An Equation of State for Electrolyte Solutions Covering Wide Ranges

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Ind. Eng. Chem. Res. 2002, 41, 3282-3297

An Equation of State for Electrolyte Solutions Covering Wide Ranges of Temperature, Pressure, and Composition Jason A. Myers and Stanley I. Sandler* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

Robert H. Wood Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19716

An equation of state has been developed for electrolyte solutions over wide ranges of temperature, pressure, and composition. The equation is in terms of the Helmholtz free energy and contains three terms accounting for the various interactions in solution: a Peng-Robinson term for the interactions between uncharged species, a Born term for the charging free energy of ions, and a mean spherical approximation term for the electrostatic interactions between ions. The equation of state was fitted to experimental data for 138 aqueous electrolyte solutions at 25 °C and 1 bar and to data for aqueous NaCl, NaBr, CaCl2, Li2SO4, Na2SO4, K2SO4, and Cs2SO4 solutions from 0 to 300 °C and from 1 to 120 bar. The equation does well in correlating activity coefficients, osmotic coefficients, densities, and free energies of hydration over the entire range of conditions. The simplicity and wide range of applicability of the new equation of state make it particularly well suited for use in engineering applications. Introduction Electrolyte solutions are encountered in a wide variety of industrial processes such as wastewater treatment, extraction, seawater desalinization, and distillation and also in geological processes. A good understanding of the thermodynamics of electrolyte solutions is necessary when working with such systems. For example, process development, process optimization, process control, and the interpretation of some geological phenomena involving dissolved salts all require knowledge of thermodynamic properties. Much experimental work has been done on measuring the thermodynamic properties of various electrolyte systems, but for engineering purposes, a good model capable of describing the behavior of these systems is desired. Electrolyte solutions are considerably more difficult to model than most other solutions encountered in engineering applications. The reason is that there are a number of additional factors to consider when dealing with solutions of dissolved electrolytes that exist as ions, and these ions can interact both with each other and with the solvent. Because ions are charged particles, there are electrostatic interactions that are different from the dispersion and hard-core interactions taken into account by models such as the Peng-Robinson and van der Waals equations. At low electrolyte concentrations, only long-range electrostatic forces are significant, while at high electrolyte concentrations, short-range attractive and repulsive forces become important as well. Another difficulty in modeling electrolyte solutions is that the chemistry of the solution changes at different conditions. At low temperatures and high densities, strong electrolytes are fully ionized. At high temperatures and low densities, the ions tend to associate, * To whom correspondence should be addressed. E-mail: [email protected]. Phone: (302) 831-2945. Fax: (302) 8314466.

forming weakly bonded ion pairs that behave not as ions but as dipoles dissolved in the solvent. At moderate temperatures and densities, a mixture of ion pairs and ions will exist. An accurate engineering model for electrolyte solutions must take all of these interactions into account. Liu and Watanasiri1 give an excellent overview of the main issues encountered when modeling electrolyte systems. Many different models have been proposed for describing the thermodynamic properties of electrolyte solutions. Chen et al.2 developed a local composition NRTL activity coefficient model for single, completely dissociated electrolytes in a single solvent. Their model is in terms of the excess Gibbs free energy and is only useful in predicting activity coefficients. Chen and Evans3 and Mock et al.4 extended the local composition model to mixed-electrolyte and mixed-solvent systems. Chen et al.5 also extended the local composition model to partially dissociated salts. Pitzer et al.6 derived a semiempirical equation of state for aqueous NaCl solutions that is valid from 0 to 300 °C but is only useful for fully ionized NaCl solutions. Anderko and Pitzer7 developed a theoretical equation of state for fully ionpaired NaCl solutions at temperatures above 300 °C. However, this model cannot be used at lower temperatures where NaCl is fully ionized. Lvov and Wood8 developed a volumetric equation of state for aqueous NaCl solutions over a wide range of temperature, pressure, and concentration that is only useful for calculating densities. Several fundamental equations of state for electrolyte solutions have been published, the most notable being the equations of Jin and Donohue,9,10 Fu¨rst and Renon,11 and Wu and Prausnitz.12 The aim of this work is to develop an equation of state that can predict the thermodynamic properties of fully ionized electrolyte solutions over wide ranges of temperature, pressure, and composition. This work differs from most other studies in that we do not focus on a

10.1021/ie011016g CCC: $22.00 © 2002 American Chemical Society Published on Web 05/23/2002

Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3283

Figure 1. Path to the formation of an electrolyte solution at constant temperature and volume. This path is used for both the vapor and liquid phases.

specific range of conditions or type of electrolyte; we wish to have an equation of state that is as general as possible. However, we also wanted a degree of simplicity in the equation of state so that it can be used efficiently in engineering applications. The new equation of state is an expression for the Helmholtz free energy as a function of temperature, volume, and number of moles of each component in the system. From this fundamental equation of state, all thermodynamic properties of a system can be calculated from various derivatives. This new equation of state can also be used for mixtures of electrolytes and, though we do not do so here, for mixedsolvent systems, as well as for solutions of electrolytes and nonelectrolytes. Ion pairing can be included by adding the ion pairs as an additional component and using an equilibrium constant for ion pairing. However, in the present paper, we do not consider ion pairing so we limit testing of our model to temperatures at which there are no appreciable numbers of ion pairs. Thermodynamic Model Any realistic thermodynamic model of electrolyte solutions should take into account all of the interactions between the various species in the solution. Furthermore, the model should be applicable to pure fluids, single electrolyte systems, mixtures of electrolytes, and mixed-solvent systems and cover wide ranges of temperature, pressure, and composition. To develop such a model, we first consider a path to the formation of an electrolyte solution from a mixture of ideal gases. Because we want an equation of state in terms of the Helmholtz free energy, both temperature and volume remain constant on the path. Figure 1 shows the path chosen for the formation of an electrolyte solution at a constant temperature and volume. The change in the Helmholtz free energy on forming an ionic solution from an ideal gas mixture is given by the sum of the changes in the Helmholtz free energy for each of the four steps along the path shown in Figure 1.

I. A mixture containing ions and solvent molecules is initially in a hypothetical ideal gas state at a temperature T and volume V. In the ideal gas state there are no interactions between the particles in the mixture. For the first step, the charges on all of the ions are removed. The change in the Helmholtz free energy for discharging the ions, ∆ABorn dis , is calculated from the Born equation for ions in a vacuum. After all of the ions are discharged, the mixture consists of an ideal gas of hypothetical neutral ions and solvent molecules. II. Next, the excluded-volume repulsive forces and short-range attractive dispersion forces between the neutral particles in the mixture are turned on. The Peng-Robinson equation of state is used to calculate the change in the Helmholtz free energy for this transition, ∆APR. III. Next, the ions are recharged, only taking into account the ion-solvent interactions. The contribution to the Helmholtz free energy of recharging the ions, ∆ABorn chg , is calculated from the Born equation for ions at infinite dilution in a dielectric solvent. IV. In the final step, the long-range electrostatic interactions between the ions in the mixture are turned on. These interactions are accounted for by the mean spherical approximation (MSA), and the change in the Helmholtz free energy for this last step is given by ∆AMSA. The total change in the Helmholtz free energy for forming the electrolyte solution on this path is

A(T,V,n b) - AIGM(T,V,n b) ) ∆APR + ∆ABorn + ∆AMSA (1) where Born ∆ABorn ) ∆ABorn dis + ∆Achg

(2)

where T is the temperature of the system, V is the system volume, b n is the vector of the number of moles

