3282 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 forming weakly bonded ion pairs that behave not as ions
but as dipoles dissolved in the solvent. At moderate
Electrolyte solutions are encountered in a wide temperatures and densities, a mixture of ion pairs and
variety of industrial processes such as wastewater ions will exist. An accurate engineering model for
treatment, extraction, seawater desalinization, and dis- electrolyte solutions must take all of these interactions
tillation and also in geological processes. A good under- into account. Liu and Watanasiri1 give an excellent
standing of the thermodynamics of electrolyte solutions overview of the main issues encountered when modeling
is necessary when working with such systems. For electrolyte systems.
example, process development, process optimization,
process control, and the interpretation of some geological Many different models have been proposed for de-
phenomena involving dissolved salts all require knowl- scribing the thermodynamic properties of electrolyte
edge of thermodynamic properties. Much experimental solutions. Chen et al.2 developed a local composition
work has been done on measuring the thermodynamic NRTL activity coefficient model for single, completely
properties of various electrolyte systems, but for engi- dissociated electrolytes in a single solvent. Their model
neering purposes, a good model capable of describing is in terms of the excess Gibbs free energy and is only
the behavior of these systems is desired. useful in predicting activity coefficients. Chen and
Electrolyte solutions are considerably more difficult Evans3 and Mock et al.4 extended the local composition
to model than most other solutions encountered in model to mixed-electrolyte and mixed-solvent systems.
engineering applications. The reason is that there are Chen et al.5 also extended the local composition model
a number of additional factors to consider when dealing to partially dissociated salts. Pitzer et al.6 derived a
with solutions of dissolved electrolytes that exist as ions, semiempirical equation of state for aqueous NaCl solu-
and these ions can interact both with each other and tions that is valid from 0 to 300 °C but is only useful
with the solvent. Because ions are charged particles, for fully ionized NaCl solutions. Anderko and Pitzer7
there are electrostatic interactions that are different developed a theoretical equation of state for fully ion-
from the dispersion and hard-core interactions taken paired NaCl solutions at temperatures above 300 °C.
into account by models such as the Peng-Robinson and However, this model cannot be used at lower temper-
van der Waals equations. At low electrolyte concentra- atures where NaCl is fully ionized. Lvov and Wood8
tions, only long-range electrostatic forces are significant, developed a volumetric equation of state for aqueous
while at high electrolyte concentrations, short-range NaCl solutions over a wide range of temperature,
attractive and repulsive forces become important as pressure, and concentration that is only useful for
well. Another difficulty in modeling electrolyte solutions calculating densities. Several fundamental equations of
is that the chemistry of the solution changes at different state for electrolyte solutions have been published, the
conditions. At low temperatures and high densities, most notable being the equations of Jin and Donohue,9,10
strong electrolytes are fully ionized. At high tempera- Fürst and Renon,11 and Wu and Prausnitz.12
tures and low densities, the ions tend to associate, The aim of this work is to develop an equation of state
that can predict the thermodynamic properties of fully
* To whom correspondence should be addressed. E-mail: ionized electrolyte solutions over wide ranges of tem-
sandler@Udel.edu. Phone: (302) 831-2945. Fax: (302) 831- perature, pressure, and composition. This work differs
4466. 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 I. A mixture containing ions and solvent molecules is
wish to have an equation of state that is as general as initially in a hypothetical ideal gas state at a temper-
possible. However, we also wanted a degree of simplicity ature T and volume V. In the ideal gas state there are
in the equation of state so that it can be used efficiently no interactions between the particles in the mixture.
in engineering applications. The new equation of state For the first step, the charges on all of the ions are
is an expression for the Helmholtz free energy as a removed. The change in the Helmholtz free energy for
function of temperature, volume, and number of moles discharging the ions, ∆ABorn dis , is calculated from the
of each component in the system. From this fundamen- Born equation for ions in a vacuum. After all of the ions
tal equation of state, all thermodynamic properties of a are discharged, the mixture consists of an ideal gas of
system can be calculated from various derivatives. This hypothetical neutral ions and solvent molecules.
new equation of state can also be used for mixtures of II. Next, the excluded-volume repulsive forces and
electrolytes and, though we do not do so here, for mixed-
short-range attractive dispersion forces between the
solvent systems, as well as for solutions of electrolytes
neutral particles in the mixture are turned on. The
and nonelectrolytes. Ion pairing can be included by
Peng-Robinson equation of state is used to calculate
adding the ion pairs as an additional component and
the change in the Helmholtz free energy for this
using an equilibrium constant for ion pairing. However,
transition, ∆APR.
in the present paper, we do not consider ion pairing so
we limit testing of our model to temperatures at which III. Next, the ions are recharged, only taking into
there are no appreciable numbers of ion pairs. account the ion-solvent interactions. The contribution
to the Helmholtz free energy of recharging the ions,
Thermodynamic Model ∆ABorn
chg , is calculated from the Born equation for ions at
infinite dilution in a dielectric solvent.
Any realistic thermodynamic model of electrolyte IV. In the final step, the long-range electrostatic
solutions should take into account all of the interactions interactions between the ions in the mixture are turned
between the various species in the solution. Further- on. These interactions are accounted for by the mean
more, the model should be applicable to pure fluids, spherical approximation (MSA), and the change in the
single electrolyte systems, mixtures of electrolytes, and Helmholtz free energy for this last step is given by
mixed-solvent systems and cover wide ranges of tem- ∆AMSA.
perature, pressure, and composition. To develop such a
model, we first consider a path to the formation of an The total change in the Helmholtz free energy for
electrolyte solution from a mixture of ideal gases. forming the electrolyte solution on this path is
Because we want an equation of state in terms of the
Helmholtz free energy, both temperature and volume b) - AIGM(T,V,n
A(T,V,n b) ) ∆APR + ∆ABorn + ∆AMSA (1)
remain constant on the path. Figure 1 shows the path
chosen for the formation of an electrolyte solution at a where
constant temperature and volume.
The change in the Helmholtz free energy on forming
an ionic solution from an ideal gas mixture is given ∆ABorn ) ∆ABorn Born
dis + ∆Achg (2)
by the sum of the changes in the Helmholtz free
energy for each of the four steps along the path shown where T is the temperature of the system, V is the
in Figure 1. system volume, b
n is the vector of the number of moles
3284 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

of each component of the mixture, and AIGM is the parameter for each component ci with the following
Helmholtz free energy of an ideal gas mixture. The mixing rule:
details of the individual parts of this equation of state
will be discussed next. 1
Peng-Robinson Contribution. The short-range c)
N
∑i nici (7)
interactions between uncharged species in solution are
accounted for by the equation of Peng and Robinson.13
These short-range interactions include excluded-volume The Peng-Robinson equation with volume translation,
interactions and attractive (dispersion) interactions. The eq 6, is part of the electrolyte equation of state developed
residual Helmholtz free energy for the Peng-Robinson here. In general, the ai, bi, and ci parameters for each
equation of state is species will be temperature-dependent. The binary
interaction parameters, kij, may also be temperature-

[ ]
dependent or can be set to a fixed value. The addition
Na V + b(1 - x2) of a volume translation parameter to a cubic equation
∆APR(T,V,n
b) ) ln -
2x2b V + b(1 + x2) of state greatly improves volumetric predictions.

