜ M

MM dM

3

+ (1 - x˜ M)x˜ M

∑A

v,n(1

- 2x˜ M)n

n)0

Av,0 ) -4.07 × 10-3 L/mol Av,1 ) -0.23 × 10-3 L/mol Av,2 ) 0.5 × 10-3 L/mol Av,3 ) 0.7 × 10-3 L/mol (based on experimental data from refs 75-104)

Wx˜ W ) x˜ W

ΨMW, ΨWM (UNIQUAC)

(0) (1) βNa +,Cl-,W, βNa+,Cl-,W, φ (1) CNaCl,W, RNa +,Cl-,W m Ksp,NaCl,W

(0) (1) βNa +,Cl-,M, βNa+,Cl-,M, φ (1) CNaCl,M, RNa +,Cl-,M m m Ksp,NaCl,M {∆tGNaCl,WfM }

m m Ksp,NaCl {∆tGNaCl }

MW dW MW dW

MM + Mx˜ M dM MM

+ x˜ M

3

∑A

+ (1 - x˜ M)x˜ M

,n(1

- 2x˜ M)n

n)0

dM

A,0 ) -25.2 + 11170/T (K) A,1 ) -5.39 + 3560/T (K) A,2 ) -27 + 10700/T (K) A,3 ) -17 + 6250/T (K) (based on experimental data from refs 107-112) ΨMW ) exp[1.8134 - 728.69/T (K)] ΨWM ) exp[-2.1338 + 790.08/T (K)] (based on experimental VLE data from refs 60-74) NaCl + H2O (1) Silvester and Pitzer113 (where RNa +,Cl-,W ) 2) Silvester and Pitzer113 NaCl + CH3OH (1) (1) βNa +,Cl-,M ) 2.4(RNa+,Cl-,M ) 2) (0) φ βNa+,Cl-,M ) CNaCl,M ) 0 (based on experimental VLE data at T ) 298.15 K from Barthel et al.114) cf. eq 103 m ∆tGNaCl,WfM (kJ/mol) ) 20.46 + 0.1216[T (K) - 298.15] (based on experimental salt solubility data from refs 117 and 118) NaCl + H2O + CH3OH cf. eq 105 3

m m ∆tGNaCl ) x˜ M∆tGNaCl,WfM + (1 - x˜ M)x˜ M

∑A

Ksp,n(1

- 2x˜ M)n

n)0

φ (0) (1) βNa +,Cl-, βNa+,Cl-, CNaCl, (1) RNa+,Cl-

AKsp,0 ) 7.055 kJ/mol AKsp,1 ) 0.577 kJ/mol AKsp,2 ) 5.01 kJ/mol AKsp,3 ) -3.75 kJ/mol [based on emf data (standard potentials) at T ) 298.15 K from refs 116, 120, 121, and 123] (0) (0) (0) βNa ˜ M)βNa ˜ MβNa ˜ M)x˜ M +,Cl- ) (1 - x +,Cl-,W + x +,Cl-,M + 0.52970(1 - x (1) (1) (1) βNa ˜ M)βNa ˜ MβNa +,Cl- ) (1 - x +,Cl-,W + x +,Cl-,M + (1 - x˜ M)x˜ M[-3.0315 - 1.0143(1 - 2x˜ M) - 2.6378(1 - 2x˜ M)2 + 5.1646(1 - 2x˜ M)3] φ φ φ CNaCl ) (1 - x˜ M)CNaCl,W + x˜ MCNaCl,M - 0.029543(1 - x˜ M)x˜ M (1) RNa +,Cl- ) 2 m [based on emf data and salt solubility data (γ(,NaCl ) at T ) 298.15 K from refs 116-118, 120, and 121]