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Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

of each component of the mixture, and AIGM is the Helmholtz free energy of an ideal gas mixture. The details of the individual parts of this equation of state will be discussed next. Peng-Robinson Contribution. The short-range interactions between uncharged species in solution are accounted for by the equation of Peng and Robinson.13 These short-range interactions include excluded-volume interactions and attractive (dispersion) interactions. The residual Helmholtz free energy for the Peng-Robinson equation of state is

∆APR(T,V,n b) )

[

]

V + b(1 - x2) Na ln 2x2b V + b(1 + x2)

( )

NRT ln

V-b (3) V

where a is the van der Waals attraction parameter, b is the van der Waals excluded-volume parameter, V is the molar volume, N is the total number of moles, and R is the gas constant. The a and b parameters in eq 3 are specific to the mixture and can be calculated from the pure component a and b parameters with the van der Waals one-fluid mixing rules:

a)

1 N2

∑i ∑j ninj xaiaj(1 - kij)

b)

N

∑i nibi

[

(5)

]

V + c + b(1 - x2) Na ln 2x2b V + c + b(1 + x2) NRT ln

(

1 N

)

V+c-b (6) V

where c is the volume translation parameter for the mixture and is obtained from the volume translation

∑i nici

(7)

The Peng-Robinson equation with volume translation, eq 6, is part of the electrolyte equation of state developed here. In general, the ai, bi, and ci parameters for each species will be temperature-dependent. The binary interaction parameters, kij, may also be temperaturedependent or can be set to a fixed value. The addition of a volume translation parameter to a cubic equation of state greatly improves volumetric predictions. Born Contribution. The free energy required to discharge or charge an ion in solution was first calculated by Born.17 The continuum hydration model of Born treats ions as hard spheres of diameter σ in a continuum of uniform dielectric constant. The change in the Helmholtz free energy on discharging or charging ions in a system at constant temperature and volume is equal to the net work required to remove the charges from the ions (discharging) or replace the charges (charging). The sum of the changes in the Helmholtz free energies from the Born model on discharging in a vacuum and charging in a dielectric solvent is

∆A

where ai and bi are the pure-component attraction and excluded-volume parameters, respectively, kij is a binary interaction parameter, ni is the number of moles of species i, and the summations are over all neutral species (solvent molecules and uncharged ions). The binary interaction parameter, which accounts for the short-range interactions between species i and j, is symmetric (kij ) kji) and is equal to zero for i ) j. While the vapor-liquid equilibria of nonpolar systems are predicted quite well with cubic equations of state such as that of Peng and Robinson, volumetric predictions are usually not as accurate. Many attempts have been made to improve the volumetric predictions of cubic equations of state. Peneloux et al.14 have proposed a volume translation for the Redlich-Kwong-Soave equation of state that is used to shift the volume predictions by a constant value along an isotherm. Several others, including Tsai and Chen15 and de Sant’Ana et al.,16 have extended the concept of volume translation to the Peng-Robinson equation of state. The residual Helmholtz free energy for the volume-translated Peng-Robinson equation of state is given by

b) ) ∆APR(T,V,n

c)

(4)

and

1

parameter for each component ci with the following mixing rule:

Born

(T,V,n b) ) -

( )∑

Nae2 4π0

1-

1

ions

niZi2 σi

(8)

where Na is Avogadro’s number, e is the unit of elementary charge, 0 is the permittivity of free space, is the dielectric constant, and Zi is the charge number of ion i. The Born equation provides a means of calculating solvation free energies of ions in water provided that the correct ionic diameters are used. The diameters to be used in eq 8 are not the bare ion diameters but the diameters of the cavities in the solvent in which the ions reside.18,19 Empirically, the cavity diameters of anions are approximately 0.1 Å greater than the bare ion diameters, while the cavity diameters of cations are approximately 0.85 Å larger than the bare ion diameters.18 Because the Born equation is linear in the number of moles of ions, it has no effect on the ion activity coefficients, though this term is very important in calculating free energies of hydration of ions. The Born equation does have an effect on the solution density though through the density dependence of the dielectric constant. MSA Contribution. Many models have been proposed to describe the interactions between charged particles in solution. It is the long-range electrostatic interactions in electrolyte solutions that present the most difficulty. Several statistical mechanical theories such as perturbation theory and integral equation theory have been applied to electrolyte solutions. There are several different approaches to the integral equation theory of electrolyte solutions; the two most popular are the hypernetted chain (HNC) theory and the MSA. Each of these is derived from the solution of the OrnsteinZernike equation with different closures. The HNC approach has proven to be the most accurate theory for electrolyte solutions and accurately predicts the structure of electrolyte solutions as well as their thermodynamic properties. However, the HNC approach involves


numerically solving a complex system of highly nonlinear equations and therefore has limited use in engineering applications. The MSA approach has been extensively and successfully applied to electrolyte solutions and yields analytical solutions for the thermodynamic and structural properties of such systems. Although the MSA theory is not as accurate as the HNC theory, it appealing because of its relative simplicity. The MSA model accounts for the long-range chargecharge interactions between ions in an electrolyte solution and is based on the primitive model of electrolyte solutions in which ions are treated as charged hard spheres in a medium of uniform dielectric constant. The MSA model for electrolyte solutions was first solved for the restricted primitive model by Waisman and Lebowitz,20 who obtained an analytical solution using the perturbation theory. Later, Blum21 and Blum and Hoye22 solved the MSA for the primitive model and obtained analytical expressions for excess thermodynamic properties (relative to a mixture of hard spheres) including the Helmholtz free energy, activity coefficients, chemical potentials, and osmotic coefficients. The original solution of the MSA model is a complicated set of equations that must be solved iteratively to obtain useful information. However, Harvey et al.23 have derived an explicit approximation to the MSA that has been shown to be simple and accurate. The approximation assumes that all of the ions in the system have the same diameter given by the following average diameter:

∑ niσi

σ)

ions

∑

(9) ni

ions

where the summations are over all ionic species. The MSA contribution to the Helmholtz free energy of the system is

b) ) ∆AMSA(T,V,n

2Γ3RTV 3 1 + σΓ 3πNa 2

(

)

(10)

where Γ is the MSA screening parameter and κ is the Debye screening length. These quantities are

1 [x1 + 2σκ - 1] 2σ

Γ) and

κ)

(

e2Na2

)

(11)

1/2

∑ niZi

0RTV ions

2

(12)

The MSA equation reduces to the Debye-Hu¨ckel limiting law as σ approaches zero; however, the MSA equation gives much more accurate predictions of the thermodynamic properties at high electrolyte concentrations. Normally, the MSA calculation also includes a hard-sphere repulsion term, but in the present equation, this has already been included in the Peng-Robinson term. The MSA is formulated in the McMillan-Mayer framework in which the independent variables are temperature, volume, solvent chemical potential, and solute mole numbers. Our equation of state is in terms

of temperature, volume, and the number of moles of each component, and most experimental data are reported in terms of temperature, pressure, and the number of moles of each component. Thus, the MSA is incompatible with both the equation of state and the experimental data in the literature. To calculate osmotic and activity coefficients from the equation of state, the McMillan-Mayer framework must be transformed to the Lewis-Randall framework. Haynes and Newman24 have shown that the conversion between the solute activity coefficient in the two different frameworks is MM ln γLR i ) ln γi