( )
Born Contribution. The free energy required to
V-b discharge or charge an ion in solution was first calcu-
NRT ln (3)
V lated by Born.17 The continuum hydration model of Born
treats ions as hard spheres of diameter σ in a continuum
where a is the van der Waals attraction parameter, b of uniform dielectric constant. The change in the Helm-
is the van der Waals excluded-volume parameter, V is holtz free energy on discharging or charging ions in a
the molar volume, N is the total number of moles, and system at constant temperature and volume is equal to
R is the gas constant. The a and b parameters in eq 3 the net work required to remove the charges from the
are specific to the mixture and can be calculated from ions (discharging) or replace the charges (charging). The
the pure component a and b parameters with the van sum of the changes in the Helmholtz free energies from
der Waals one-fluid mixing rules: the Born model on discharging in a vacuum and
charging in a dielectric solvent is
1
a) ∑i ∑j ninj xaiaj(1 - kij)
( )∑
(4) Nae2 niZi2
N2 Born
1
∆A b) ) -
(T,V,n 1- (8)
4π0  ions σi
and
where Na is Avogadro’s number, e is the unit of
1
b)
N
∑i nibi (5) 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
where ai and bi are the pure-component attraction and calculating solvation free energies of ions in water
excluded-volume parameters, respectively, kij is a binary provided that the correct ionic diameters are used. The
interaction parameter, ni is the number of moles of diameters to be used in eq 8 are not the bare ion
species i, and the summations are over all neutral diameters but the diameters of the cavities in the
species (solvent molecules and uncharged ions). The solvent in which the ions reside.18,19 Empirically, the
binary interaction parameter, which accounts for the cavity diameters of anions are approximately 0.1 Å
short-range interactions between species i and j, is greater than the bare ion diameters, while the cavity
symmetric (kij ) kji) and is equal to zero for i ) j. diameters of cations are approximately 0.85 Å larger
While the vapor-liquid equilibria of nonpolar systems than the bare ion diameters.18
are predicted quite well with cubic equations of state Because the Born equation is linear in the number of
such as that of Peng and Robinson, volumetric predic- moles of ions, it has no effect on the ion activity
tions are usually not as accurate. Many attempts have coefficients, though this term is very important in
been made to improve the volumetric predictions of calculating free energies of hydration of ions. The Born
cubic equations of state. Peneloux et al.14 have proposed equation does have an effect on the solution density
a volume translation for the Redlich-Kwong-Soave though through the density dependence of the dielectric
equation of state that is used to shift the volume constant.
predictions by a constant value along an isotherm. MSA Contribution. Many models have been pro-
Several others, including Tsai and Chen15 and de posed to describe the interactions between charged
Sant’Ana et al.,16 have extended the concept of volume particles in solution. It is the long-range electrostatic
translation to the Peng-Robinson equation of state. The interactions in electrolyte solutions that present the
residual Helmholtz free energy for the volume-trans- most difficulty. Several statistical mechanical theories
lated Peng-Robinson equation of state is given by such as perturbation theory and integral equation
theory have been applied to electrolyte solutions. There

[ ]
are several different approaches to the integral equation
Na V + c + b(1 - x2) theory of electrolyte solutions; the two most popular are
∆APR(T,V,n
b) ) ln -
2x2b V + c + b(1 + x2) the hypernetted chain (HNC) theory and the MSA. Each

( )
of these is derived from the solution of the Ornstein-
V+c-b Zernike equation with different closures. The HNC
NRT ln (6)
V approach has proven to be the most accurate theory for
electrolyte solutions and accurately predicts the struc-
where c is the volume translation parameter for the ture of electrolyte solutions as well as their thermody-
mixture and is obtained from the volume translation namic properties. However, the HNC approach involves
 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3285

numerically solving a complex system of highly nonlin- of temperature, volume, and the number of moles of
ear equations and therefore has limited use in engineer- each component, and most experimental data are re-
ing applications. The MSA approach has been exten- ported in terms of temperature, pressure, and the
sively and successfully applied to electrolyte solutions number of moles of each component. Thus, the MSA is
and yields analytical solutions for the thermodynamic incompatible with both the equation of state and the
and structural properties of such systems. Although the experimental data in the literature. To calculate osmotic
MSA theory is not as accurate as the HNC theory, it and activity coefficients from the equation of state, the
appealing because of its relative simplicity. McMillan-Mayer framework must be transformed to
The MSA model accounts for the long-range charge- the Lewis-Randall framework. Haynes and Newman24
charge interactions between ions in an electrolyte have shown that the conversion between the solute
solution and is based on the primitive model of electro- activity coefficient in the two different frameworks is
lyte solutions in which ions are treated as charged hard
spheres in a medium of uniform dielectric constant. The
ViθPEX
MSA model for electrolyte solutions was first solved for ln γLR MM
i ) ln γi - (13)
the restricted primitive model by Waisman and Lebow- RT
itz,20 who obtained an analytical solution using the
perturbation theory. Later, Blum21 and Blum and where V h θi is the partial molar volume of ion i and PEX is
Hoye22 solved the MSA for the primitive model and the excess pressure in the McMillan-Mayer framework
obtained analytical expressions for excess thermody- (i.e., the pressure from the MSA equation). They also
namic properties (relative to a mixture of hard spheres) show that contribution of the excess pressure term to
including the Helmholtz free energy, activity coef- the activity coefficient of a 1:1 electrolyte in water at
ficients, chemical potentials, and osmotic coefficients. 25 °C is very small. The magnitude of this term
The original solution of the MSA model is a complicated increases with increasing ion diameter and concentra-
set of equations that must be solved iteratively to obtain tion, but for most electrolyte solutions, it is less than
useful information. However, Harvey et al.23 have 3% of the total activity coefficient. Therefore, we neglect
derived an explicit approximation to the MSA that has the difference between the McMillan-Mayer and Lewis-
been shown to be simple and accurate. The approxima- Randall frameworks when using the MSA in our equa-
tion assumes that all of the ions in the system have the tion of state, especially because our equation contains
same diameter given by the following average diameter: several empirical parameters which will compensate for
this small difference.
∑ niσi
ions
σ) (9) Thermodynamic Properties
∑
ions
ni
Because the equation of state above is in terms of the
Helmholtz free energy, any thermodynamic property of
where the summations are over all ionic species. The a system can be calculated from its various temperature,
MSA contribution to the Helmholtz free energy of the volume, and mole number derivatives. In general,
system is thermodynamic properties are usually calculated as
functions of the system temperature and pressure,
though temperature and volume are the independent
2Γ3RTV 3
∆AMSA(T,V,n
b) ) -
3πNa
1 + σΓ
2 ( ) (10) variables in the Helmholtz free energy. Therefore, all
calculations with the equation of state must be per-
formed as a function of temperature and volume, while
where Γ is the MSA screening parameter and κ is the most experimental data are measured and reported as
Debye screening length. These quantities are functions of temperature and pressure. To be able to fit
the equation of state to experimental data, one needs
1 to be able to compute various thermodynamic properties
Γ) [x1 + 2σκ - 1] (11)
2σ from the Helmholtz free energy equation of state as
functions of temperature and pressure. This is ac-
and complished by first calculating the volume of the system

