Chemical Modeling of Aqueous Systems II - American Chemical Society

Chapter 4

Models for Aqueous Electrolyte Mixtures for Systems Extending from Dilute Solutions to Fused Salts 1

Downloaded by UNIV ILLINOIS URBANA on July 27, 2013 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch004

Roberto T. Pabalan and Kenneth S. Pitzer Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720

Models based on general equations for the excess Gibbs energy of aqueous electrolyte mixtures provide a thermodynamically consistent basis for evaluating and predicting aqueous electrolyte properties. Upon appropriate differentiation, these equations yield expressions for osmotic and activity coefficients, excess enthalpies, heat capacities, and volumes. Thus, a wide array of experimental data are available from which model parameters and their temperature or pressure dependence can be evaluated. The most commonly used model for systems of moderate concentration is the ion-interaction approach developed by Pitzer (1) and coworkers. For more concentrated electrolyte solutions, including those extending to the fused salt, an alternate model based on a Margules expansion and commonly used for nonelectrolytes was proposed by Pitzer and Simonson (5). These two models are discussed and examples of parameter evaluations are given for some geologically relevant systems to high temperatures and pressures. Applications of the models to calculations of solubility equilibria are also shown for the systems NaCl-Na SO -NaOH-H O and NaCl-KCl-H O to temperatures up to 350°C. 2

4

2

2

Understanding various geochemical processes and industrial problems requires a thorough knowledge of the thermodynamic properties of aqueous electrolyte solutions. Chemical models that accurately describe the excess properties of aqueous electrolytes over wide ranges of temperature, pressure, and concentration, and which allow prediction of these properties for complex mixtures based on parameters evaluated from simple systems are essential to this understanding. Models based on general equations for the excess Gibbs energy of the aqueous solution provide a thermodynamically consistent basis for evaluating and predicting aqueous electrolyte properties. Upon appropriate differentiation, these equations yield other quantities, including osmotic and activity coefficients, excess enthalpies, entropies, heat capacities, and volumes. Thus, a wide array of experimental techniques provide data from which model parameters and their temperature or pressure dependence can be evaluated. The purpose of this paper is to review two thermodynamic models for calculating aqueous electrolyte properties and give examples of parameter evaluations to high temperatures and pressures as well as applications to solubility calculations. The first model [the ion-interaction model of Pitzer (1) and coworkers] has been discussed extensively elsewhere (1-4) and will be reviewed only briefly here, while more detail will be given for an alternate model using a Margules expansion as proposed by Pitzer and Simonson (5). Current address: Southwest Research Institute, 6220 Culebra Road, San Antonio, T X 78228-0510 0097-6156/90/0416-0044$06.00/0 © 1990 American Chemical Society

In Chemical Modeling of Aqueous Systems II; Melchior, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

4.

PABALAN & PITZER

45

Models for Aqueous Electrolyte Mixtures

ION-INTERACTION OR VIRIAL COEFFICIENT APPROACH For systems ranging from dilute to moderate concentrations, the most commonly used model is based on a virial expansion developed by Pitzer (1). For a system with one solute, M X , having v cations of charge z and v anions of charge z , the excess Gibbs energy can be written as M

M

x

x

2

G'Vn^T = f(I) + 2v v [m B (I) + m \ ^ p ^ M

x

(1)