ViθPEX RT

(13)

where V h θi is the partial molar volume of ion i and PEX is the excess pressure in the McMillan-Mayer framework (i.e., the pressure from the MSA equation). They also show that contribution of the excess pressure term to the activity coefficient of a 1:1 electrolyte in water at 25 °C is very small. The magnitude of this term increases with increasing ion diameter and concentration, but for most electrolyte solutions, it is less than 3% of the total activity coefficient. Therefore, we neglect the difference between the McMillan-Mayer and LewisRandall frameworks when using the MSA in our equation of state, especially because our equation contains several empirical parameters which will compensate for this small difference. Thermodynamic Properties Because the equation of state above is in terms of the Helmholtz free energy, any thermodynamic property of a system can be calculated from its various temperature, volume, and mole number derivatives. In general, thermodynamic properties are usually calculated as functions of the system temperature and pressure, though temperature and volume are the independent variables in the Helmholtz free energy. Therefore, all calculations with the equation of state must be performed as a function of temperature and volume, while most experimental data are measured and reported as functions of temperature and pressure. To be able to fit the equation of state to experimental data, one needs to be able to compute various thermodynamic properties from the Helmholtz free energy equation of state as functions of temperature and pressure. This is accomplished by first calculating the volume of the system at the specified temperature and pressure from the equation of state and then using this volume in the calculations of other thermodynamic properties. Of primary importance to engineers is phase equilibria, for which the fugacity coefficient of each species in a mixture is the important quantity. Although the fugacity coefficient is usually expressed as a function of temperature and pressure,25 here it is necessary to have an expression for this quantity in terms of temperature and volume. This expression can be derived by first calculating the difference between the ideal gas chemical potential as a function of the system temperature and pressure and the ideal gas chemical potential as a function of the system temperature and volume and then substituting this result into the original equation for the fugacity coefficient. The difference in chemical potentials and the resulting equation for the fugacity

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coefficient as a function of temperature and volume are given below:

GiIGM(T,P,n b) - GiIGM(T,V,n b) ) RT ln Z

(14)

and

φi(T,V,n b) )

{

}

Gi(T,V,n b) - GiIGM(T,V,n b) 1 exp Z RT

(15)

where Z ) PV/NRT is the compressibility factor and b n indicates the vector of species mole numbers. When performing phase equilibrium calculations for electrolyte systems, the standard state of each component in the system must carefully be considered. Often the standard state for a liquid is the pure liquid at the temperature of the system and 1 bar. However electrolytes, especially salts, usually do not exist as pure liquids, and therefore, a standard state based on the pure substance as a liquid is unrealistic. For dilute species including electrolytes, the standard state can be taken to be a hypothetical ideal 1 m solution. With this, the activity coefficient of a solvent species in an electrolyte solution is

b) ) γi(T,P,n

φi (T,P,n b)

(16)

φipure(T,P)

where φ h ipure(T,P) is the fugacity coefficient of pure species i at the temperature and pressure of the system. The molality-based activity coefficient for a solute species is then calculated as

b) ) γim(T,P,n

(

)[

φi(T,P,n b) 1 1 + νmMsolvent φ (T,P,n )0) i i

]

(17)

where ν is the total number of molecules upon complete dissociation of one molecule of salt, m is the molality, Msolvent is the molecular mass of the solvent in kg/mol, and φ h i(T,P,ni)0) is the fugacity coefficient of species i at infinite dilution. Because activity coefficients of the individual ionic species in an electrolyte solution cannot be directly measured, an average activity coefficient for the dissolved salt is defined. For a solution consisting of a single salt, the mean ionic activity coefficient is defined as m ν+ m ν- 1/ν γm ( ) [(γ+ ) + (γ- ) ]

(18)

where ν+ and ν- are the number of cations and anions, respectively, in one molecule of the salt. In terms of eq 17, this becomes

b) ) γm ( (T,P,n

[

(

) ][

1 × 1 + νmMsolvent

φ+(T,P,n b)

φ+ (T,P,n+,-)0)

ν+/ν

φ-(T,P,n b)

]

φ-(T,P,n+,-)0)

∆m h Gi(T,P) ) RT ln φi(T,P,ni)0) +

( )

RT ln(Msolventmθ) + RT ln

P (21) Pθ

where the second term is the conversion from the mole fraction to the molality standard state (mθ ) 1 mol/kg) and the third term is a correction for the difference between the system pressure and the standard state pressure (Pθ ) 1 bar). Application of the Equation of State to Aqueous Electrolyte Solutions Pure Water. Because we are focusing on aqueous electrolyte solutions, the equation of state should accurately predict the properties of pure water in the limit of an infinitely diluted salt solution. The parameters for pure water were fitted to volumetric and vapor-pressure data generated from the equation of state of the International Association for the Properties of Water and Steam.26 This 56-parameter equation of state predicts the thermodynamic properties of water with extremely high accuracy. The data used in the regression included vapor pressures from 10 °C to the critical temperature (374.15 °C) and liquid densities from 10 °C to the critical temperature and pressures up to 250 bar. For pure water, there are no ionic interactions; thus, we only need to consider the attractive dispersion and repulsive interactions between the molecules. Because the Born and MSA terms of the equation of state are zero for a system without ions, the properties of pure water are only dependent on the Peng-Robinson term. There are three parameters in the volume-translated Peng-Robinson equation for each neutral species: the attraction parameter, a, the excluded-volume parameter, b, and the volume translation parameter, c. All three parameters were assumed to be temperaturedependent to give a good fit over the entire temperature and pressure range. A Levenberg-Marquardt nonlinear least-squares routine with the aforementioned data gave the following set of best-fit parameters:

aH2O ) 1.26944 - 0.89381Tr + 0.16937Tr2 Pa‚m6/mol (22) bH2O ) 15.6345 + 6.14518Tr - 5.2795Tr2 cm3/mol (23)

ν-/ν

(19)

The osmotic coefficient is

ln[xsolventγsolvent(T,P,n b)] Φ(T,P,n b) ) νmMsolvent

Another property that is important in electrolyte solutions is the free energy of hydration. This quantity is defined as the Gibbs free energy required to transfer an ion from an ideal gas at the temperature of interest and 1 bar to an ideal 1 m solution at the temperature and pressure of interest. The free energy of hydration is a key property in solvation thermodynamics and is required when calculating salt dissociation and ion-pair formation equilibrium constants from an equation of state. The free energy of hydration is calculated from the equation of state as follows:

(20)

cH2O ) -2.7227 + 11.4201Tr - 6.0157Tr2 cm3/mol (24) where Tr ) T/647.29 K. With this set of parameters for pure water, the equation of state accurately predicts the vapor pressure and liquid densities of water over a wide range of