( )
at the specified temperature and pressure from the
e2Na2 1/2
equation of state and then using this volume in the
κ)
0RTV ions
∑ niZi 2
(12) calculations of other thermodynamic properties.
Of primary importance to engineers is phase equilib-
ria, for which the fugacity coefficient of each species in
The MSA equation reduces to the Debye-Hückel limit- a mixture is the important quantity. Although the
ing law as σ approaches zero; however, the MSA fugacity coefficient is usually expressed as a function
equation gives much more accurate predictions of the of temperature and pressure,25 here it is necessary to
thermodynamic properties at high electrolyte concentra- have an expression for this quantity in terms of tem-
tions. Normally, the MSA calculation also includes a perature and volume. This expression can be derived
hard-sphere repulsion term, but in the present equation, by first calculating the difference between the ideal gas
this has already been included in the Peng-Robinson chemical potential as a function of the system temper-
term. ature and pressure and the ideal gas chemical potential
The MSA is formulated in the McMillan-Mayer as a function of the system temperature and volume and
framework in which the independent variables are then substituting this result into the original equation
temperature, volume, solvent chemical potential, and for the fugacity coefficient. The difference in chemical
solute mole numbers. Our equation of state is in terms potentials and the resulting equation for the fugacity
3286 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

coefficient as a function of temperature and volume are Another property that is important in electrolyte
given below: solutions is the free energy of hydration. This quantity
is defined as the Gibbs free energy required to transfer
GiIGM(T,P,n
b) - GiIGM(T,V,n
b) ) RT ln Z (14) an ion from an ideal gas at the temperature of interest
and 1 bar to an ideal 1 m solution at the temperature
and and pressure of interest. The free energy of hydration
is a key property in solvation thermodynamics and is

b) )
φi(T,V,n
1
Z
exp{ b) - GiIGM(T,V,n
Gi(T,V,n
RT
b)
} (15)
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:
where Z ) PV/NRT is the compressibility factor and b n
indicates the vector of species mole numbers.
When performing phase equilibrium calculations for ∆m
h Gi(T,P) ) RT ln φi(T,P,ni)0) +
electrolyte systems, the standard state of each compo-
nent in the system must carefully be considered. Often
the standard state for a liquid is the pure liquid at the
RT ln(Msolventmθ) + RT ln ( )
P
Pθ
(21)

temperature of the system and 1 bar. However electro- where the second term is the conversion from the mole
lytes, especially salts, usually do not exist as pure fraction to the molality standard state (mθ ) 1 mol/kg)
liquids, and therefore, a standard state based on the and the third term is a correction for the difference
pure substance as a liquid is unrealistic. For dilute between the system pressure and the standard state
species including electrolytes, the standard state can be pressure (Pθ ) 1 bar).
taken to be a hypothetical ideal 1 m solution. With this,
the activity coefficient of a solvent species in an elec- Application of the Equation of State to Aqueous
trolyte solution is Electrolyte Solutions

b)
φi (T,P,n Pure Water. Because we are focusing on aqueous
b) )
γi(T,P,n (16) electrolyte solutions, the equation of state should ac-
φipure(T,P) curately predict the properties of pure water in the limit
of an infinitely diluted salt solution. The parameters for
where φ h ipure(T,P) is the fugacity coefficient of pure pure water were fitted to volumetric and vapor-pressure
species i at the temperature and pressure of the system. data generated from the equation of state of the
The molality-based activity coefficient for a solute International Association for the Properties of Water
species is then calculated as and Steam.26 This 56-parameter equation of state

( )[ ]
predicts the thermodynamic properties of water with
1 b)
φi(T,P,n extremely high accuracy. The data used in the regres-
γim(T,P,n
b) ) (17) sion included vapor pressures from 10 °C to the critical
1 + νmMsolvent φ (T,P,n )0) temperature (374.15 °C) and liquid densities from 10
i i
°C to the critical temperature and pressures up to 250
where ν is the total number of molecules upon complete bar. For pure water, there are no ionic interactions;
dissociation of one molecule of salt, m is the molality, thus, we only need to consider the attractive dispersion
Msolvent is the molecular mass of the solvent in kg/mol, and repulsive interactions between the molecules. Be-
and φh i(T,P,ni)0) is the fugacity coefficient of species i cause the Born and MSA terms of the equation of state
at infinite dilution. Because activity coefficients of the are zero for a system without ions, the properties of pure
individual ionic species in an electrolyte solution cannot water are only dependent on the Peng-Robinson term.
be directly measured, an average activity coefficient for There are three parameters in the volume-translated
the dissolved salt is defined. For a solution consisting Peng-Robinson equation for each neutral species: the
of a single salt, the mean ionic activity coefficient is attraction parameter, a, the excluded-volume param-
defined as eter, b, and the volume translation parameter, c. All
three parameters were assumed to be temperature-
γm m ν+ m ν- 1/ν
( ) [(γ+ ) + (γ- ) ] (18) dependent to give a good fit over the entire temperature
and pressure range. A Levenberg-Marquardt nonlinear
where ν+ and ν- are the number of cations and anions, least-squares routine with the aforementioned data gave
respectively, in one molecule of the salt. In terms of eq the following set of best-fit parameters:
17, this becomes

( )
aH2O ) 1.26944 - 0.89381Tr + 0.16937Tr2 Pa‚m6/mol
1 (22)
γm b) )
( (T,P,n ×
1 + νmMsolvent
bH2O ) 15.6345 + 6.14518Tr - 5.2795Tr2 cm3/mol

[ ][ ]
ν+/ν ν-/ν (23)
b)
φ+(T,P,n b)
φ-(T,P,n
(19)
φ+ (T,P,n+,-)0) φ-(T,P,n+,-)0) cH2O ) -2.7227 + 11.4201Tr - 6.0157Tr2 cm3/mol
(24)
The osmotic coefficient is
where Tr ) T/647.29 K.
b)]
ln[xsolventγsolvent(T,P,n With this set of parameters for pure water, the
b) ) -
Φ(T,P,n (20) equation of state accurately predicts the vapor pressure
νmMsolvent and liquid densities of water over a wide range of
 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3287