MX

where n is the number of kilograms of solvent, f(I) is a Debye-Huckel function, m is the molality and I is the ionic strength. B j ^ and C ^ are, respectively, second and third virial coefficients representing short-range interactions between ions taken two and three at a time, and which can be determined from experimental data on single electrolyte solutions. Two additional virial coefficients (3> and y) arise for ternary and more complex systems, but these can be evaluated from data on simple mixtures (2-4,6). The working equations for these solution properties are derived by differentiation of Equation 1, and are given by Pitzer (2). In the case of systems for which a variety of experimental methods have been used, it is advantageous to develop general equations based on all critically evaluated data. Examples are the equations for the thermodynamic properties of NaCl(aq) to 300°C and 1 kb (7,8), of KCl(aq) to 325°C and 500 bars (9), of the alkali chlorides to 250°C at saturation pressure (10), and of the alkali sulfates to 225°C at saturation pressure (11). For many electrolytes, experimental activity/composition data at high temperatures are limited and parameter evaluations have to rely on other types of measurements. A good example is Na S0 (aq) for which a chemical model was developed by Holmes and Mesmer (11)fromtheir isopiestic measurements to 225°C and other literature data. Solubility calculations (12), however, indicated that their model can be improved by including additional experiments at temperatures above 180 C. These improvements were provided by heat capacity measurements (13) from 140 - 300°C at a pressure of 200 bars. Details of the parameter evaluations for Na S0 (aq) at temperaturesfrom25°to 300 C and pressures to 200 bars are described in Pabalan and Pitzer (H), and applications of the derived model to solubility calculations are given below. Prediction of mineral solubilities in electrolyte mixtures is an important use of chemical models for electrolyte solutions and is a stringent test of equations for activity and osmotic coefficients. Recent studies have shown that the ion-interaction model can be used successfully to predict solubility equilibria to high temperatures (12-14). In the examples given below, solubilities of the four stable sodium sulfate minerals are calculated and compared with experimental values. NajSOâq) activity and osmotic coefficients, as well as standard state Gibbs energies, were calculated using the ion-interaction model (13). Gibbs energies of the solids were taken from the literature (15,16). In Figure 1, predicted solubilities of sodium sulfate solids in water to 350°C are compared to experimental values tabulated by Linke (17). Although the calculated values are higher than the experimental data below 24TC, the differences are not large, averaging 0.08 m Na S0 . For solubility calculations in ternary and more complex systems, the ion-interaction model requires two additional terms, and \j/ (6). Whereas both these parameters undoubtedly vary with temperature, it was found adequate for solubility calculations to keep O at its 25°C value and to assign simple temperature functions to y (12,13). For the ternary systems N a ^ O ^ N a C l - ^ O and N a ^ O ^ N a O H - ^ O , comparisons of calculated and experimental solubilities up to 300°C are given by Pabalan and Pitzer (12,13). More stringent tests of the model are solubility calculations in the quaternary system Na S0 -NaCl-NaOH-H 0. Predicted and experimental solubilities at 200° and 300 C are compared in Figure 2. Other examples of solubility calculations using the ion-interaction approach are given in Refs. 4,12,14, and 18-26.


w

2

4

6

8

2

2

4

4

e

2

4

2

M A R G U L E S EXPANSION M O D E L An alternative to the ion-interaction approach is the Margules expansion model for systems which may range in concentration from dilute solutions to the fused salt. Aqueous solutions miscible over the whole concentration range at moderate temperatures are relatively uncommon, but there are electrolytes of geochemical and industrial interest which become extremely soluble in water at high temperatures and pressures. Although the ion-interaction model can represent electrolyte properties to very high ionic strengths by using additional virial terms (28), this treatment becomes more complex and unsatisfactory In Chemical Modeling of Aqueous Systems II; Melchior, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


46

CHEMICAL MODELING OF AQUEOUS SYSTEMS II

Figure 1. Solubilities of mirabilite (NajSO^ 10H O) and thenardite (Na S0 ) in water as a function of temperature. The symbols are experimental data tabulated by Linke (17) and the curves are predicted values. (Reproduced with permission from Ref. 13. Copyright 1988 Pergamon Press.) 2

2

4


4.

PABALAN & PITZER



NaOH

Mole % Figure 2. Calculated and experimental solubilities of thenardite (NaÔ^ at fixed molalities of NaOH in the quaternary system NaCl-NajSO^NaOH-Hp at: a) 200°C, and b) 300°C. The symbols are experimental data from Schroeder et al. (27). (Reproduced with permission from Ref. 13. Copyright 1988 Pergamon Press.) #

American Chemical Society Library In Chemical Modeling of Aqueous Systems II; Melchior, D., et al.;

ACS Symposium Series; American Chemical 115S 16th St., Society: N.W. Washington, DC, 1990.