temperature and pressure resulting in average relative deviations of only 0.22% and 0.83%, respectively. These deviations are sufficiently small to ensure that the equation of state can accurately predict the behavior of dilute aqueous electrolyte solutions. The largest deviations between the equation of state predictions and the experimental data occur near the critical temperature of water where most cubic equations of state fail to accurately predict thermodynamic properties. The use of a temperature-dependent volume translation parameter in the Peng-Robinson equation of state minimizes the errors in densities near the critical point. Several models have been proposed for calculating the dielectric constant of pure water needed for the Born and MSA terms. Uematsu and Franck27 have developed a 10-parameter model that is accurate over a temperature range of 273-1273 K and pressures up to 1600 MPa. Fernandez et al.28 have developed a more accurate 12-parameter model that gives the dielectric constant of pure water as a function of temperature and density over a temperature range of 238-1273 K and pressures up to 1200 MPa. We used the Uematsu and Franck model because it is simpler and has fewer parameters, and calculations required about half as much time as those with the model of Fernandez et al. Aqueous Electrolyte Solutions at 25 °C and 1 bar. Both the Born and MSA contributions to the equation of state depend on the dielectric constant of the solvent of the electrolyte solution. It is known that ions in solution polarize the surrounding solvent molecules, thus decreasing the dielectric constant;29 however, it is difficult to accurately model this effect with a simple analytical equation. Therefore, the dielectric constant of the pure solvent, water, has been used for all calculations here. Although aqueous electrolyte solutions are by far the most common types of electrolyte solutions encountered in natural and industrial processes, solutions with other types of solvents do exist. Harvey and Prausnitz30 have developed an expression for estimating the dielectric constant of fluid mixtures that is useful when dealing with nonaqueous and mixed solvent systems. The Uematsu and Franck equation for the dielectric constant of pure water is a function of temperature and water density. Thus, it is necessary to first obtain a value for the density of pure water to be used in this equation before other equation of state calculations are performed. One way to do this is to calculate the density of water from the Peng-Robinson equation because the properties of pure water are dependent only on this term. Unfortunately, if this method is used, it is impossible to express the equation of state exclusively in terms of the variables T, V, and n because the dielectric constant will be a function of the solution to the cubic Peng-Robinson equation for water. Consequently, it will be impossible to obtain analytical derivatives of the Helmholtz free energy with respect to temperature, volume, and mole numbers. Another drawback of using this method to calculate the density of water is that this calculation is rather time-consuming. An alternate method is to approximate the density of water with a simple expression in terms of the variables of the equation of state: T, V, and n. We found that the approximation that works best with our equation is

FH2O ) nH2OMH2O/V

(25)

where nH2O is the number of moles of water in the solution and MH2O is the molecular mass of water in kg/mol. In a solution of a single electrolyte dissolved in water, there are three distinct species: cations, anions, and water molecules. At high temperatures and high salt concentrations, individual ions may associate to form ion pairs. In this work, we focused on a range of temperature, pressure, and concentration in which ion pairing can be neglected. That is, we assumed that the dissolved salt is fully ionized in each system we investigated. We first focused on aqueous electrolyte solutions at 25 °C and 1 bar. The equation of state proposed here has five parameters for each ionic species: the attraction and excludedvolume parameters, a and b, the ion-water binary interaction parameter, kiw, the volume translation parameter, c, and the ion diameter, σ, in the Born and MSA terms. Because experimental activity coefficients for electrolyte solutions are specific to salts rather than the constituent ions, the effects of the individual ions cannot be separated when determining the equation of state parameters for a specific salt solution. Therefore, for each salt species, the parameters for the cations and anions are set equal to a single value. Because we did not include density data in the regressions at 25 °C, we set the volume translation parameter for each salt, c, equal to zero. Furthermore, we found that the ionwater binary interaction parameter, kiw, did not have a significant effect on the equation of state correlations for most salt solutions at 25 °C. Therefore, we set this parameter equal to zero for all salts. Consequently, there are only three adjustable parameters (a, b, and σ) for each salt used to fit to the 25 °C and 1 bar data. We first compared the results from our model to the NRTL model of Chen et al.5 That model has two adjustable parameters for each salt, and because it is an excess free energy model, it cannot be used to predict solution densities. In general, each ion has a number of water of hydration molecules associated with it. Chen et al. expanded the NRTL model to account for the hydration effects that occur in electrolyte solutions by adding a third hydration parameter for each ion to improve the fit of the data. We used mean ionic activity coefficients for 138 salt solutions tabulated by Robinson and Stokes31 in separate data regressions to determine the optimum equation of state parameters for each electrolyte. This set includes the 116 salts used by Chen et al. to determine the NRTL parameters and 22 other salts. The results for each of the 138 salts are presented in Table 1 and are compared to the results from the NRTL model with hydration. The maximum molality (up to 6 m), for which activity coefficients were calculated for each salt, is also reported. Each of the three adjustable parameters for each salt is well determined with a value much larger than the range of its 95% confidence limit. The only apparent trend in the parameters was that salts with large (small) values of a also had large (small) values of b and σ. A correlation between b and σ is to be expected because each parameter is a measure of ion size. As mentioned earlier, we have used salt-specific parameters instead of ion-specific parameters because it is not possible to separate the effects of each ion when analyzing experimental activity coefficient data. Each parameter for a salt should be dependent on both the cation

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Table 1. % AADs in Mean Ionic Activity Coefficients at 25 °C electrolyte

a (Pa‚m6/mol)

b (cm3/mol)

AgNO3 CsAc CsBr CsCl CsI CsNO3 CsOH HBr HCl HClO4 HI HNO3 KAc KBrO3 KCl KClO3 KCNS KF KH adipate KH malonate KH succinate KTol KH2PO4 KI KNO3 KOH KBr LiAc LiBr LiCl LiClO4 LiI LiNO3 LiOH LiTol NaAc NaBr NaBrO3 NaCl NaClO3 NaClO4 NaCNS NaF Na formate NaH malonate Na propionate NaH succinate NaTol NaI NaNO3 NaOH NH4Cl NH4NO3 RbAc RbBr RbCl RbI RbNO3 TlAc

0.0384 12.0035 0.2267 0.2037 0.0885 0.7412 13.2115 18.5145 10.8072 18.8935 19.6013 4.9972 9.6989 2.5265 0.7332 0.2391 0.3709 1.9845 2.9218 0.3529 0.7082 0.6069 0.8458 1.3641 0.0790 4.9778 0.7522 3.1638 10.5086 8.0018 12.9217 20.7897 5.6595 0.2401 1.4791 7.1602 3.2858 0.6920 2.4611 0.7813 0.9612 4.0473 1.1814 1.7453 0.2392 11.4792 0.6232 0.2775 5.4462 0.1189 3.8268 0.5300 0.0000059 9.2959 0.3066 0.4340 0.3356 0.0346 0.0209

12.9306 68.1606 13.1350 11.2541 8.6376 29.5030 68.0463 82.1459 64.6144 92.2856 87.6413 46.0054 61.0982 18.8586 19.6245 19.2805 10.5410 31.2904 41.0558 18.1751 22.8839 24.6390 30.9256 21.9101 14.4821 43.4108 16.8917 34.2560 68.1208 58.9605 70.9185 90.2289 46.3514 8.1861 29.1960 52.3677 38.3654 24.0680 34.6464 22.2293 20.7826 38.9232 27.9280 25.0364 12.9744 66.6921 19.7096 16.4907 45.4724 10.5855 43.7018 12.9909 2.9832 59.9059 12.1865 15.0284 13.3635 12.5468 1.3019

Cs2SO4 K2SO4 K2CrO4 Li2SO4 Na2CrO4 Na2SO4 Na2S2O3 (NH4)2SO4 Rb2SO4

0.8660 0.4036 0.6387 1.1478 1.8841 0.4962 1.0029 0.0684 0.8289

28.3819 22.6983 24.8318 29.1075 38.5592 24.7164 30.2521 12.1388 28.8106

BaBr2 BaCl2 Ba(ClO4)2 BaI2 Ba(NO3)2 BaAc2

5.0044 2.6572 5.4075 11.1626 0.4079 0.3790

47.6234 35.3305 45.7915 67.4211 25.4403 10.4986

σ (Å) 1-1 Electrolytes 2.548 5.827 2.728 2.681 2.933 2.894 5.041 4.791 4.686 5.514 4.577 4.764 5.498 4.348 3.925 4.276 4.157 3.800 4.399 3.798 3.699 5.791 2.889 4.551 2.797 3.682 4.004 4.472 4.401 4.443 4.573 5.347 4.855 2.010 4.920 5.264 4.278 3.845 4.322 4.436 4.426 5.258 3.961 4.746 3.698 5.445 3.626 5.042 4.495 3.744 4.297 4.041 2.926 5.079 3.485 3.450 3.358 2.441 2.825 1-2 Electrolytes 4.290 3.493 4.288 4.408 5.013 4.038 4.389 3.108 4.249 2-1 Electrolytes 5.059 4.993 5.358 5.219 3.592 5.432