temperature and pressure resulting in average relative where nH2O is the number of moles of water in the
deviations of only 0.22% and 0.83%, respectively. These solution and MH2O is the molecular mass of water in
deviations are sufficiently small to ensure that the kg/mol.
equation of state can accurately predict the behavior of In a solution of a single electrolyte dissolved in water,
dilute aqueous electrolyte solutions. The largest devia- there are three distinct species: cations, anions, and
tions between the equation of state predictions and the water molecules. At high temperatures and high salt
experimental data occur near the critical temperature concentrations, individual ions may associate to form
of water where most cubic equations of state fail to ion pairs. In this work, we focused on a range of
accurately predict thermodynamic properties. The use temperature, pressure, and concentration in which ion
of a temperature-dependent volume translation param- pairing can be neglected. That is, we assumed that the
eter in the Peng-Robinson equation of state minimizes dissolved salt is fully ionized in each system we inves-
the errors in densities near the critical point. tigated. We first focused on aqueous electrolyte solutions
Several models have been proposed for calculating the at 25 °C and 1 bar.
dielectric constant of pure water needed for the Born The equation of state proposed here has five param-
and MSA terms. Uematsu and Franck27 have developed eters for each ionic species: the attraction and excluded-
a 10-parameter model that is accurate over a temper- volume parameters, a and b, the ion-water binary
ature range of 273-1273 K and pressures up to 1600 interaction parameter, kiw, the volume translation pa-
MPa. Fernandez et al.28 have developed a more accurate rameter, c, and the ion diameter, σ, in the Born and
12-parameter model that gives the dielectric constant MSA terms. Because experimental activity coefficients
of pure water as a function of temperature and density for electrolyte solutions are specific to salts rather than
over a temperature range of 238-1273 K and pressures the constituent ions, the effects of the individual ions
up to 1200 MPa. We used the Uematsu and Franck cannot be separated when determining the equation of
model because it is simpler and has fewer parameters, state parameters for a specific salt solution. Therefore,
and calculations required about half as much time as for each salt species, the parameters for the cations and
those with the model of Fernandez et al. anions are set equal to a single value. Because we did
Aqueous Electrolyte Solutions at 25 °C and 1 not include density data in the regressions at 25 °C, we
bar. Both the Born and MSA contributions to the set the volume translation parameter for each salt, c,
equation of state depend on the dielectric constant of equal to zero. Furthermore, we found that the ion-
the solvent of the electrolyte solution. It is known that water binary interaction parameter, kiw, did not have a
ions in solution polarize the surrounding solvent mol- significant effect on the equation of state correlations
ecules, thus decreasing the dielectric constant;29 how- for most salt solutions at 25 °C. Therefore, we set this
ever, it is difficult to accurately model this effect with a parameter equal to zero for all salts. Consequently,
simple analytical equation. Therefore, the dielectric there are only three adjustable parameters (a, b, and
constant of the pure solvent, water, has been used for σ) for each salt used to fit to the 25 °C and 1 bar data.
all calculations here. Although aqueous electrolyte We first compared the results from our model to the
solutions are by far the most common types of electrolyte NRTL model of Chen et al.5 That model has two
solutions encountered in natural and industrial pro- adjustable parameters for each salt, and because it is
cesses, solutions with other types of solvents do exist. an excess free energy model, it cannot be used to predict
Harvey and Prausnitz30 have developed an expression solution densities. In general, each ion has a number
for estimating the dielectric constant of fluid mixtures of water of hydration molecules associated with it. Chen
that is useful when dealing with nonaqueous and mixed et al. expanded the NRTL model to account for the
solvent systems. hydration effects that occur in electrolyte solutions by
The Uematsu and Franck equation for the dielectric adding a third hydration parameter for each ion to
constant of pure water is a function of temperature and improve the fit of the data.
water density. Thus, it is necessary to first obtain a We used mean ionic activity coefficients for 138 salt
value for the density of pure water to be used in this solutions tabulated by Robinson and Stokes31 in sepa-
equation before other equation of state calculations are rate data regressions to determine the optimum equa-
performed. One way to do this is to calculate the density tion of state parameters for each electrolyte. This set
of water from the Peng-Robinson equation because the includes the 116 salts used by Chen et al. to determine
properties of pure water are dependent only on this the NRTL parameters and 22 other salts. The results
term. Unfortunately, if this method is used, it is for each of the 138 salts are presented in Table 1 and
impossible to express the equation of state exclusively are compared to the results from the NRTL model with
in terms of the variables T, V, and n because the hydration. The maximum molality (up to 6 m), for which
dielectric constant will be a function of the solution to activity coefficients were calculated for each salt, is also
the cubic Peng-Robinson equation for water. Conse- reported.
quently, it will be impossible to obtain analytical
derivatives of the Helmholtz free energy with respect Each of the three adjustable parameters for each salt
to temperature, volume, and mole numbers. Another is well determined with a value much larger than the
drawback of using this method to calculate the density range of its 95% confidence limit. The only apparent
of water is that this calculation is rather time-consum- trend in the parameters was that salts with large
ing. An alternate method is to approximate the density (small) values of a also had large (small) values of b and
of water with a simple expression in terms of the σ. A correlation between b and σ is to be expected
variables of the equation of state: T, V, and n. We found because each parameter is a measure of ion size. As
that the approximation that works best with our equa- mentioned earlier, we have used salt-specific param-
tion is eters instead of ion-specific parameters because it is not
possible to separate the effects of each ion when analyz-
ing experimental activity coefficient data. Each param-
FH2O ) nH2OMH2O/V (25) eter for a salt should be dependent on both the cation
3288 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

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

% AAD max molality
electrolyte a (Pa‚m6/mol) b (cm3/mol) σ (Å) NRTL modela this model (mol/kg)
1-1 Electrolytes
AgNO3 0.0384 12.9306 2.548 0.77 0.20 6
CsAc 12.0035 68.1606 5.827 0.79 0.65 3.5
CsBr 0.2267 13.1350 2.728 0.52 0.31 5
CsCl 0.2037 11.2541 2.681 0.52 0.28 6
CsI 0.0885 8.6376 2.933 0.52 0.17 3
CsNO3 0.7412 29.5030 2.894 0.24 0.15 1.4
CsOH 13.2115 68.0463 5.041 b 0.46 0.9
HBr 18.5145 82.1459 4.791 0.75 0.09 3
HCl 10.8072 64.6144 4.686 0.53 0.63 6
HClO4 18.8935 92.2856 5.514 1.19 0.20 6
HI 19.6013 87.6413 4.577 0.62 2.61 3
HNO3 4.9972 46.0054 4.764 0.80 0.06 3
KAc 9.6989 61.0982 5.498 0.66 0.61 3.5
KBrO3 2.5265 18.8586 4.348 0.14 0.08 0.5
KCl 0.7332 19.6245 3.925 0.23 0.08 4.5
KClO3 0.2391 19.2805 4.276 0.24 0.08 0.7
KCNS 0.3709 10.5410 4.157 0.13 0.13 5
KF 1.9845 31.2904 3.800 0.36 0.08 4
KH adipate 2.9218 41.0558 4.399 0.12 0.09 1
KH malonate 0.3529 18.1751 3.798 0.36 0.52 5
KH succinate 0.7082 22.8839 3.699 0.21 0.22 4.5
KTol 0.6069 24.6390 5.791 2.47 0.35 3.5
KH2PO4 0.8458 30.9256 2.889 0.22 0.08 1.8
KI 1.3641 21.9101 4.551 0.38 0.20 4.5
KNO3 0.0790 14.4821 2.797 0.58 0.13 3.5
KOH 4.9778 43.4108 3.682 0.48 0.60 6
KBr 0.7522 16.8917 4.004 b 0.09 5.5
LiAc 3.1638 34.2560 4.472 0.22 0.30 4
LiBr 10.5086 68.1208 4.401 0.95 0.37 6
LiCl 8.0018 58.9605 4.443 0.88 0.25 6
LiClO4 12.9217 70.9185 4.573 1.32 4.26 4
LiI 20.7897 90.2289 5.347 1.78 1.29 3
LiNO3 5.6595 46.3514 4.855 0.35 1.55 6
LiOH 0.2401 8.1861 2.010 3.00 1.01 4
LiTol 1.4791 29.1960 4.920 0.97 0.58 4.5
NaAc 7.1602 52.3677 5.264 0.79 0.61 3.5
NaBr 3.2858 38.3654 4.278 0.36 0.09 4
NaBrO3 0.6920 24.0680 3.845 0.10 0.08 2.5
NaCl 2.4611 34.6464 4.322 0.44 0.08 6
NaClO3 0.7813 22.2293 4.436 0.30 0.09 3.5
NaClO4 0.9612 20.7826 4.426 0.46 0.07 6
NaCNS 4.0473 38.9232 5.258 1.22 0.36 4
NaF 1.1814 27.9280 3.961 0.02 0.02 1
Na formate 1.7453 25.0364 4.746 0.71 0.31 3.5
NaH malonate 0.2392 12.9744 3.698 0.36 0.26 5
Na propionate 11.4792 66.6921 5.445 1.01 0.76 3
NaH succinate 0.6232 19.7096 3.626 0.35 0.20 5
NaTol 0.2775 16.4907 5.042 0.99 0.44 4
NaI 5.4462 45.4724 4.495 0.57 0.19 3.5
NaNO3 0.1189 10.5855 3.744 0.12 0.11 6
NaOH 3.8268 43.7018 4.297 0.63 0.67 6
NH4Cl 0.5300 12.9909 4.041 0.06 0.29 6
NH4NO3 0.0000059 2.9832 2.926 0.93 0.16 6
RbAc 9.2959 59.9059 5.079 0.70 0.83 3.5
RbBr 0.3066 12.1865 3.485 0.16 0.07 5
RbCl 0.4340 15.0284 3.450 0.15 0.07 5
RbI 0.3356 13.3635 3.358 0.18 0.12 5
RbNO3 0.0346 12.5468 2.441 0.85 0.19 4.5
TlAc 0.0209 1.3019 2.825 0.92 0.23 6
1-2 Electrolytes
Cs2SO4 0.8660 28.3819 4.290 0.47 0.22 1.8
K2SO4 0.4036 22.6983 3.493 b 0.28 0.7
K2CrO4 0.6387 24.8318 4.288 1.37 0.25 3.5
Li2SO4 1.1478 29.1075 4.408 0.81 0.27 3
Na2CrO4 1.8841 38.5592 5.013 2.23 0.41 4
Na2SO4 0.4962 24.7164 4.038 0.26 0.16 4
Na2S2O3 1.0029 30.2521 4.389 0.89 0.17 3.5
(NH4)2SO4 0.0684 12.1388 3.108 0.92 0.25 4
Rb2SO4 0.8289 28.8106 4.249 0.47 0.15 1.8
2-1 Electrolytes
BaBr2 5.0044 47.6234 5.059 1.63 0.16 2
BaCl2 2.6572 35.3305 4.993 1.33 0.17 1.8
Ba(ClO4)2 5.4075 45.7915 5.358 3.22 4.75 5
BaI2 11.1626 67.4211 5.219 2.02 0.30 2
Ba(NO3)2 0.4079 25.4403 3.592 0.07 0.11 0.4
BaAc2 0.3790 10.4986 5.432 b 4.21 3.5
 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3289