222


Table 5 (Continued) CO2 + H2O Rumpf and Maurer5 (cf. eq 122)

m kH,CO (T,psW) 2,W (∞) vCO 2,W

Brelvi and O’Connell133

(0) βCO , µCO2,CO2,CO2,W 2,CO2,W

(0) Rumpf and Maurer5 (βCO ) µCO2,CO2,CO2,W ) 0) 2,CO2,W

m kH,CO (T,psM) 2,M

CO2 + CH3OH m s ln[kH,CO (T,p ,M M) (MPa)] ) -118.349 + 1022.94/T (K) + 2

∞ vCO 2,M

22.4222 ln[T (K)] - 0.0455865T (K) (based on experimental gas solubility data from Xia et al.134 and from refs. 135 to 154) ∞ vCO (cm3/mol) ) -0.63 - 0.4555[T (K)] 2,M

(0) βCO , µCO2,CO2,CO2,M 2,CO2,M

(0) βCO ) 0.065421 - 25.828/T (K) 2,CO2,M µCO2,CO2,CO2,M ) -1.3275 × 10-3 + 0.4333/T (K) (based on experimental gas solubility data from Xia et al.134)

CO2 + H2O + CH3OH m s m (T,p ˜ M) (MPa)] ) (1 - x˜ M) ln[kH,CO (T,psW) (MPa)] + ln[kH,CO WM,x 2 2,W

m kH,CO (T,psMW,x˜ M) 2 (∞) vCO 2

m (T,psM) (MPa)] + (1 - x˜ M)x˜ M[ - 5.807 + 1907.7/T x˜ M ln[kH,CO 2,M ∞ ∞ ∞ 3 vCO2 (cm /mol) ) (1 - x˜ M)vCO (cm3/mol) + x˜ MvCO (cm3/mol) + 2,W 2,M 3 (1 - x˜ M)x˜ M(10 ){[0.6585 - 243.2/T (K)] +

(0) βCO , 2,CO2

(0) βCO 2,CO2

(K)]

[-3.153 + 1079/T (K)](1 - 2x˜ M) + [-0.9378 + 421.1/T (K)](1 - 2x˜ M)2} (0) (0) ) (1 - x˜ M)βCO + x˜ MβCO + (1 - x˜ M)x˜ M[-0.29602 + 71.949/T (K)] 2,CO2,W 2,CO2,M µCO2,CO2,CO2 ) (1 - x˜ M)µCO2,CO2,CO2,W + x˜ MµCO2,CO2,CO2,M + (1 - x˜ M)x˜ M[0.0002955 + 0.037792/T (K)] (based on experimental gas solubility data from Xia et al.134)

µCO2,CO2,CO2

as results from

Nomenclature

as ) vs/{exp[-0.42704(F˜ s - 1) + 2.089(F˜ s - 1)2 0.42367(F˜ s - 1)3] - 1} (A6) F˜ s is the reduced density of the solvent and is calculated from the molar volume vs and the characteristic parameter of the solvent

F˜ s ) v+ s /vs

(A7)

bs is calculated from

bs ) -2.4467 + 2.12074F˜ s

(for 2.0 e F˜ s < 2.785) (A8)

bs ) 3.02214 - 1.87085F˜ s + 0.71955F˜ s2 (for 2.785 e F˜ s e 3.2) (A9) ∞ To estimate vCO , the molar volume of liquid water 2,W was approximated by the molar volume of saturated liquid water and was taken from Saul and Wagner,57 + and vCO and v+ W were adopted from ref 5 (according to 2 + Edwards et al.,157 vCO ) 80 cm3/mol and v+ W ) 46.4 2 3 cm /mol). ∞ The following numerical values result for vCO : 2,W 3 3 33.4 cm /mol at 313.75 K, 36.4 cm /mol at 354.35 K, and 41.0 cm3/mol at 395.0 K. Compilation of Parameters. Table 5 gives an overview of all correlation equations and parameters determined in the present work as well as the sources for correlation equations and parameters adopted from the literature.

