thermodynamic modeling of concentrated multicomponent aqueous electrolyte and non-electrolyte...

Pergamon Chemical Engineering Science, Vol. 50, No. 12, pp. 1953 1959, 1995 Copyright © 1995 Elsevier Science Ltd

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0009-2509(95)00047-X

THERMODYNAMIC MODELING OF CONCENTRATED MULTICOMPONENT AQUEOUS ELECTROLYTE AND

NON-ELECTROLYTE SOLUTIONS

ALEXANDER K O L K E R and JUAN J. DE PABLO Department of Chemical Engineering, University of Wisconsin Madison, 1415 Johnson Drive, Madison,

WI 53706, U.S.A.

(Received 12 October 1994; accepted in revised form 9 January 1995)

Abstraet--A new approach is presented for calculation of activity coefficients in multicomponent aqueous electrolyte and non-electrolyte solutions. The approach is based on an integration of the Gibbs-Duhem equation. It is shown that using standard thermodynamic properties for the pure components and for the components at infinite dilution in water, it is possible to predict joint solubilities of both electrolytes and non-electrolytes. To illustrate the usefulness of the method proposed in this work, calculations of solubility are presented for the two-electrolyte systems H20 NaCI KC1, H20-NaNO3-KNO3, H20 CaC12 NaC1, and for a system containing both an electrolyte and a non-electrolyte H20-sucrose NaCI. The solubility of carbon dioxide at different pressures in water containing various electrolytes [-Ca(NO3)2, NaCI, CaCI2, N a 2 S O 4 ] is also calculated.

I. I N T R O D U C T I O N

Most chemical industries comprise processes that in- volve mult icomponent electrolyte solutions. Damage prevention of industrial equipment requires accurate knowledge of the concentration of electrolytes throughout a process. Recent environmental concerns require that the concentration of electrolytes in final products or in waste streams be controlled with pre- cision. Prediction of the thermodynamic properties of mult icomponent electrolyte solutions therefore poses an important challenge for engineers.

In practice, the equilibrium thermodynamic properties of mult icomponent solutions are often determined within the framework provided by Pitzer (1991) or that provided by Chen et al. (1982). Pitzer's equations are based on a virial expansion for the osmotic pressure. The coefficients appearing in such an expansion are determined by regression of experimental data. The model of Chen et al. (1982) is based on an extension of the NRTL model of Renon and Prausnitz (1968), which was originally conceived for non-electrolyte solutions. The parameters appearing in such a model are also determined by regression of experimental data. Both formalisms lead to results of com- parable quality, and are generally included in com- mercial process simulators (e.g. ASPEN, PROCESS).

For mult icomponent solutions, both formalisms require binary and ternary experimental data for all components in the mixture. Unfortunately, such data may not be available for a number of systems of engineering importance. Clearly, new methods that require only limited experimental information but that lead to reasonable predictions of the equilibrium thermodynamic properties of electrolyte solutions would be of great use in chemical engineering practice.

Recently, we have proposed a novel interpretation of the NRTL model which appears to provide good results for phase equilibrium in binary electrolyte solutions (Kolker and de Pablo, 1995). Encouraged by the success of this new method, in this paper we extend it to mult icomponent solutions and apply it to several systems of engineering importance. Our extension is based on an integration of the Gibbs -Duhem equation. Such an integration is described in Section 2. In Section 3 we review briefly the model and the methods described in our previous paper (Kolker and de Pablo, 1995), which we refer to as (I) throughout the remainder of this work. Section 4 discusses the calculation of solubilities in aqueous solutions containing both two electrolytes and an electrolyte and a non-electrolyte, and Section 5 presents our calculations of solubility of a volatile solute (carbon dioxide) in several aqueous electrolyte solutions.