Table 1. (Continued)
% AAD max molality
electrolyte a (Pa‚m6/mol) b (cm3/mol) σ (Å) NRTL modela this model (mol/kg)
2-1 Electrolytes
CaBr2 15.8804 89.4555 5.906 5.72 2.11 6
CaCl2 7.2623 60.1464 4.990 3.06 0.71 6
Ca(ClO4)2 18.3813 85.2520 5.889 3.67 3.90 6
CaI2 17.7616 90.1023 5.723 2.46 0.18 2
Ca(NO3)2 1.1068 20.1824 4.708 1.84 0.21 6
Cd(NO3)2 3.4414 35.8613 5.538 b 0.52 2.5
CdCl2 0.0000020 18.2814 1.569 b 21.28 6
CdBr2 0.0000010 25.1028 1.231 b 26.53 4
CdI2 0.0000144 43.0445 0.902 b 45.56 2.5
CoBr2 14.9928 76.5799 5.658 2.45 4.70 5
CoCl2 7.3404 53.2723 5.219 1.94 2.65 4
CoI2 6.9741 52.1105 3.208 b 18.92 2
Co(NO3)2 5.8125 47.1703 4.879 2.39 0.73 5
CuCl2 2.0046 26.9883 6.600 b 4.59 6
Cu(NO3)2 5.8125 47.1703 5.051 2.62 1.71 6
FeCl2 7.0986 58.7403 5.034 1.64 0.22 2
MgAc2 1.1304 20.0419 3.906 1.08 0.96 4
MgBr2 15.3623 83.9197 5.283 4.16 0.27 5
MgCl2 11.5149 75.6245 5.313 3.65 0.39 5
MgI2 39.6775 116.7610 6.280 b 0.50 5
Mg(ClO4)2 48.0065 129.1700 6.824 b 0.48 4
Mg(NO3)2 7.3415 53.1808 5.068 2.88 0.34 5
MnCl2 4.5477 41.6882 5.788 1.90 6.80 6
NiCl2 7.9816 55.2474 5.152 2.85 3.89 5
Pb(ClO4)2 7.3738 53.1612 5.350 2.95 2.68 6
Pb(NO3)2 0.0542 15.5291 2.722 1.45 0.19 2
SrBr2 9.8770 70.2552 5.382 2.35 0.16 2
SrCl2 6.1191 55.8518 5.094 2.73 0.30 4
Sr(ClO4)2 13.9512 73.7878 6.121 2.76 6.45 6
SrI2 16.5394 88.1006 5.835 2.57 0.23 2
Sr(NO3)2 1.7576 18.3618 4.834 2.00 0.23 4
UO2Cl2 11.1507 65.7514 6.298 3.45 2.24 3
UO2(ClO4)2 162.6549 268.2230 11.143 7.26 1.69 5.5
UO2(NO3)2 11.5511 66.9808 7.541 b 15.65 5.5
ZnBr2 4.0927 39.1711 9.705 b 4.24 6
ZnCl2 2.4795 40.5478 7.479 7.31 1.18 6
ZnI2 2.9942 33.1947 9.508 b 14.06 6
Zn(ClO4)2 31.4520 120.4010 6.202 4.54 0.38 2
Zn(NO3)2 7.7135 54.3961 5.572 2.94 2.15 6
2-2 Electrolytes
BeSO4 0.9095 27.4868 3.185 1.83 1.86 4
MgSO4 0.6670 25.4272 3.226 1.45 2.17 3.5
MnSO4 0.6374 28.6029 3.261 1.50 1.78 4
NiSO4 4.0658 27.6524 3.214 1.26 1.76 2.5
CuSO4 0.4099 13.7100 3.166 1.33 1.56 1.4
ZnSO4 0.8459 32.8818 3.217 1.50 2.05 3.5
CdSO4 0.4461 24.5582 3.177 1.94 1.53 3.5
UO2SO4 0.6317 15.3058 3.137 b 1.71 6
3-1 Electrolytes
AlCl3 17.9117 97.3688 6.109 4.57 0.61 1.8
CeCl3 6.9281 60.7677 5.382 3.97 0.44 2
CrCl3 10.6489 72.6823 5.580 2.69 0.49 1.2
Cr(NO3)3 7.3251 58.3339 5.371 2.88 0.21 1.4
EuCl3 8.3589 66.7114 5.573 4.18 0.27 2
LaCl3 6.8821 60.3325 5.432 3.96 0.16 2
NdCl3 7.0823 61.5979 5.347 3.87 0.22 2
PrCl3 6.9115 60.9113 5.380 3.82 0.32 2
ScCl3 9.2870 68.2657 5.339 3.46 0.29 1.8
SmCl3 7.6646 63.7411 5.446 3.97 0.23 2
YCl3 8.7553 68.3844 5.414 4.07 0.29 2
1-3 Electrolyte
K3Fe(CN)6 0.8151 26.8172 4.860 b 0.12 1.4
3-2 Electrolytes
Al2(SO4)3 1.5724 37.6690 4.253 b 2.81 1
Cr2(SO4)3 3.6241 58.0423 5.531 b 0.52 1.2
4-1 Electrolytes
Th(NO3)4 2.9651 33.2948 9.012 b 7.87 5
ThCl4 19.7484 99.3895 9.676 b 1.73 1.6
1-4 Electrolyte
K4Fe(CN)6 3.4615 24.1887 5.208 b 0.28 0.9
a 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 anion. Obtaining parameters specific to each ion would
parameter values for salts with a common cation or require the simultaneous regression of activity coef-
3290 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

Figure 2. Activity coefficient of aqueous UO2(ClO4)2 and UO2- Figure 3. Activity coefficient of aqueous electrolyte solutions at
SO4 at 25 °C and 1 bar. 25 °C and 1 bar.