47


48

CHEMICAL MODELING O F AQUEOUS SYSTEMS II

at very high ionic strengths. In addition, the molality for a pure fused salt is infinite. An alternate model is required, therefore, for systems extending to the fused salt and for other systems of very high, but limited, solubility. The model discussed below is analogous to models used for nonelectrolyte solutions. Various expressions have been used to describe the excess Gibbs energy of nonelectrolytes (29), and the Margules expansion has been used successfully by Adler et al. (30) for several nonelectrolyte systems. The theoretical basis for extending this approach to electrolyte solutions was discussed by Pitzer (31). Briefly, in electrolyte solutions where there is substantial ionic concentration, the long-range interionic forces are effectively screened to short-range by the pattern of alternating charges. Thus all of the interparticle forces are effectively short-range and the system properties can be calculated by methods similar to those for nonelectrolytes. In the dilute range, however, the alternating charge pattern and its accompanying screening effect is lost and the long-range nature of electrostatic forces must be considered. As with the ion-interaction model, this effect is described by an extended Debye-Huckel treatment. The Gibbs energy per mole of solution can be written in terms of the sum of short-range and electrical effects: G

S

'"/£n = g" = g + g i

D H

( 2 )

i

A choice must be made for the reference state for the solute: either the pure liquid (possibly supercooled), or the solute at infinite dilution in the solvent. The latter differs from the conventional solute standard state only in the use of the mole fraction scale rather than molality units. The activity coefficient of a symmetrical salt M X is either 0 a s x

M7M&)->

i-»

0

( 3 )

ln (YMYX) -> 0 as XJ -> 1,

4

()

where the asterisk denotes the infinitely dilute reference state on a mole fraction basis, and x is the mole fraction of the solvent on an ionized basis. For example, l

Xj = n /(n +2n ) 1

1

(5)

2

with n and n the numbers of moles of solvent and solute, respectively. In addition, the mole fractions are defined as: 1

2

x = x = n ^ n ^ 2n ) = (l-x^/2 ,

(6)

x = x +x = 1-Xj

(7)

M

x

2

2

M

x

and the ionic strength on a mole fraction basis is given by: Ix = (

(

W l x zf

8

)

;

i with z the charge on the ith ion. For the present case with z = 1, ^ = x*/2. A system of equations for electrolytes based on the reference states expressed in Equations 3 and 4 was developed in detail for singly-charged ions by Pitzer and Simonson (5). Although they considered both types of reference states for the solute, most of their working equations are for the pure liquid reference state. This reference state was used by Pitzer and L i (32) for a study of the NaCl-H 0 system extending to 550°C. For the present research limited to 350°C, however, it seemed better to use the infinitely dilute reference state, and the equations below are derived on that basis. The short-range {

2


4.

PABALAN & PITZER


49

contribution to the excess Gibbs energy of an aqueous solution with a single solute M X can be written in terms of the Margules expression: s

2

g /RT = - x ( W 2

1 > I

x-x U 1

l i M X

)

(9)

where W and U are specific to each solute M X and are functions of temperature and pressure. The Debye-Huckel term representing the long-range electrostatic contribution is given by: 1 M X

1 M X

DH

1/2

g /RT = -(4A I /p) ln(l+pl ). x

x

(10)

x

The Debye-Huckel parameter A is related to the usual parameter molality basis) by

(for the osmotic coefficient on a


x

A =^ %

(11)

x

where Q. is the number of moles of solvent per kg.(~55.51 for water). Appropriate differentiations of the Gibbs energy with respect to the numbers of moles of the solvent and M X (at constant T and P) yield the equations for the activity coefficients: ln

Y l

= 2A I x

2A

3/2 x

/(l+pI

2

1/2 x

ln

2

) + x [W 2

1+

J

1/2

LIMX

+ (l-2 )U Xl

1>MX

1/2

ln(YMTx = * { ( / P ) ( P x ) + K (l-2I )/(l+pI x

2

+ 2(x -l)W 1

2

1>MX

-f4x x U 2

1

1)MX

],

m x

(12)

)}

.

(13)

The activities of the solvent and the solute are then given by: (14)

ln a = lntx^), {

ln

a M X

(15)