% AAD NRTL modela this model

max molality (mol/kg)

0.77 0.79 0.52 0.52 0.52 0.24 b 0.75 0.53 1.19 0.62 0.80 0.66 0.14 0.23 0.24 0.13 0.36 0.12 0.36 0.21 2.47 0.22 0.38 0.58 0.48 b 0.22 0.95 0.88 1.32 1.78 0.35 3.00 0.97 0.79 0.36 0.10 0.44 0.30 0.46 1.22 0.02 0.71 0.36 1.01 0.35 0.99 0.57 0.12 0.63 0.06 0.93 0.70 0.16 0.15 0.18 0.85 0.92

0.20 0.65 0.31 0.28 0.17 0.15 0.46 0.09 0.63 0.20 2.61 0.06 0.61 0.08 0.08 0.08 0.13 0.08 0.09 0.52 0.22 0.35 0.08 0.20 0.13 0.60 0.09 0.30 0.37 0.25 4.26 1.29 1.55 1.01 0.58 0.61 0.09 0.08 0.08 0.09 0.07 0.36 0.02 0.31 0.26 0.76 0.20 0.44 0.19 0.11 0.67 0.29 0.16 0.83 0.07 0.07 0.12 0.19 0.23

6 3.5 5 6 3 1.4 0.9 3 6 6 3 3 3.5 0.5 4.5 0.7 5 4 1 5 4.5 3.5 1.8 4.5 3.5 6 5.5 4 6 6 4 3 6 4 4.5 3.5 4 2.5 6 3.5 6 4 1 3.5 5 3 5 4 3.5 6 6 6 6 3.5 5 5 5 4.5 6

0.47 b 1.37 0.81 2.23 0.26 0.89 0.92 0.47

0.22 0.28 0.25 0.27 0.41 0.16 0.17 0.25 0.15

1.8 0.7 3.5 3 4 4 3.5 4 1.8

1.63 1.33 3.22 2.02 0.07 b

0.16 0.17 4.75 0.30 0.11 4.21

2 1.8 5 2 0.4 3.5

Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3289 Table 1. (Continued) electrolyte

a (Pa‚m6/mol)

b (cm3/mol)

CaBr2 CaCl2 Ca(ClO4)2 CaI2 Ca(NO3)2 Cd(NO3)2 CdCl2 CdBr2 CdI2 CoBr2 CoCl2 CoI2 Co(NO3)2 CuCl2 Cu(NO3)2 FeCl2 MgAc2 MgBr2 MgCl2 MgI2 Mg(ClO4)2 Mg(NO3)2 MnCl2 NiCl2 Pb(ClO4)2 Pb(NO3)2 SrBr2 SrCl2 Sr(ClO4)2 SrI2 Sr(NO3)2 UO2Cl2 UO2(ClO4)2 UO2(NO3)2 ZnBr2 ZnCl2 ZnI2 Zn(ClO4)2 Zn(NO3)2

15.8804 7.2623 18.3813 17.7616 1.1068 3.4414 0.0000020 0.0000010 0.0000144 14.9928 7.3404 6.9741 5.8125 2.0046 5.8125 7.0986 1.1304 15.3623 11.5149 39.6775 48.0065 7.3415 4.5477 7.9816 7.3738 0.0542 9.8770 6.1191 13.9512 16.5394 1.7576 11.1507 162.6549 11.5511 4.0927 2.4795 2.9942 31.4520 7.7135

89.4555 60.1464 85.2520 90.1023 20.1824 35.8613 18.2814 25.1028 43.0445 76.5799 53.2723 52.1105 47.1703 26.9883 47.1703 58.7403 20.0419 83.9197 75.6245 116.7610 129.1700 53.1808 41.6882 55.2474 53.1612 15.5291 70.2552 55.8518 73.7878 88.1006 18.3618 65.7514 268.2230 66.9808 39.1711 40.5478 33.1947 120.4010 54.3961

BeSO4 MgSO4 MnSO4 NiSO4 CuSO4 ZnSO4 CdSO4 UO2SO4

0.9095 0.6670 0.6374 4.0658 0.4099 0.8459 0.4461 0.6317

27.4868 25.4272 28.6029 27.6524 13.7100 32.8818 24.5582 15.3058

AlCl3 CeCl3 CrCl3 Cr(NO3)3 EuCl3 LaCl3 NdCl3 PrCl3 ScCl3 SmCl3 YCl3

17.9117 6.9281 10.6489 7.3251 8.3589 6.8821 7.0823 6.9115 9.2870 7.6646 8.7553

97.3688 60.7677 72.6823 58.3339 66.7114 60.3325 61.5979 60.9113 68.2657 63.7411 68.3844

K3Fe(CN)6

0.8151

26.8172

Al2(SO4)3 Cr2(SO4)3

1.5724 3.6241

37.6690 58.0423

Th(NO3)4 ThCl4

2.9651 19.7484

33.2948 99.3895

3.4615

24.1887

K4Fe(CN)6 a

% AAD NRTL modela this model

σ (Å) 2-1 Electrolytes 5.906 4.990 5.889 5.723 4.708 5.538 1.569 1.231 0.902 5.658 5.219 3.208 4.879 6.600 5.051 5.034 3.906 5.283 5.313 6.280 6.824 5.068 5.788 5.152 5.350 2.722 5.382 5.094 6.121 5.835 4.834 6.298 11.143 7.541 9.705 7.479 9.508 6.202 5.572 2-2 Electrolytes 3.185 3.226 3.261 3.214 3.166 3.217 3.177 3.137 3-1 Electrolytes 6.109 5.382 5.580 5.371 5.573 5.432 5.347 5.380 5.339 5.446 5.414 1-3 Electrolyte 4.860 3-2 Electrolytes 4.253 5.531 4-1 Electrolytes 9.012 9.676 1-4 Electrolyte 5.208

max molality (mol/kg)

5.72 3.06 3.67 2.46 1.84 b b b b 2.45 1.94 b 2.39 b 2.62 1.64 1.08 4.16 3.65 b b 2.88 1.90 2.85 2.95 1.45 2.35 2.73 2.76 2.57 2.00 3.45 7.26 b b 7.31 b 4.54 2.94

2.11 0.71 3.90 0.18 0.21 0.52 21.28 26.53 45.56 4.70 2.65 18.92 0.73 4.59 1.71 0.22 0.96 0.27 0.39 0.50 0.48 0.34 6.80 3.89 2.68 0.19 0.16 0.30 6.45 0.23 0.23 2.24 1.69 15.65 4.24 1.18 14.06 0.38 2.15

6 6 6 2 6 2.5 6 4 2.5 5 4 2 5 6 6 2 4 5 5 5 4 5 6 5 6 2 2 4 6 2 4 3 5.5 5.5 6 6 6 2 6

1.83 1.45 1.50 1.26 1.33 1.50 1.94 b

1.86 2.17 1.78 1.76 1.56 2.05 1.53 1.71

4 3.5 4 2.5 1.4 3.5 3.5 6

4.57 3.97 2.69 2.88 4.18 3.96 3.87 3.82 3.46 3.97 4.07

0.61 0.44 0.49 0.21 0.27 0.16 0.22 0.32 0.29 0.23 0.29

1.8 2 1.2 1.4 2 2 2 2 1.8 2 2

b

0.12

1.4

b b

2.81 0.52

1 1.2

b b

7.87 1.73

5 1.6

b

0.28

0.9

b

% AAD is for the NRTL model with hydration. Chen et al. do not report results for these salts.

and anion of the salt, which explains the variation in parameter values for salts with a common cation or

anion. Obtaining parameters specific to each ion would require the simultaneous regression of activity coef-

3290


Figure 2. Activity coefficient of aqueous UO2(ClO4)2 and UO2SO4 at 25 °C and 1 bar.