ficient data for all 138 salt solutions, which would be able parameter per salt. They report a 6% AAD in
very difficult and time-consuming. activity coefficients and a 2% deviation in specific
In general, our equation of state does an excellent job volumes of 50 aqueous salt solutions. Although our
in correlating the activity coefficients of most electro- equation of state contains more parameters than theirs,
lytes at room temperature. The correlations for the 1-1, we are able to calculate activity coefficients with a much
1-2, and 3-1 type salts were especially good. For most greater accuracy (0.83% AAD for 116 salt solutions).
of the salt solutions, our model has a lower percentage Furthermore, the Jin and Donohue equation of state is
of average absolute deviation (% AAD) than the NRTL fairly complicated, containing 10 terms, while our
activity coefficient model. It should be remembered simpler equation contains only 3 terms. Fürst and
though that the NRTL model has two binary parameters Renon11 have developed a fundamental equation of state
for each salt and one hydration number for each ion with a nonelectrolyte part based on a Redlich-Kwong-
while our model has three parameters for each salt. Soave type equation of state, a short-range ionic part,
With the same number of parameters, our model has and a long-range electrostatic part based on the MSA.
the advantage of giving better correlations of activity Their model contains a total of four parameters per salt
coefficients. (cation and anion diameters, an interaction parameter
There were a few salts for which the equation of state between cation and solvent, and an interaction param-
did not accurately represent the activity coefficients over eter between cation and anion). Furthermore, they
the entire concentration range of the experimental data. developed correlations between three of the parameters,
Specifically, the correlations for several of the zinc and essentially reducing their equation of state to a one-
cadmium salts had relatively high % AAD values. parameter model. They report an average root-mean-
Because our model does not take into account the partial square deviation of 1.4% in osmotic coefficients for 87
dissociation of salts or the formation of ion pairs and different aqueous salt solutions. Our model does better
higher complexes, it is possible that these effects might with an average deviation of 0.46% for the osmotic
contribute to the high errors for these salts. Chen et coefficients of 116 salt solutions. Neither Jin and
al.5 were able to accurately capture the correct trend in Donohue nor Fürst and Renon report results for tem-
activity coefficients for several of these systems by peratures and pressures other than 25 °C and 1 bar.
including partial dissociation in their NRTL model. The One of the advantages of our equation of state is that,
activity coefficient errors for the systems for which they as we show below, it has been tested and shown to be
included this were less than 2% compared to about 10% accurate at much higher temperatures and pressures.
without dissociation. We plan to include partial dis- Thermodynamic Properties over a Wide Range
sociation and ion-pair formation equilibria in future of Temperature and Pressure. There have been
work on our equation of state. many previous studies of the thermodynamics of elec-
The equation of state performs well in correlating the trolyte solutions over wide ranges of temperature and
activity coefficients for highly nonideal electrolyte solu- pressure. For example, Archer32,33 compiled thermody-
tions. Figure 2 shows the activity coefficient results from namic data for aqueous NaCl solutions from many
our model for two electrolytes: UO2(ClO4)2 that exhibits sources and generated a comprehensive and very ac-
large positive deviations from ideality and UO2SO4 that curate equation of state valid from 0 to 300 °C specific
exhibits large negative deviations. Although we only fit to NaCl. Archer34 also made a similar study for aqueous
the equation of state parameters to data up to 6 m, our NaBr solutions. Holmes et al.35 compiled data for CaCl2
model also does well when extrapolated to much higher solutions and used a Pitzer ion-interaction model to
concentrations. Figure 3 shows the activity coefficient develop a general equation valid from 0 to 250 °C.
predictions of the equation of state over a wide range Ananthaswamy et al.36 developed a model for the
of concentrations for several electrolytes. Even up to 15 thermodynamic properties of aqueous CaCl2 in the
m, our equation of state can still accurately predict temperature range 0-100 °C. For aqueous solutions of
activity coefficients. the alkali-metal sulfates, Holmes and Mesmer37 have
Several others have reported equation of state predic- developed a model valid from 0 to 250 °C. Several other
tions of the properties of aqueous electrolyte solutions studies of interest have been reported.38-41 However,
at 25 °C and 1 bar. Jin and Donohue9,10 have developed these models have several disadvantages. They have a
a fundamental equation of state based on the perturbed- large number of parameters, are often difficult to extend
anisotropic-chain theory that contains only one adjust- 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 Table 3. Best-Fit Parameter Values for Electrolytes
Data Regression
electrolyte a(1) a(2) b (cm3/mol) ksw σ(1) σ(2)
T range pressure molality
NaCl 0.932 -207.9 9.90 -0.2540 5.695 -551.3
electrolyte property (°C) range (bar) range ref
NaBr 1.360 -280.3 14.3 -0.1888 8.114 -1242
NaCl γ(, Φ, F 0-300 1-85.8 0-6 33 CaCl2 1.694 -326.7 13.4 -0.4461 6.871 -897.7
∆m h
h G
25 1 0 29 Li2SO4 1.619 -352.7 9.46 -0.1167 4.835 -358.1
NaBr γ(, Φ, F 0-325 1-120 0-8 34 Na2SO4 2.737 -661.3 15.5 0.1670 4.850 -432.2
∆m h 25 1 0 29 K2SO4 1.520 -392.2 10.6 0.2020 4.566 -388.7
h G
CaCl2 γ(, Φ 25-250 1-39.8 0-4 35 Cs2SO4 1.005 -247.3 10.2 0.1215 4.132 -168.6
F 50-325 1-389 0-6.4 41
∆m h
h G
25 1 0 29 state only to activity coefficient data at 25 °C, while in
Li2SO4 γ(, Φ 0-225 1-25.6 0-3 37 the second case, we fitted to activity coefficients, osmotic
Na2SO4 γ(, Φ 0-225 1-25.6 0-3 37 coefficients, free energies of hydration, and densities
K2SO4 γ(, Φ 0-225 1-25.6 0-2.5 37 from 0 to 300 °C. In the second case, we also added an
Cs2SO4 γ(, Φ 0-225 1-25.6 0-6 37
additional adjustable parameter, ksw, which has a large
effect on the value of the van der Waals attraction
developed, and are also relatively complicated. In parameter, a. The excluded-volume parameter, b, is also
contrast, the model presented here is relatively simple, different in each of the cases because the solution
has a small number of parameters, and can be used for densities, which were not used in the 25 °C fit, are
any type of system and with any number of components. strongly dependent on the values of this parameter. The
To establish this, we examined the following seven equation of state does not give very good predictions of
aqueous salt solutions over wide temperature, pressure, densities at 25 °C with the set of parameters in Table
and concentration ranges for which comprehensive 1, while it does give reasonable predictions with the set
studies have been previously performed: NaCl, which of parameters in Table 3. The ion diameter, σ, is also
is a premier example of a 1-1 type electrolyte, NaBr, changed slightly in each case, although not as much as
CaCl2, which is a premier example of a 2-1 type the other two parameters. The different data types and
electrolyte, and four alkali-metal sulfates, which are different ranges of conditions have a significant effect
good examples of 1-2 type electrolytes. For NaCl, NaBr, on the set of best-fit parameters.
and CaCl2, we used activity coefficients, osmotic coef-
The results show that all of the average ion diameters
ficients, densities, and free energies of hydration in a
increase with increasing temperature. As mentioned, we
nonlinear least-squares regression analysis to determine
are using average ion cavity diameters in the equation
the set of best-fit equation of state parameters for these
of state rather than bare ion diameters, so the cavity
salts. For the four alkali-metal sulfates, we only used
diameter accounts for the diameter of the bare ion as
available activity and osmotic coefficients in the least-
well as the distance to the first solvation shell of the
squares analysis. Table 2 lists the sources of data and
ion. Our results agree with previous work42 that has
the range of conditions of each data type used in the
shown that the average cation-oxygen distance in
regression analysis.
aqueous electrolyte solutions increases with increasing
As mentioned in the previous section, for each salt
temperature.
species, the parameters for the cations and anions are
The results from the least-squares regression calcula-
given a single average value. The van der Waals
tions are given in the following figures and tables.
attraction parameter and the average ion diameter for
Figures 4-9 show the activity coefficients and osmotic
each electrolyte were given the following simple tem-
coefficients of NaCl, NaBr, and CaCl2 as functions of
perature dependence to improve the fit over a wide
temperature and salt molality. The equation of state
temperature range:
results for the densities of the electrolyte solutions are
shown in Figures 10-12. The lines result from the
a ) a(1) + a(2)/T (26) equation of state, and the points are the experimental
data reported in the references in Table 2. In these
σ ) σ(1) + σ(2)/T (27) results for temperatures of less than or equal to 100 °C,
the pressure is 1 bar, and for temperatures greater than
We found that including the salt-water binary inter- 100 °C, the pressure is equal to the saturation pressure
action parameter, ksw, as a temperature-independent at the specified temperature.
adjustable parameter greatly improved the equation of The equation of state accurately reproduces the
state correlations over the entire temperature range for activity and osmotic coefficients in the 0-300 °C tem-
each of the seven systems. Furthermore, the volume perature range and the 1-120 bar pressure range of
translation parameter, c, did not have a significant effect the NaCl, NaBr, and CaCl2 solutions. The largest
on the results. Therefore, we set this parameter equal deviations occur at the lower temperatures where the
to zero for all salts. The set of best-fit parameters is activity and osmotic coefficients have the largest values.
given in Table 3 where a is in Pa‚m6/mol and σ is in Å. Although thermodynamic data at higher concentrations
Each of the parameters has a value much greater than than those shown in the figures do exist, these data were
the range of its 95% confidence limit. In general, the not used in the data regression. This is because, as the
set of parameters in Table 3 calculated at 25 °C will be concentration of salt increases, ions may associate to
different than that reported in Table 1 for the same salt form ion pairs, and this is not presently included in the
at 25 °C. For example, for NaCl, Table 1 gives a ) 2.461 equation of state. Ion-pair formation becomes more
while Table 3 gives a ) 0.235 at 25 °C. This is not a pronounced at higher temperatures where the dielectric
failure of the model, nor is it indicative of too many constant is small. For example, activity coefficient data
parameters in the model. The differences arise because for aqueous CaCl2 solutions from 0 to 300 °C are
we are fitting to different sets of experimental data in reported for concentrations up to 30 m,38 but it is
each case. In the first case, we fitted the equation of believed that the calcium and chloride ions form ion
3292 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