2. T H E R M O D Y N A M I C B A C K G R O U N D

We propose to calculate activity coefficients for each salt by integrating the Gibbs -Duhem equation for a mult icomponent system. The general form of this equation (temperature and pressure are constant) is

mldpl + ~ mid#i = 0 (1) i=2

where m~ = const is the molality of pure water ( = 55.508), mi is the molality of solute i, and n is the number of components (1 solvent and n -- 1 solutes). Chemical potentials (#i and p~) are related to activities (in differential form) by

d#i = RTd In ai(m2 . . . . . m,). (2)

The activities are functions of n - 1 independent variables (molalities). We may therefore write the follow-

1953 CES 50-12-H

1954

ing general expressions:

" {0 In a i \ d l n a i = k ~ 2 t ~ m k ) ~ dmk. (3)

Substituting expressions (2) and (3) into eq. (1), we find

01n al

k mj# k

~. ~. {~ in ai" ~ +i~2mlk~_2~ ~mk ;.,~,, ,,dmk =O (4)

and, after some rearrangements, we arrive at

4 r f631na~'~ 4 f~lna~'~ ] , ,__2., ), . , + 2-, m i / ~ / [Omk = O. i= ~ \ cm~ /tmj, k ]

(5)

Equation (5) is satisfied if all the coefficients in par- entheses vanish, thereby leading to the following n - 1 equations:

f? In a t~ ~ ft~ In a~'~ rn~ - - + 2.. m i l - - I = O,

~. Omk )r,b,,~, i=2 \C~mk.]m,~-~, k = 2 . . . . . n. (6)

Next, we proceed to convert eqs (6) into a form in which each of them contains only the activity of one of the salts, ak, and the activity of water, at. Thermodyn- amic consistency requires that the partial derivatives be related by (McKay, 1953)

(0 In a i / O m k ) m ~ # ~ = (C ~ In a~/Omi)mj. ,. (7)

Substituting eqs (7) into eqs (6), we have

~mi(O In ak/t~mi)m~. , + m x (c~ In a~/Omk)m~ ~ ~ = O, i=2

k = 2 . . . . . n (8)

To integrate these n - 1 equations, we introduce a new set of independent variables, namely the mole fraction of the salt i in the mixture of solutes (on a water-free basis), zi = m i / ~ ~= 2mi (with ~ ~= 2 zi = 1), and the mole fraction of water, x t =m~/(m~ + ~2~= 2 toO. We replace molalities by these new independent variables to get

(O In ak/c3mi)m~ ,,, = (63 in ak/63x 1) . . . . . . . . . (63x 1/c3ml)m~ ¢,,

n - 1

+ ~ (0 in ak/Oz~)x, (Oz/Omi)m~ ~, j=2

(9)

where the partial derivatives of xt and z~ can be written explicitly as

(Ox~/Om,) . . . . = -- x2/m~

(Oz~/Om~),,~ , , = x t (1 -- zi)/(x,mt)

(cgz/Omi) . . . . = -- ZjXl/(Xsml)

and where x~ denotes the mole fraction of all solutes combined, i.e. x, = 1 - x~.

ALEXANDER KOLKER and JUAN J. DE PABLO

Substituting expressions (9) into eqs (8), we arrive at a much simpler set of equations, namely

(63 In ak/OXO ........ , _ ml (d In al/~mk)m~, ~, X1Xs

k = 2 . . . . . n. (10)

Note that eqs (10) contain a derivative of the activity of water and, in contrast to eqs (8), only one solute-activity derivative. In order to facilitate the sub- sequent integration of eqs (10), we express the derivatives of the activity of water with respect to molalities in terms of derivatives with respect to xl and z~. After some algebraic manipulations we arrive at:

(d In ak/dXl) . . . . . . . . . = -- (t? In al/t~Xl) . . . . . . . . . Xl/Xs

+ (~In al/dZk)x,(1 -- Zk)/X 2

--(j="2~x kZJalnal /OzJ) /x2,

k = 2 . . . . . n. (11)

Activities, ak, and activity coefficients, Yk, are related by

ak=XkTk, k = 1 . . . . . n. (12)

Note that the mole fraction of solute Xk is related to xs through Xk = ZkX~. Substituting this definition into eqs (11) and integrating them by parts from xt to 1 at Zk = const ¢ 1, we arrive at expressions for the activity coefficients of each solute.