Figure 3. Activity coefficient of aqueous electrolyte solutions at 25 °C and 1 bar.

ficient data for all 138 salt solutions, which would be very difficult and time-consuming. In general, our equation of state does an excellent job in correlating the activity coefficients of most electrolytes at room temperature. The correlations for the 1-1, 1-2, and 3-1 type salts were especially good. For most of the salt solutions, our model has a lower percentage of average absolute deviation (% AAD) than the NRTL activity coefficient model. It should be remembered though that the NRTL model has two binary parameters for each salt and one hydration number for each ion while our model has three parameters for each salt. With the same number of parameters, our model has the advantage of giving better correlations of activity coefficients. There were a few salts for which the equation of state did not accurately represent the activity coefficients over the entire concentration range of the experimental data. Specifically, the correlations for several of the zinc and cadmium salts had relatively high % AAD values. Because our model does not take into account the partial dissociation of salts or the formation of ion pairs and higher complexes, it is possible that these effects might contribute to the high errors for these salts. Chen et al.5 were able to accurately capture the correct trend in activity coefficients for several of these systems by including partial dissociation in their NRTL model. The activity coefficient errors for the systems for which they included this were less than 2% compared to about 10% without dissociation. We plan to include partial dissociation and ion-pair formation equilibria in future work on our equation of state. The equation of state performs well in correlating the activity coefficients for highly nonideal electrolyte solutions. Figure 2 shows the activity coefficient results from our model for two electrolytes: UO2(ClO4)2 that exhibits large positive deviations from ideality and UO2SO4 that exhibits large negative deviations. Although we only fit the equation of state parameters to data up to 6 m, our model also does well when extrapolated to much higher concentrations. Figure 3 shows the activity coefficient predictions of the equation of state over a wide range of concentrations for several electrolytes. Even up to 15 m, our equation of state can still accurately predict activity coefficients. Several others have reported equation of state predictions of the properties of aqueous electrolyte solutions at 25 °C and 1 bar. Jin and Donohue9,10 have developed a fundamental equation of state based on the perturbedanisotropic-chain theory that contains only one adjust-

able parameter per salt. They report a 6% AAD in activity coefficients and a 2% deviation in specific volumes of 50 aqueous salt solutions. Although our equation of state contains more parameters than theirs, we are able to calculate activity coefficients with a much greater accuracy (0.83% AAD for 116 salt solutions). Furthermore, the Jin and Donohue equation of state is fairly complicated, containing 10 terms, while our simpler equation contains only 3 terms. Fu¨rst and Renon11 have developed a fundamental equation of state with a nonelectrolyte part based on a Redlich-KwongSoave type equation of state, a short-range ionic part, and a long-range electrostatic part based on the MSA. Their model contains a total of four parameters per salt (cation and anion diameters, an interaction parameter between cation and solvent, and an interaction parameter between cation and anion). Furthermore, they developed correlations between three of the parameters, essentially reducing their equation of state to a oneparameter model. They report an average root-meansquare deviation of 1.4% in osmotic coefficients for 87 different aqueous salt solutions. Our model does better with an average deviation of 0.46% for the osmotic coefficients of 116 salt solutions. Neither Jin and Donohue nor Fu¨rst and Renon report results for temperatures and pressures other than 25 °C and 1 bar. One of the advantages of our equation of state is that, as we show below, it has been tested and shown to be accurate at much higher temperatures and pressures. Thermodynamic Properties over a Wide Range of Temperature and Pressure. There have been many previous studies of the thermodynamics of electrolyte solutions over wide ranges of temperature and pressure. For example, Archer32,33 compiled thermodynamic data for aqueous NaCl solutions from many sources and generated a comprehensive and very accurate equation of state valid from 0 to 300 °C specific to NaCl. Archer34 also made a similar study for aqueous NaBr solutions. Holmes et al.35 compiled data for CaCl2 solutions and used a Pitzer ion-interaction model to develop a general equation valid from 0 to 250 °C. Ananthaswamy et al.36 developed a model for the thermodynamic properties of aqueous CaCl2 in the temperature range 0-100 °C. For aqueous solutions of the alkali-metal sulfates, Holmes and Mesmer37 have developed a model valid from 0 to 250 °C. Several other studies of interest have been reported.38-41 However, these models have several disadvantages. They have a large number of parameters, are often difficult to extend to systems other than the one for which they have been

Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3291 Table 2. Sources of Thermodynamic Properties Used in Data Regression electrolyte NaCl NaBr CaCl2

Li2SO4 Na2SO4 K2SO4 Cs2SO4

property

T range (°C)

pressure range (bar)

molality range

ref

γ(, Φ, F ∆m h h G γ(, Φ, F ∆m h h G γ(, Φ F ∆m h h G γ(, Φ γ(, Φ γ(, Φ γ(, Φ

0-300 25 0-325 25 25-250 50-325 25 0-225 0-225 0-225 0-225

1-85.8 1 1-120 1 1-39.8 1-389 1 1-25.6 1-25.6 1-25.6 1-25.6

0-6 0 0-8 0 0-4 0-6.4 0 0-3 0-3 0-2.5 0-6

33 29 34 29 35 41 29 37 37 37 37

developed, and are also relatively complicated. In contrast, the model presented here is relatively simple, has a small number of parameters, and can be used for any type of system and with any number of components. To establish this, we examined the following seven aqueous salt solutions over wide temperature, pressure, and concentration ranges for which comprehensive studies have been previously performed: NaCl, which is a premier example of a 1-1 type electrolyte, NaBr, CaCl2, which is a premier example of a 2-1 type electrolyte, and four alkali-metal sulfates, which are good examples of 1-2 type electrolytes. For NaCl, NaBr, and CaCl2, we used activity coefficients, osmotic coefficients, densities, and free energies of hydration in a nonlinear least-squares regression analysis to determine the set of best-fit equation of state parameters for these salts. For the four alkali-metal sulfates, we only used available activity and osmotic coefficients in the leastsquares analysis. Table 2 lists the sources of data and the range of conditions of each data type used in the regression analysis. As mentioned in the previous section, for each salt species, the parameters for the cations and anions are given a single average value. The van der Waals attraction parameter and the average ion diameter for each electrolyte were given the following simple temperature dependence to improve the fit over a wide temperature range:

a ) a(1) + a(2)/T

(26)

σ ) σ(1) + σ(2)/T

(27)

We found that including the salt-water binary interaction parameter, ksw, as a temperature-independent adjustable parameter greatly improved the equation of state correlations over the entire temperature range for each of the seven systems. Furthermore, the volume translation parameter, c, did not have a significant effect on the results. Therefore, we set this parameter equal to zero for all salts. The set of best-fit parameters is given in Table 3 where a is in Pa‚m6/mol and σ is in Å. Each of the parameters has a value much greater than the range of its 95% confidence limit. In general, the set of parameters in Table 3 calculated at 25 °C will be different than that reported in Table 1 for the same salt at 25 °C. For example, for NaCl, Table 1 gives a ) 2.461 while Table 3 gives a ) 0.235 at 25 °C. This is not a failure of the model, nor is it indicative of too many parameters in the model. The differences arise because we are fitting to different sets of experimental data in each case. In the first case, we fitted the equation of