= ln(x x Y^y ). M

x

x

The parameter p in Equations 10, 12, and 13 is related to the distance of closest approach of ions. To keep the equations for the thermodynamic properties of electrolyte mixtures simple, it is desirable to have the same value of p for a wide variety of salts and for a wide range of temperature and pressure. The functional forms of Equations 12 and 13 are relatively insensitive to variations in p values. It has been found satisfactory Q_,5) to take a standard value for p and let the short-range force terms accommodate any composition dependency of p. In calculations for metal nitrates in water from 100-163°C (33), p was given a fixed value of 14.9. For the systems considered in this study, a constant value of 15.0 was found to be satisfactory. The Margules expansion model has been tested on some ionic systems over very wide ranges of composition, but over limited ranges of temperature and pressure (33,34). In this study, the model is applied over a wider range of temperature and pressure, from 25-350°C and from 1 bar or saturation pressure to 1 kb. NaCl and KC1 are major solute components in natural fluids and there are abundant experimental data from which their fit parameters can be evaluated. Models based on the ion-interaction approach are available for NaCl(aq) and KCl(aq) (8,9), but these are accurate only to about 6 molal. Solubilities of NaCl and KC1 in water, however, reach 12 and 20 m, respectively, at 350°C, and ionic strengths of NaCl-KCl-H 0 solutions reach more than 30 m at this temperature (35). The objective of this study is to describe the thermodynamic properties, particularly the osmotic and activity coefficients, of NaCl(aq) and KCl(aq) to their respective saturation concentrations in binary salt-H 0 mixtures and in ternary NaCl-KCl-H 0 systems, and to apply the Margules expansion model to solubility calculations to 350°C. For purposes of developing general equations for the thermodynamic properties of electrolyte solutions, it is useful to recalculate experimental values to a single reference pressure. This allows experimental data for different solution properties (e.g., activities, enthalpies, and heat capacities) whose relationships with each other are defined on an isobaric basis, to be considered in the overall regression 2

2

2


50

CHEMICAL MODELING OF AQUEOUS SYSTEMS II

of the model equations. This procedure has been shown to be useful for NaCl(aq) (8), KCl(aq) (9), and Na S0 (aq) (13). The parameters required to recalculate thermodynamic data from the experimental to the reference pressure can be determined from a regression of volumetric data. For the Margules expansion model, the pressure dependence of the excess Gibbs energy, osmotic coefficient and activity coefficient can be derived from volumetric data as shown below. The apparent molal volume, V , of the solution is given by: 2

4

1/2

*V = ( A / P ) l n ( l + pl ) + V - 2RTx [W; v

2

2

>MX

- x , ^ ]

0*0

where V° is the partial molal volume of the salt at infinite dilution. Also, 2


A

v x

= ^RTOÔA^PJt ,

Wl,Mx = 0 W

1 > M X

(16b)

/aP) ,

d6c)

/9P) .

(

T

and UI,MX = O U . 1

m x

T

v

1 0

, 6 D

>

res

Equations 16c and 16d relate the volumetric parameters W and \J\ ^ P sure dependence of the parameters Wj ^ and U ^ for the excess Gibbs energy and osmotic/activity coefficients. In this study values of ^ and U* ^ for NaCl(aq) were evaluated at temperatures from 25' to 350°C, and pressures from 1 bar or saturation pressure to 1 kb, based on apparent molal volumes calculated from the equations of Rogers and Pitzer (7), plus additional experimental data not considered in their study (36-38). Initial regressions of Equation 16 to isothermal and isobaric sets of experimental values indicated the P and T dependence of V , ^ and U* j ^ . Excellent fits are normally obtained to isobaric-isothermal sets of data. Examples are shown in Figure 3 for apparent molal volumes at 350°C and pressures to 1 kb. The curves are calculated values at the indicated pressures using parameters listed in Table I. The next step is to perform a simultaneous regression of NaCl(aq) apparent molal volumes from 25-350°C. Over this wide range of temperature, however, and particularly above 300°C, standard-state properties based on the infinitely dilute reference state exhibit a very complex behavior (7,8), which is related to various peculiarities of the solvent. Thus in their representation of NaCl(aq) volumetric properties, Rogers and Pitzer (7) adopted a reference composition of a "hydrated fused salt," NaCl • 10H O, to minimize the P and T dependence of the reference state volume and to adequately fit volumetric data to 300°C and 1 kb. In this study the (supercooled) fused salt is used as the reference state. The equation for the apparent molal volume on this basis can be easily derived from that for the excess Gibbs energy of Pitzer and Simonson (5), and is given by: lMK

iMX

l

2

2

* = (A , /p)ln[(l + p l f K i + p(I ) )] + V ' + 2 R T ( W ; v

1/2

v

x

x

fs

2

Xl

)MX

+ x U] \ 2

(17)

Chemical Modeling of Aqueous Systems II - American Chemical Society

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