For components k = 2 . . . . . n - 1 we find

f2 lnyk = lny~ ° -- x l / x s lny l -- lnyl/x2sdXl 1

-- (1 - Zk) f ] (O In yt/OZk)/X~ dx l

n--1 I i + ~ zj (alnyl/c?zj)xl/x~dxl (13)

j= 2 , j v~ k t

For the "last" solute, component n, we get

;J lny. = lny2 - xl/x~ln71 -- lnT~/x~dxl 1

n-l=2 ~ + ~ z j (c~lnTt/~zj)~/x2dxl (14) J

where In yF (and In y~) denotes the activity coefficient of component k at infinite dilution (at Xl = 1) in pure water.

Equations (13) and (14) are exact thermodynamic relations. They do not depend on any molecular model but are merely an integral form of the Gibbs -Duhem equation. Their usefulness resides in the fact that the activity coefficient of each solute can be calculated from the activity coefficient of water and its dependence on zj (j = 2, ..., n - 1). The activity coefficient of water in a mult icomponent solution, ln,/l, and its derivatives with respect to a solute's mole fraction in the anhydrous mixture (0 In ?q/ctzi)~ can be obtained either from experiment or from a model. In

Thermodynamic modeling

the latter case, a solution theory is required; ulti- mately, the accuracy of the calculations depends on the accuracy of such a theory.

3. C A L C U L A T I O N O F THE ACTIVITY C O E F F I C I E N T O F

WATER

To calculate In ?1 in a multicomponent solution we apply a method originally developed for binary solutions. While a detailed description can be found in (I), for convenience Appendix A gives a condensed ver- sion of the necessary equations. In this work we treat a multicomponent system as pseudobinary, i.e. we consider the system to consist of a water (component 1) and a mixture of all dissolved salts, which is treated as one unique complex pseudocomponent 2.

The calculation of ln?l in the framework of this method requires estimation of several molecular characteristics of the mixture (see I and Appendix A). As initial data, functions n°z(z2 . . . . . z, l) and 022(z2 . . . . . z, 1) for the mixture of solutes are necessary. We define a function n°2(z2 . . . . . z,-1), which corresponds to the average number of nearest neighbors of complex pseudocomponent 2 in the absence of water, as a linear combination of nO:

n

n°2 = X zkn°k i=2

where nO denotes the average number of nearest neighbors of pure component k. We define a function g22(z2 . . . . . z._ 1), which represents a Gibbs free energy difference between pure complex pseudocomponent 2, and the same pseudocomponent at infinite dilution in the solvent as

. 2 2 = zk,, ° - z.¢(z l (15) k=2 k=2

where kt ° is the standard chemical potential of component k in the anhydrous mixture in a "supercooled fused salt" reference state, and #~' is the standard chemical potential of the same component in a reference state corresponding to a "hypothetical ideal solution of unit concentration" (Robinson and Stokes, 1965). The former can be written as

It°(zk) = p°(z k = t) + RTln?k,~Zk (16)

where In ?k,~ are the activity coefficients of the salts in their anhydrous mixture (x2--* 1). Their values depend on the interaction energies of the salts with each other. An ideal mixture of salts is formed when all In 7k,s are equal to zero.

Equation (15) can be rewritten in the following form:

022 ~--- ~ Zkgkk -1" gEs (17) k=2

where 9ES is the excess Gibbs energy of the system on a solvent-free basis (anhydrous mixture)

9ES = R T ~ Zk lnTk.~, (18) k=2

1955

and where gkk are determined as described in Appen- dix A (eq. (A4)]. Note that our definitions of n°2 and 922 for a complex pseudocomponent follow naturally from our interpretation of these functions for a binary system (I).

4. RESULTS O F C A L C U L A T I O N S

The model presented in this work has been used to predict solubilities in ternary systems. While the equations obtained in previous sections are applicable for calculating the activity coefficients of a wide class of systems, for simplicity, we restrict our attention to the calculation of branches of solubility for components which do not form crystal hydrates or double salts in the corresponding solid phases. Generalization to systems that form crystal hydrates or double salts in solid phases will be presented in a forthcoming paper.