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 Table 4. Free Energies of Hydration Calculated from the
temperature range. Modeling of this effect would require Equation of State at 25 °C and 1 bar
the use of ion-pair formation equilibrium constants in exptl ∆m h
h G calcd ∆m h
h G % deviation
the equation of state. These constants have been electrolyte (kJ/mol) (kJ/mol) of EOS
measured for some electrolytes,43,44 and work is under- NaCl -728.5 -731.5 0.41
way to extend the equation of state developed here to NaBr -714.6 -719.1 0.63
solutions that include ion pairs. CaCl2 -2227.6 -2237.8 0.46
The equation of state captures the correct trend in
the densities of NaCl, NaBr, and CaCl2 solutions, Experimental values were taken from the tables of
although the predictions are less accurate as the con- Friedman and Krishnan,29 and the calculated values are
centration increases. With cubic equations of state such from eq 21.
as the Peng-Robinson, it is often difficult to accurately Finally, the results for the four alkali-metal sulfate
predict densities. In addition to activity coefficients, solutions are presented in Figures 13-20. The lines are
osmotic coefficients, and densities, the model was also from the equation of state, and the points are the data
fit to free energies of hydration at 25 °C and 1 bar. The calculated from the equations of Holmes and Mesmer.37
results of the fit to the experimental data are shown in Again, for temperatures of less than or equal to 100 °C,
Table 4. The values reported are the sums of the free the pressure is 1 bar, and for temperatures greater than
energies of hydration of the individual ions of each salt. 100 °C, the saturation pressure is used at the specified
 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3293

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.

Table 5. AADs of Equation of State Predictions

temperature. The equation of state reproduces the
thermodynamic properties of all four salt solutions quite % AAD
well over the entire range of temperature and composi- electrolyte γ( Φ F ∆m h
h G
tion. NaCl 2.01 1.03 1.51 0.41
The AADs of the equation of state calculations for NaBr 2.60 1.63 1.24 0.63
each property of the seven salt solutions are listed in CaCl2 4.45 2.83 1.97 0.46
Li2SO4 3.45 1.42 a a
Table 5. These deviations are reasonable considering the Na2SO4 2.57 1.97 a a
small number of adjustable parameters of the equation K2SO4 1.59 1.07 a a
of state and the ranges of temperature, pressure, and Cs2SO4 4.09 2.63 a a
composition investigated. When the equation of state a Equation of state not fitted to these data.
is fitted to a system for which only a limited amount of
data are available, it is possible to reduce the number
With this correlation and with the other five parameters
of adjustable parameters by correlating the excluded-
in Table 3 readjusted, the average deviations in activity
volume parameter, b, with the ion diameter, σ, using
coefficients, osmotic coefficients, densities, and free
the following equation: 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
b ) 0.802(σ(1) + σ(2)/298.15)3 - 32.761 cm3/mol (28) 0.50% to 0.86%, respectively. Hence, it is reasonable to
3294 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

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.

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 equa-
tion 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 analyti-
cal 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
Figure 18. Osmotic coefficient of aqueous Na2SO4. properties of other salt solutions, we can only compare
results for the activity coefficient of NaCl-water solu-
use this correlation if the experimental data are scarce. tions. A comparison between their graphs and Figure 4
Otherwise, it is recommended to treat all of the param- shows that we achieve a similar accuracy with our
eters in Table 3 as adjustable to obtain better correla- equation of state.
tions. Our proposed equation of state does have several
Wu and Prausnitz12 have developed a similar equa- limitations. The Born contribution to the equation of
tion of state that includes a Peng-Robinson term, a state gives the free energy change for charging a hard
Born term, an MSA term, and a term from the SAFT sphere in a dielectric medium. Although the Born
theory to account for hydrogen bonding in the solution. equation provides good predictions of the free energies
Their model contains a total of five adjustable param- of hydration for most systems, it fails at very low
eters for water and two parameters for each salt species, solution densities. The reason for this is that the Born
with one of the salt parameters being temperature- equation ignores the compressibility of the solvent. In
dependent. Although their equation of state might electrolyte solutions, the electric field of the ions will
appear to be quite similar to ours, there are a number cause the solvent molecules to be compressed, thus
of differences between the two. First, their Born term changing the dielectric constant of the solution. In high-
does not account for the Helmholtz free energy on density solutions, the effect of this compression of the
discharging ions in a vacuum because their reference solvent by the ions is small and can be neglected, while
system consists of a mixture of neutral particles while at low densities (F < about 0.5 g/cm3), the effect is large
ours consists of an ideal gas mixture of ions and solvent and cannot be ignored. Wood et al.45 have developed a
 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3295