Table 3. Best-Fit Parameter Values for Electrolytes electrolyte

a(1)

a(2)

b (cm3/mol)

ksw

NaCl NaBr CaCl2 Li2SO4 Na2SO4 K2SO4 Cs2SO4

0.932 1.360 1.694 1.619 2.737 1.520 1.005

-207.9 -280.3 -326.7 -352.7 -661.3 -392.2 -247.3

9.90 14.3 13.4 9.46 15.5 10.6 10.2

-0.2540 -0.1888 -0.4461 -0.1167 0.1670 0.2020 0.1215

σ(1)

σ(2)

5.695 -551.3 8.114 -1242 6.871 -897.7 4.835 -358.1 4.850 -432.2 4.566 -388.7 4.132 -168.6

state only to activity coefficient data at 25 °C, while in the second case, we fitted to activity coefficients, osmotic coefficients, free energies of hydration, and densities from 0 to 300 °C. In the second case, we also added an additional adjustable parameter, ksw, which has a large effect on the value of the van der Waals attraction parameter, a. The excluded-volume parameter, b, is also different in each of the cases because the solution densities, which were not used in the 25 °C fit, are strongly dependent on the values of this parameter. The equation of state does not give very good predictions of densities at 25 °C with the set of parameters in Table 1, while it does give reasonable predictions with the set of parameters in Table 3. The ion diameter, σ, is also changed slightly in each case, although not as much as the other two parameters. The different data types and different ranges of conditions have a significant effect on the set of best-fit parameters. The results show that all of the average ion diameters increase with increasing temperature. As mentioned, we are using average ion cavity diameters in the equation of state rather than bare ion diameters, so the cavity diameter accounts for the diameter of the bare ion as well as the distance to the first solvation shell of the ion. Our results agree with previous work42 that has shown that the average cation-oxygen distance in aqueous electrolyte solutions increases with increasing temperature. The results from the least-squares regression calculations are given in the following figures and tables. Figures 4-9 show the activity coefficients and osmotic coefficients of NaCl, NaBr, and CaCl2 as functions of temperature and salt molality. The equation of state results for the densities of the electrolyte solutions are shown in Figures 10-12. The lines result from the equation of state, and the points are the experimental data reported in the references in Table 2. In these results for temperatures of less than or equal to 100 °C, the pressure is 1 bar, and for temperatures greater than 100 °C, the pressure is equal to the saturation pressure at the specified temperature. The equation of state accurately reproduces the activity and osmotic coefficients in the 0-300 °C temperature range and the 1-120 bar pressure range of the NaCl, NaBr, and CaCl2 solutions. The largest deviations occur at the lower temperatures where the activity and osmotic coefficients have the largest values. Although thermodynamic data at higher concentrations than those shown in the figures do exist, these data were not used in the data regression. This is because, as the concentration of salt increases, ions may associate to form ion pairs, and this is not presently included in the equation of state. Ion-pair formation becomes more pronounced at higher temperatures where the dielectric constant is small. For example, activity coefficient data for aqueous CaCl2 solutions from 0 to 300 °C are reported for concentrations up to 30 m,38 but it is believed that the calcium and chloride ions form ion

3292


Figure 4. Activity coefficient of aqueous NaCl.

Figure 7. Osmotic coefficient of aqueous NaCl.

Figure 5. Activity coefficient of aqueous NaBr.

Figure 8. Osmotic coefficient of aqueous NaBr.

Figure 6. Activity coefficient of aqueous CaCl2.

Figure 9. Osmotic coefficient of aqueous CaCl2.

pairs in concentrated solutions somewhere in this temperature range. Modeling of this effect would require the use of ion-pair formation equilibrium constants in the equation of state. These constants have been measured for some electrolytes,43,44 and work is underway to extend the equation of state developed here to solutions that include ion pairs. The equation of state captures the correct trend in the densities of NaCl, NaBr, and CaCl2 solutions, although the predictions are less accurate as the concentration increases. With cubic equations of state such as the Peng-Robinson, it is often difficult to accurately predict densities. In addition to activity coefficients, osmotic coefficients, and densities, the model was also fit to free energies of hydration at 25 °C and 1 bar. The results of the fit to the experimental data are shown in Table 4. The values reported are the sums of the free energies of hydration of the individual ions of each salt.

Table 4. Free Energies of Hydration Calculated from the Equation of State at 25 °C and 1 bar electrolyte

h exptl ∆m h G (kJ/mol)

calcd ∆m h h G (kJ/mol)

% deviation of EOS

NaCl NaBr CaCl2

-728.5 -714.6 -2227.6

-731.5 -719.1 -2237.8

0.41 0.63 0.46

Experimental values were taken from the tables of Friedman and Krishnan,29 and the calculated values are from eq 21. Finally, the results for the four alkali-metal sulfate solutions are presented in Figures 13-20. The lines are from the equation of state, and the points are the data calculated from the equations of Holmes and Mesmer.37 Again, for temperatures of less than or equal to 100 °C, the pressure is 1 bar, and for temperatures greater than 100 °C, the saturation pressure is used at the specified


Figure 10. Density of aqueous NaCl.

Figure 13. Activity coefficient of aqueous Li2SO4.

Figure 11. Density of aqueous NaBr.

Figure 14. Activity coefficient of aqueous Na2SO4.

Figure 12. Density of aqueous CaCl2.

Figure 15. Activity coefficient of aqueous K2SO4.

temperature. The equation of state reproduces the thermodynamic properties of all four salt solutions quite well over the entire range of temperature and composition. The AADs of the equation of state calculations for each property of the seven salt solutions are listed in Table 5. These deviations are reasonable considering the small number of adjustable parameters of the equation of state and the ranges of temperature, pressure, and composition investigated. When the equation of state is fitted to a system for which only a limited amount of data are available, it is possible to reduce the number of adjustable parameters by correlating the excludedvolume parameter, b, with the ion diameter, σ, using the following equation:

b ) 0.802(σ(1) + σ(2)/298.15)3 - 32.761 cm3/mol (28)

Table 5. AADs of Equation of State Predictions % AAD electrolyte

γ(

Φ

F

∆m h h G

NaCl NaBr CaCl2 Li2SO4 Na2SO4 K2SO4 Cs2SO4

2.01 2.60 4.45 3.45 2.57 1.59 4.09

1.03 1.63 2.83 1.42 1.97 1.07 2.63

1.51 1.24 1.97 a a a a

0.41 0.63 0.46 a a a a

a

Equation of state not fitted to these data.

With this correlation and with the other five parameters in Table 3 readjusted, the average deviations in activity coefficients, osmotic coefficients, densities, and free energies of hydration over all seven salts increase from 2.97% to 3.64%, 1.80% to 2.19%, 1.57% to 1.59%, and 0.50% to 0.86%, respectively. Hence, it is reasonable to

3294


Figure 16. Activity coefficient of aqueous Cs2SO4.

Figure 19. Osmotic coefficient of aqueous K2SO4.

Figure 17. Osmotic coefficient of aqueous Li2SO4.

Figure 20. Osmotic coefficient of aqueous Cs2SO4.