We give examples for the following two-electrolyte systems: H20(1)-KCI(2)-NaCI(3), H20(1)-KNO3(2) -NaNO3(3), and H20(1)-CaC12(2)-NaCI(3). We also present results for a system containing both an electrolyte and a non-electrolyte: H20(1)-sucrose(2)- NaCl(3). Note that we use the same equations for these different types of systems; most conventional models cannot be applied directly to both of them.

Equations (13) and (14) for the activity coefficients of the components of a ternary system (n = 3) take the following form:

f2 ln72 = l n T ~ - x l / x s l n 7 1 - ln? l /x~dx l l

-- (1 - z2) (0 In ?l/Oz2)x,/x~ dxl (19) i

f2 In 73 = In 7~ - x l /xs In 71 - In 71/x~ dxl 1

f2 + z2 (~? In 71/OzE)x,/x~ dx l . (20) I

To calculate 022 we need to determine the excess Gibbs energy of a supercooled fused anhydrous mixture of salts, 0~- Unfortunately, direct experimental data for this quantity are not available. There is, however, experimental information about the heat of mixing, hES, of fused alkali-metal nitrates and halides (Lumsden, 1966). In most cases, such fused mixtures are described well by regular solution theory. In such a theory, the excess Gibbs free energy of mixing, 9ES, is given by

gES : h Es = A 2 3 z 2 z 3.

Extrapolation with respect to temperature of the activity coefficients given by Lumsden (1966) (we assume that ln?~oc 1/T) leads to the value of A 2 3 ~ - - 720 cal/mol for the mixture KNO3-NaNO3 a n d A23 ~ - - 100 cal/mol for the mixture KC1-NaC1. These numbers reflect the fact that these anhydrous mixtures of salts are not ideal but exhibit small nega- tive deviations from ideality.

1956 ALEXANDER KOLKER a n d JUAN J. DE PABLO

For the system CaC12(2)-NaCI(3), the activity coefficients extrapolated to the standard temperature 5 25°C have the form (Lumsden, 1966)

4 In 72,s = -- 7.6z2/(2 - - g3) 2,

lny3.s = - 15.2z~/(2 - - 23) 2 E~ 3

To calculate g~ we use eq. (18). ,~ To obtain gf for the anhydrous system NaCI(2)- 2-

sucrose(3), we used data for the standard Gibbs energy of transfer of NaC1 from water to an aqueous 1 sucrose solution (Wang et al., 1993). These authors measured the dependence of this quantity on the con- 0 centration of sucrose in solution up to 30% wt; they found it to be linear. Assuming that a linear extrapolation up to pure 100% sucrose can be made, we obtained the standard Gibbs free energy of transfer of NaCI to pure sucrose. From this quantity, we then determined the activity coefficient of NaCI at infinite dilution in sucrose and, consequently, the value A23 ~ - 6 kcal/mol (assuming that regular solution theory is valid for this binary system). 7

The individual component activities for the aqueous ternary system on the branches of solubility, In a2 and In a3, must be constant and satisfy

lnak = -- {AHo.k(1 - T/To.k) + ~AHi,k(1 - T/Ti,k) i

-- ACv.k[To.k -- T(1 + In TO.k/T)]}/RT (21)

for components 2 and 3, respectively (I). Here, AHo,k and To,k are the solute's enthalpy and temperature of fusion and ACp,k is the change of heat capacity upon fusion. The second term in the right-hand side of eq. (21) has to be added if i phase transitions occur in the crystal structure of the salt, each with enthalpy change AHi at temperature Ti. Enforcing these criteria per- mits calculation of the joint solubility.

Figures 1-4 show experimental data and the results of our solubility predictions. In general, the predicted results are in reasonable agreement with experiment. We should point out, however, that the description of

• ~ ' ,;, ' 6 " 6 1'0 1'2 N a N O 3 , m 3

Fig. 2. Solubility isotherm for the system water-KNO 3 (2)- NaNO 3 (3) at 25°C: (11) shows experimental data (Linke and Seidell, 1965); ( ) gives the results of our calculations.