ai ) activity of species i AF,n ) Redlich-Kister parameter (for describing property F; n ) 0, 1, 2, ...) Aφ ) Debye-Hu¨ckel parameter b ) parameter in Pitzer’s modified Debye-Hu¨ckel term B ) second virial coefficient Cp ) molar heat capacity (at constant pressure) Cφ ) third osmotic virial coefficient in Pitzer’s GE equation d ) specific density e ) electron charge f ) function fi ) fugacity of component i F ) any property dependent on the solvent mixture composition (cf. eq 70) g ) function G ) Gibbs energy (or molar Gibbs energy) GE ) Gibbs excess energy h ) function H ) molar enthalpy I ) ionic strength (on the molality scale) kB ) Boltzmann constant kH,j ) Henry’s constant of component j in the solvent (mixture) under consideration Kr ) equilibrium constant for chemical reaction r Ksp ) solubility product mi ) true molality of species i m j i ) stoichiometric molality of component i m ˜ i ) mass of solvent component i m° ) reference molality (m° ) 1 mol/kg) Mi ) molar mass of component i M/i ) relative molar mass of solvent component i divided by 1000 M* ) (mean) relative molar mass of the solute-free solvent mixture divided by 1000 ni ) amount of substance of species i nT ) total amount of substance NA ) Avogadro’s number p ) total pressure

Ind. Eng. Chem. Res., Vol. 44, No. 1, 2005 223 pi ) partial pressure of component i psi ) vapor pressure of component i p° ) reference pressure (e.g., the standard pressure, p° ) 0.1 MPa) qi ) UNIQUAC parameter of pure component i (depending on the external surface area) ri ) UNIQUAC parameter of pure component i (depending on the molecular size) ri ) ionic radius of species i R ) universal gas constant RD ) mean radius of gyration S ) molar entropy T ) absolute temperature T° ) reference temperature (e.g., the standard temperature, T° ) 298.15 K) vi ) (partial) molar volume of component i v+ ) characteristic parameter x ) function xi ) true mole fraction of species i x˜ i ) mole fraction of component i in the solute-free solvent mixture yi ) vapor-phase mole fraction of component i z ) coordination number (z ) 10) z ) compressibility factor zi ) number of charges on the solute species i Greek Letters E R(k) ij ) binary parameters in Pitzer’s G equation (k ) 1 and 2) dependent on the solvent mixture E β(k) ij ) parameters in Pitzer’s G equation (k ) 0, 1, and 2) describing binary interactions (between solute species i and j in the solvent mixture) γi ) activity coefficient of species i γ( ) mean ionic activity coefficient ∆r ) reaction property ∆s ) standard deviation ∆sol ) solution property ∆t ) transfer property ) relative dielectric constant 0 ) permittivity of vacuum η ) association parameter θ˜ i ) area fraction of component i in the solute-free solvent mixture λij ) second virial coefficient in Pitzer’s GE equation µ ) molecular dipole moment µi ) chemical potential of species i µijk ) third virial coefficient in Pitzer’s GE equation ) parameter describing ternary interactions (between solute species i, j, and k in the solvent mixture) νi,r ) stoichiometric factor of reactant i in reaction r F˜ ) reduced density φ ) osmotic coefficient φi ) fugacity coefficient of component i φ˜ i ) segment fraction of component i in the solute-free solvent mixture ψij ) UNIQUAC parameter for interactions between solvent components i and j

Subscripts c ) critical property conv ) conversion i, j, k ) species i, j, and k M ) methanol sat. ) saturation W ) water Superscripts c ) on the molarity scale m ) on the molality scale ° ) in a reference state

ref ) reference state s ) saturated vapor property x ) on the mole fraction scale ∞ ) infinite dilution in the solvent mixture

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Received for review May 26, 2004 Revised manuscript received September 27, 2004 Accepted October 18, 2004 IE049543Y

Model for the Gibbs Excess Energy of Mixed-Solvent (Chemical

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