Figure 18. Osmotic coefficient of aqueous Na2SO4.

use this correlation if the experimental data are scarce. Otherwise, it is recommended to treat all of the parameters in Table 3 as adjustable to obtain better correlations. Wu and Prausnitz12 have developed a similar equation of state that includes a Peng-Robinson term, a Born term, an MSA term, and a term from the SAFT theory to account for hydrogen bonding in the solution. Their model contains a total of five adjustable parameters for water and two parameters for each salt species, with one of the salt parameters being temperaturedependent. Although their equation of state might appear to be quite similar to ours, there are a number of differences between the two. First, their Born term does not account for the Helmholtz free energy on discharging ions in a vacuum because their reference system consists of a mixture of neutral particles while ours consists of an ideal gas mixture of ions and solvent

molecules. The main implication of this is that one can calculate free energies of hydration of ions with our equation of state but not with that of Wu and Prausnitz. Another difference is that the Wu and Prausnitz equation of state uses the full analytical form of the MSA while our equation of state uses a simplified form. While the full form of the MSA might be more accurate, it requires an iterative solution to obtain the screening parameter, Γ, making it impossible to calculate analytical derivatives of the equation of state explicitly in terms of T, V, and n. Finally, Wu and Prausnitz report results only for the solubility of methane in NaCl-water solutions at 125 °C and for the activity coefficient of NaCl between 25 and 300 °C. Because they do not report results for densities or for the other thermodynamic properties of other salt solutions, we can only compare results for the activity coefficient of NaCl-water solutions. A comparison between their graphs and Figure 4 shows that we achieve a similar accuracy with our equation of state. Our proposed equation of state does have several limitations. The Born contribution to the equation of state gives the free energy change for charging a hard sphere in a dielectric medium. Although the Born equation provides good predictions of the free energies of hydration for most systems, it fails at very low solution densities. The reason for this is that the Born equation ignores the compressibility of the solvent. In electrolyte solutions, the electric field of the ions will cause the solvent molecules to be compressed, thus changing the dielectric constant of the solution. In highdensity solutions, the effect of this compression of the solvent by the ions is small and can be neglected, while at low densities (F < about 0.5 g/cm3), the effect is large and cannot be ignored. Wood et al.45 have developed a


Figure 21. Osmotic coefficient of aqueous (1 - y)NaCl + yLiCl at 25 °C and 1 bar. Equation of state parameters for each salt are taken from Table 1. Experimental data are from Robinson et al.46

compressible continuum model that takes this into account. Their model accurately predicts the coordination number of ions in low-density solutions where the Born model fails but is not well-suited for use in an engineering equation of state because it requires numerical integration. We, therefore, suggest that our proposed equation of state not be used to calculate free energies of hydration for very low-density solutions. A similar problem arises when dealing with electrolyte solutions of very high concentration. The model of an electrolyte solution as a mixture of ions in a dielectric continuum is not valid at very high ion concentrations as a result of using the primitive model of electrolyte solutions. An improvement to the equation of state would be to use a model in which the solvent is treated as a system of discrete molecules, not as a continuum. However, an equation of state based on such a model would be much more complex than our current equation of state. Finally, we fitted the equation of state parameters to activity and osmotic coefficients and to densities over a wide temperature range and to concentrations up to 8 m. As we showed for several aqueous electrolyte solutions at 25 °C and 1 bar, we were able to extrapolate activity coefficients to concentrations as high as 15 m. Nonetheless, we suggest that, because of the limitations of the primitive model at very high concentrations, our proposed equation of state should not be used for systems with salt concentrations greater than about 15 m. Mixed Electrolyte Solutions. As we mentioned in the beginning of this paper, our equation of state should be applicable to both single salt systems and systems with mixed salts. Although we have obtained parameters specific to each salt using only experimental data for single salt solutions, we can use these parameters directly to predict the thermodynamic properties of mixed salt solutions. The predictions of our equation of state for two aqueous mixed-salt systems are shown in the following two figures. Figure 21 shows the osmotic coefficient of aqueous mixtures of NaCl and LiCl at 25 °C and 1 bar as a function of the fraction of LiCl. The results are given for four different isopiestic molalities. The points are the experimental data taken from Robinson et al.,46 and the lines are the equation of state predictions with parameters for each salt taken from Table 1. Figure 22 shows the osmotic coefficient of aqueous mixtures of NaCl and CaCl2 at 110 °C and 1.45 bar as a function of total molality (mNaCl + mCaCl2). The points are the experimental data taken from Holmes et

Figure 22. Osmotic coefficient of aqueous (1 - y)NaCl + yCaCl2 at 110 °C and 1.45 bar. Equation of state parameters for each salt are taken from Table 3. Experimental data are from Holmes et al.47

al.,47 and the lines are the equation of state predictions with parameters for each salt taken from Table 3. The equation of state does an excellent job predicting the osmotic coefficients of both mixed-salt solutions without any additional parameters. A more comprehensive study of our equation of state applied to mixed-salt systems will be given in the future. Conclusions The aim of this work has been to develop a fundamental equation of state for fully ionized electrolyte solutions that is applicable over wide ranges of temperature, pressure, and composition. The new equation of state is successful in correlating the activity coefficients of most aqueous electrolyte solutions at 25 °C and 1 bar and concentrations up to 6 m, and it is also accurate for extrapolating activity coefficients to higher concentrations. The model is also successful in correlating the thermodynamic properties of aqueous NaCl, NaBr, CaCl2, and alkali-metal sulfate solutions over wide temperature and pressure ranges. Our equation of state differs significantly from equations of state that are valid for only one salt across a wide temperature range in that it is relatively simple, has few adjustable parameters, can be used with any salt, and should be applicable to mixtures of salts and mixed-solvent systems. Furthermore, because we used an equation based on the total Helmholtz free energy instead of an activity coefficient model, we are also able to calculate densities as well as the pressure dependencies of activity and osmotic coefficients. The number of adjustable parameters in our equation is not unreasonable, given the large range of temperature, pressure, and concentration over which the equation is valid. Furthermore, only two of the parameters are temperature-dependent, and the temperature dependences are relatively simple. A limitation of the equation of state is that it cannot be used for systems with significant ion pairing (systems of weak electrolytes) unless the necessary chemical equilibria are included. Because of its relative simplicity, the new equation should be well suited for use in engineering applications. These results presented are encouraging, and further development of this model is underway to extend its range of applicability to other systems including salt mixtures, mixedsolvent systems, and systems with both ions and ion pairs.

3296


Acknowledgment The authors thank Ricardo Macias for some exploratory calculations. Financial support for this research was provided by the National Science Foundation (CTS0083709) and the Department of Energy (DE-FG0285ER-13436 and DE-FG02-89ER-14080). Nomenclature a ) van der Waals attraction parameter A ) Helmholtz free energy b ) van der Waals excluded-volume parameter c ) volume translation parameter e ) elementary charge G ) Gibbs free energy k ) binary interaction parameter m ) molality M ) molecular mass n ) number of moles N ) total number of moles in the system Na ) Avogadro’s number P ) pressure R ) gas constant T ) temperature V ) volume Zi ) ion charge number Z ) compressibility factor Greek Symbols ∆h ) change on hydration ) dielectric constant 0 ) permittivity of free space γ ) activity coefficient Γ ) MSA screening parameter κ ) Debye screening length ν ) number of molecules in one molecule of salt φ ) fugacity coefficient Φ ) osmotic coefficient F ) density σ ) ion diameter Superscripts Born ) Born contribution to the equation of state IGM ) ideal gas mixture m ) hypothetical ideal 1 m standard state MSA ) mean spherical approximation contribution to the equation of state PR ) Peng-Robinson contribution to the equation of state Subscripts chg ) charging process dis ) discharging process ij ) binary pair of i and j r ) reduced quantity (e.g., temperature) ( ) mean ionic property

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Received for review December 14, 2001 Revised manuscript received April 1, 2002 Accepted April 3, 2002 IE011016G

An Equation of State for Electrolyte Solutions Covering Wide Ranges

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