Z

6 , , , i I ",, •

5 ',,, " , , •

4

3

2

1 -

0 0 .0 OiS 1'.0 • 1'.5 ' 210 " 2'.5 " 3'.0 " 315

m 2, CaCI 2

Fig. 3. Solubility of NaCI in aqueous solutions of CaCI 2 at 25°C: (11) shows experimental data (Linke and Seidell, 1965); ( ) gives the results of our calculations; ( . . . . ) gives

the calculated solubility assuming #ff = 0.

._1 o Z

E

i

1 2 3 4

m 2 , KCL

Fig. 1. Solubility isotherm for the system water- KCI(2)-NaCI(3) at 25°C: (11) shows experimental data (Linke and Seidell, 1965); ( ) gives the results of our

calculations.

6 .

z 4 -

c

2 -

- - J L

0

m 3 , s u c r o s e

Fig. 4. Solubility isotherm for the system water-sucrose (2)-NaCI(3) at 25°C: (11) shows experimental data (Linke and Seidell, 1965); ( ) gives the results of our calcu-

lations.


a supercooled fused anhydrous mixture of salts by regular solution theory is only a first approximation which happens to work for the systems studied here. For other mixtures, a more refined treatment of solute-solute interactions is likely to be necessary.

To illustrate how sensitive our results could be to the value of g~, we have also calculated the solubility of NaC1 in aqueous solutions of CaCI2, assuming that g~ = 0. This calculated solubility corresponds to the dashed line in Fig. 3; these results indicate that ne- glecting solute-solute interactions can in some cases introduce appreciable errors into the predictions of the model.

5. S O L U B I L I T Y O F C O 2 IN A Q U E O U S S O L U T I O N S

C O N T A I N I N G A N E L E C T R O L Y T E

Vapor-liquid equilibrium is determined by the con- dition that the chemical potential of each component be equal in the vapor and in the liquid phases, i.e.

I~°v + RTlnPy2~02 0 = #2,1iq + RTln72x2 (for C02)

(22)

#o,~ + RTlnP(1 - - 22)(/)1 -~ ~/10,1iq + R T l n a l

(for water)

where P is the total pressure, q~ and tp2 are the fugacity coefficients of water and CO2, x2 and Y2 are the mole fractions of carbon dioxide in water and in the vapor, respectively, 72 is its activity coefficient in water, and a~ is the activity of water. The values of #o and #Oq are the standard chemical potentials of carbon dioxide and water in the vapor phase and in the pure liquid state. Following the approach that we have taken in our previous work (I), we have chosen the standard state of carbon dioxide in water to be hypothetical superheated pure liquid CO2 at the temperature and pressure of the system. The advantage of this standard state compared to a standard state at infinite dilution is the independence of the former on the presence of other solutes.

Equations (22) lead to the following equation for the solubility of the gas:

I n ~2X2 = (]xO,v - - ~°liq)/RT + In P t p 2 Y2 (23)

where

Y2 = 1 -- e x p [-(/A0,1iq - - l~°,~)/RT] al/Ptpl

,~ 1 -- exp [(AG°aiq -- AG°v) /RT]/P.

The mole fraction of water in the vapor phase is very small at elevated pressures of carbon dioxide. The activity of water in the binary system CO2-H20 is close to unity. For simplicity, we assume that the ratio al/~ol ~ 1. (At elevated pressures, ~01 could deviate significantly from unity. However, the exponent in the second term in the expression for Y2 is much smaller than unity. At elevated pressures, this term, which is inversely proportional to P, becomes even smaller making Y2 close to unity. Therefore, our assumption that the ratio al/~ot ~ 1 at all pressures does not introduce an appreciable error in Y2).

1957

The difference between the chemical potentials of gaseous and superheated liquid CO2 on the right- hand of the eq. (23) is equal to

o o = AGO,~ _ AGO,liq #2 ,v - - ~2,1iq

= aHo(1 - T/To) -- AH,(1 - T/Tp)

-- Acp[T o -- T(1 + In To~T)]

- - v ° ( P - - er) (24)

Here AG°v and AG°.nq are the Gibbs energies of formation in the gaseous phase and in the superheated liquid state, AH o and AHp are the heats of sublimation and melting of solid carbon dioxide, Acp is the difference between the heat capacity of liquid and gaseous carbon dioxide, To and Tp are sublimation and melting temperatures, v ° is the molar volume of liquid carbon dioxide and P, is the reference pressure (usu- ally 1 atm).

The activity coefficient of carbon dioxide is given by eq. (20). At elevated pressures, a Poynting correction is employed according to

( ~ : - v o) ln~2 = lny2(P,) + - - (P - Pr)

R T

where f2 is the partial molar volume of carbon dioxide in water.

To calculate the fugacity coefficient of carbon dioxide we use the equation of state of (Nakamura et al., 1976). Values of AHo, AHp, To, Tp, Acp for CO2 are taken from the literature (Angus et al., 1976; Vukalovich and Altunin, 1968).

Note that in calculating the solubility of CO2 we have not taken into account the formation of ions due to the reaction

CO2 + H20~'-~H + + HCO3.

Its equilibrium constant is very small and for a molality greater than 10 -4 the effect of dissociation disap- pears (Edwards et ak, 1978).

We have applied eqs (19), (23) and (24) to calculate the solubility of carbon dioxide in water containing electrolytes, Ca(NO3)2, CaCI2, Na2SO4 and NaC1. The last two systems have been studied extensively by Corti et al. (1990) using Pitzer's equations. Within the framework of our approach, we need to determine the excess Gibbs energy of carbon dioxide-salt anhydrous mixtures. Unfortunately, in contrast to mixtures of salts, even indirect thermodynamic information for such a quantity is not available. If we apply regular solution theory we need to determine only one adjustable parameter, AEa/RT , for each of these mixtures. To do so, we have no other alternative than to use at least one experimental value for the solubility of carbon dioxide. The values of the parameter A23/RT are given in Table 1. (Note that this parameter does not depend on pressure.) Figures 5 and 6 show the results of our calculations for the solubility of carbon dioxide at different concentrations of electrolyte along with the corresponding experimental data. Again, calculated results are in reasonable agreement with experiment.

1958

Table 1. Values of the parameter A2a/RT at 25°C for anhydrous carbon dioxide-supercooled fused salt mixtures

Salt NaC1 Ca(NO3)2 CaC12 Na2SO,,

A23/RT - 0.2 - 6.1 - 9.89 10.5

3.0.

2 . 8 .

2 . 6 -

~ 2.4-

,~ 2.2- O 0 2.0-

1.8.

~: 1.6.

1 .4 .

1.2.

1.0

2

o~.o " o~.5 " 11o " ~5 " 210 " 2~5 . 3)0 3~5

molality of the electrolyte, rn 3

Fig. 5. Solubility of carbon dioxide in water containing electrolytes: (1) NaC1 (ll L (2) CaC12 (Q), (3) Na2SO 4 (A), (4) Ca(NO3) z (!?); the pressure is 1 atm, the temperature is 25°C. (11) shows experimental data (Onda et al., 1970);

( ) gives the results of our calculations.

ALEXANDER KOLKER and JUAN J. DE PABLO

lytes in aqueous solutions ana for the solubility of a gas (carbon dioxide) in water containing several electrolytes at different pressures.

In all cases, our predictions are in reasonable agreement with experiment. Note that our model relies on exact thermodynamic equations for the activity coefficients for mult icomponent systems. These equations are not dependent on any molecular model. Further- more, our model does not require empirical parameters but it requires instead the excess Gibbs energy of anhydrous solute mixtures. In this work, for simplicity, we use regular solution theory to describe such mixtures; a more elaborate model could in principle be used.

The main practical difficulty faced by our method is the determination of the excess Gibbs energy for a hypothetical anhydrous liquid mixture of solutes. Such data are not always available. However, g~ has a clear physical significance (as opposed to empirical parameters required by conventional models). This raises the possibility of calculating it either from a molecular theory or, as it has been shown in this paper, from available independent experimental data.

In practice, it may prove useful to solve the reverse problem of using the data for joint solubility in water to determine an expression for the excess Gibbs energy of the corresponding anhydrous solute mixture. This procedure may be particularly useful when a specific application requires information regarding solute-solute interactions.

1.0.

0 . 9 .

0 . 8 .

0.7.

0 0.6.

_~ 0 . 5 .

~ 0.4.

0.3.

0.2.

0.1.

0.0

molality of the electrolyte, m 3

Fig. 6. Solubility of carbon dioxide in water containing electrolytes: (1) NaC1 (m), (2) CaCl 2 (O); the pressure is 47.3 atm, the temperature is 25°C. ( I ) shows experimental data (Malinin and Kyrovskaya, 1975); ( ) gives the

results of our calculations.

CONCLUSIONS

We have developed a new method for the calculation of the thermodynamic properties of multicomponent aqueous electrolyte and non-electrolyte solutions. To illustrate the usefulness of the method, we have presented results of calculations for the joint solubility of a variety of electrolytes and non-electro-

Acknowledgement~The authors are grateful to the Nation- al Science Foundation for financial support. JJdP is grateful to the Camille and Henry Dreyfus Foundation for a New Faculty Award.

R E F E R E N C E S

Angus, S., Armstrong, B. and de Reuck, K., 1976, Interna- tional Thermodynamic Tables of the Fluid State Carbon Dioxide. IUPAC Vol. 3. Pergamon Press, Oxford.

Barta, L. and Bradley, D., 1985, Extension of the specific interaction model to include gas solubilities in high temperature brines. Geokhim. Cosmochim. Acta 49, 195-210.

Chen, C.-C., Britt, H., Boston, J. and Evans, L., 1982, Local composition model for excess Gibbs energy of electrolyte systems. A.I.Ch.E. J. 28, 588-596.

Corti, H., de Pablo, J. and Prausnitz, J., 1990, Phase equilibria for aqueous systems containing salts and carbon dioxide. Application of Pitzer's theory for electrolyte solutions. J. phys. Chem. 94, 7876-7880.

Edwards, T., Maurer, J., Newman, J. and Prausnitz, J., 1978, Vapour-liquid equilibria in multicomponent aqueous solutions of volatile weak electrolytes. A.I.Ch.E.J. 24, 966-976.

Koiker, A. and de Pablo, J., 1995, Thermodynamic modeling of concentrated binary aqueous electrolyte and non-electrolyte solutions. A.I.Ch.E.J. (in press).

Linke, W. and Seidell A., 1965, Solubilities of Inorganic and Metal Organic Compounds, 4th Edition, Vols 1 and 2. Washington, DC.

Lumsden, J., 1966, Thermodynamics of Molten Salts Mix- tures. Academic Press, London.

Malinin, S. and Kurovskaya, N., 1975, Investigation of CO 2 solubility in solutions of chlorides at elevated temperatures and pressures. Geokhimiya 4, 547-550.

McKay, A., 1953, Activities and activity coefficients in ternary systems. Trans. Faraday Soc. 49, 237-246.


Nakamura, R., Breedreld, G. and Prausnitz, J., 1976 Ther- modynamic properties of gas mixtures containing com- mon polar and nonpolar components. Ind. Enono Chem. Process Des. Dev. 15, 557-567.

Onda, K., Sada, E., Kobayashi, T., Kito, S. and Ito, K., 1970, Salting-out parameters of gas solubility in aqueous salt solutions. J. Chem. Enono Japan 3, 18-24.

Pitzer, K., Ed., 1991, Activity coefficients in Electrolyte Solu- tions, 2nd Edition. CRC Press, Boca Raton.

Pitzer, K. and Simonson, J. M., 1986, Thermodynamics of multicomponent, miscible, ionic systems: theory and equations. J. phys. chem. 90, 3005-3009.

Prausnitz, J., Lichtenthaler, R. and de Azevedo, E., 1986, Molecular Thermodynamics o f Fluid Phase Equilibria, 2nd Edition. Prentice-Hall, Englewood Cliffs, NJ.

Rcnon, H. and Prausnitz, J., 1968, Local compositions in thermodynamic excess functions for liquid mixtures. A.I.Ch.E.J. 14, 135-144.

Robinson, R. and Stokes, R., 1959, Electrolyte Solutions, 2nd Edition. Butterworths, London.

Vukalovich, M. and Altunin, V., 1968, Thermophysical Prop- erties of Carbon Dioxide. Collet's Publisher Ltd, London.

Wagman, D., Evans, W., Parker, V., Schumm, R. and Nuttal, R., 1981, Selected values of chemical thermodynamic properties. NBS Technical Note 270-3, 270-4, 270-8.

Wang, J., Liu, W., Bai, T. and Lu, J., 1993, Standard Gibbs energies of transfer of some electrolytes from water to aqueous sucrose solutions at 298.15 K. J. chem. Soc. Fara- day Trans. 89, 1741-1744.

APPENDIX A

Here we give equations for the activity coefficients of a binary system in the framework of our approach (I). A detailed description of their use has already been presented (I) and need not be given here again.

These equations are:

lny 1 = lny~ + lnylD n

lny2 = lnT~ + lny2Dn

where

R T l n y f = - [z12(A12 + A'12xlx2)/(xlA12 + x2) 2

+ z21(A~l + x2A'21)/(x2A21 -{- X 1 ) 2 ] X 2 2 (hl)

RTlnT~ = _ [r~2(A~2 - A~2x~)/(xlA~2 + x2) 2

~- "t '21(A21 - - X l X 2 A I 2 1 ) / ( x 2 A 2 1 -{- x 1 ) 2 J x 2

1959

and where ln~'lDn and 1n72o H are the Debye-Huckel contri- butions to the activity coefficient of water and that of the electrolyte, respectively. In these equations

A21 = e x p ( z 2 1 / ~ 2 1 R T ) , z21 = 021 - 011 , ~21 = n21 -~ n l l

Ax2 = e x p ( z 1 2 / o q 2 R T ) , r 1 2 = 012 - - 0 2 2 , ~12 = h i 2 + / 1 2 2 (A2)

The 0is are Gibbs free energies of interaction between species of type i and a central species of type j. The numbers nii are the average numbers of nearest neighbors of type i which interact with a central species of type j (i, j = 1, 2). These equations differ from the commonly used ones (Prausnitz et al., 1986) in three respects: (i) 012 :~ 021, (ii) a12 ~ ct21 and (iii) a~j are not regarded as adjustable parameters because they can be expressed in terms of n o (and the latter can be calculated from molecular information).

To calculate g2x, 012, 011, and n°2 in terms of 022, n°l, and n2°2 we use the following set of four equations:

z21exp(z21/n°lRT) + z12 = 022

ZleeXp(z21/n°2RT) + z21 = gH (A3)

0 o n12/nll = exp [(g12/n°2 -- Oa d n ° l ) / g T ]

n°l[exp(z21/n°lRT ) - 2] + n°2[exp( - z~Jn°egT) - 2] = 0

where

022 = ( AGo - AG~m) - R T lnml + AH0(1 - T/To)

- Acv[T0 - T(1 + In To~T)]

+ ~AHk(1 - T/Tk) + 2RTAx/pln(1 + p i l l2 ) (A4) k

Here AG~, is thc Gibbs free energy of formation of the salt at infinite dilution in the molal scale at the temperature of the system, AG O is the Gibbs free energy of formation of the pure crystal salt at the temperature of the system, m, is the molality of pure water ( = 55.51), AHo and T o are the salt's cnthalpy and temperature of fusion and Acp is the change of heat capacity upon fusion. The term ~kAHk(1 -- T/Tk) has to be added if k phase transitions occur in the crystal structure of the salt, each with cnthalpy change AHk at temperature T k. Parameters Ax and p in the expression for the Debyc-Huckel contribution have been given by Pitzer and Simonson (1986). The value of n°2 is the average number of molecules of type 1 interacting with a molecule of type 2 at infinitc dilution (i.e. it is considered to be a hydration number), and the value of n°x is the hydration number for pure water (n°l ~ 4.6 at 25°C).

thermodynamic modeling of concentrated multicomponent aqueous electrolyte and non-electrolyte...

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