Figure 21. Osmotic coefficient of aqueous (1 - y)NaCl + yLiCl Figure 22. Osmotic coefficient of aqueous (1 - y)NaCl + yCaCl2
at 25 °C and 1 bar. Equation of state parameters for each salt are at 110 °C and 1.45 bar. Equation of state parameters for each salt
taken from Table 1. Experimental data are from Robinson et al.46 are taken from Table 3. Experimental data are from Holmes et
al.47
compressible continuum model that takes this into
account. Their model accurately predicts the coordina- al.,47 and the lines are the equation of state predictions
tion number of ions in low-density solutions where the with parameters for each salt taken from Table 3. The
Born model fails but is not well-suited for use in an equation of state does an excellent job predicting the
engineering equation of state because it requires nu- osmotic coefficients of both mixed-salt solutions without
merical integration. We, therefore, suggest that our any additional parameters. A more comprehensive study
proposed equation of state not be used to calculate free of our equation of state applied to mixed-salt systems
energies of hydration for very low-density solutions. will be given in the future.
A similar problem arises when dealing with electro-
lyte solutions of very high concentration. The model of Conclusions
an electrolyte solution as a mixture of ions in a dielectric
continuum is not valid at very high ion concentrations The aim of this work has been to develop a funda-
as a result of using the primitive model of electrolyte mental equation of state for fully ionized electrolyte
solutions. An improvement to the equation of state solutions that is applicable over wide ranges of temper-
would be to use a model in which the solvent is treated ature, pressure, and composition. The new equation of
as a system of discrete molecules, not as a continuum. state is successful in correlating the activity coefficients
However, an equation of state based on such a model of most aqueous electrolyte solutions at 25 °C and 1 bar
would be much more complex than our current equation and concentrations up to 6 m, and it is also accurate
of state. Finally, we fitted the equation of state param- for extrapolating activity coefficients to higher concen-
eters to activity and osmotic coefficients and to densities trations. The model is also successful in correlating the
over a wide temperature range and to concentrations thermodynamic properties of aqueous NaCl, NaBr,
up to 8 m. As we showed for several aqueous electrolyte CaCl2, and alkali-metal sulfate solutions over wide
solutions at 25 °C and 1 bar, we were able to extrapolate temperature and pressure ranges.
activity coefficients to concentrations as high as 15 m. Our equation of state differs significantly from equa-
Nonetheless, we suggest that, because of the limitations tions of state that are valid for only one salt across a
of the primitive model at very high concentrations, our wide temperature range in that it is relatively simple,
proposed equation of state should not be used for has few adjustable parameters, can be used with any
systems with salt concentrations greater than about 15 salt, and should be applicable to mixtures of salts and
m. mixed-solvent systems. Furthermore, because we used
Mixed Electrolyte Solutions. As we mentioned in an equation based on the total Helmholtz free energy
the beginning of this paper, our equation of state should instead of an activity coefficient model, we are also able
be applicable to both single salt systems and systems to calculate densities as well as the pressure depend-
with mixed salts. Although we have obtained param- encies of activity and osmotic coefficients. The number
eters specific to each salt using only experimental data of adjustable parameters in our equation is not unrea-
for single salt solutions, we can use these parameters sonable, given the large range of temperature, pressure,
directly to predict the thermodynamic properties of and concentration over which the equation is valid.
mixed salt solutions. The predictions of our equation of Furthermore, only two of the parameters are temper-
state for two aqueous mixed-salt systems are shown in ature-dependent, and the temperature dependences are
the following two figures. Figure 21 shows the osmotic relatively simple. A limitation of the equation of state
coefficient of aqueous mixtures of NaCl and LiCl at 25 is that it cannot be used for systems with significant
°C and 1 bar as a function of the fraction of LiCl. The ion pairing (systems of weak electrolytes) unless the
results are given for four different isopiestic molalities. necessary chemical equilibria are included. Because of
The points are the experimental data taken from its relative simplicity, the new equation should be well
Robinson et al.,46 and the lines are the equation of state suited for use in engineering applications. These results
predictions with parameters for each salt taken from presented are encouraging, and further development of
Table 1. Figure 22 shows the osmotic coefficient of this model is underway to extend its range of applicabil-
aqueous mixtures of NaCl and CaCl2 at 110 °C and 1.45 ity to other systems including salt mixtures, mixed-
bar as a function of total molality (mNaCl + mCaCl2). The solvent systems, and systems with both ions and ion
points are the experimental data taken from Holmes et pairs.
3296 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002

Acknowledgment (6) Pitzer, K. S.; Peiper, J. C.; Busey, R. H. Thermodynamic

Properties of Aqueous Sodium Chloride Solutions. J. Phys. Chem.
The authors thank Ricardo Macias for some explor- Ref. Data 1984, 13, 1.
atory calculations. Financial support for this research (7) Anderko, A.; Pitzer, K. Equation-of-State Representation of
was provided by the National Science Foundation (CTS- Phase Equilibria and Volumetric Properties of the System NaCl-
0083709) and the Department of Energy (DE-FG02- H2O above 573 K. Geochim. Cosmochim. Acta 1993, 57, 1657.
(8) Lvov, S. N.; Wood, R. H. Equation of State of Aqueous NaCl
85ER-13436 and DE-FG02-89ER-14080). Solutions over a Wide Range of Temperatures, Pressures and
Concentrations. Fluid Phase Equilib. 1990, 60, 273.
Nomenclature (9) Jin, G.; Donohue, M. D. An Equation of State for Electrolyte
Solutions. 1. Aqueous Systems Containing Strong Electrolytes.
a ) van der Waals attraction parameter Ind. Eng. Chem. Res. 1988, 27, 1073.
A ) Helmholtz free energy (10) Jin, G.; Donohue, M. D. An Equation of State for Electro-
b ) van der Waals excluded-volume parameter lyte Solutions. 2. Single Volatile Weak Electrolytes in Water. Ind.
c ) volume translation parameter Eng. Chem. Res. 1988, 27, 1737.
e ) elementary charge (11) Fürst, W.; Renon, H. Representation of Excess Properties
of Electrolyte Solutions Using a New Equation of State. AIChE J.
G ) Gibbs free energy
1993, 39, 335.
k ) binary interaction parameter (12) Wu, J.; Prausnitz, J. M. Phase Equilibria for Systems
m ) molality Containing Hydrocarbons, Water, and Salt: An Extended Peng-
M ) molecular mass Robinson Equation of State. Ind. Eng. Chem. Res. 1998, 37, 1634.
n ) number of moles (13) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equa-
N ) total number of moles in the system tion of State. Ind. Eng. Chem. Fundam. 1976, 15, 59.
Na ) Avogadro’s number (14) Peneloux, A.; Rauzy, E.; Freze, R. A Consistent Correction
P ) pressure for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7.
R ) gas constant (15) Tsai, J.; Chen, Y. Application of a Volume-Translated
T ) temperature Peng-Robinson Equation of State on Vapor-Liquid Equilibrium
Calculations. Fluid Phase Equilib. 1998, 145, 193.
V ) volume (16) de Sant’Ana, H. B.; Ungerer, P.; de Hemptinne, J. C.
Zi ) ion charge number Evaluation of an Improved Volume Translation for the Prediction
Z ) compressibility factor of Hydrocarbon Volumetric Properties. Fluid Phase Equilib. 1999,
154, 193.
Greek Symbols (17) Born, M. Volumen und Hydrationswarme der Ionen. Z.
∆h ) change on hydration Phys. 1920, 1, 45.
 ) dielectric constant (18) Latimer, W. M.; Pitzer, K. S.; Slansky, C. M. The Free
0 ) permittivity of free space Energy of Hydration of Gaseous Ions and the Absolute Potential
γ ) activity coefficient of the Normal Calomel Electrode. J. Chem. Phys. 1939, 7, 108.
(19) Rashin, A. A.; Honig, B. Reevaluation of the Born Model
Γ ) MSA screening parameter of Ion Hydration. J. Phys. Chem. 1985, 89, 5588.
κ ) Debye screening length (20) Waisman, E.; Lebowitz, J. L. Exact Solution of an Integral
ν ) number of molecules in one molecule of salt Equation for Structure of a Primitive Model of Electrolytes. J.
φ ) fugacity coefficient Chem. Phys. 1970, 52, 4307.
Φ ) osmotic coefficient (21) Blum, L. Mean Spherical Model for Asymmetric Electro-
F ) density lytes. I. Method of Solution. Mol. Phys. 1975, 30, 1529.
σ ) ion diameter (22) Blum, L.; Hoye, J. S. Mean Spherical Model for Asym-
metric Electrolytes. 2. Thermodynamic Properties and the Pair
Superscripts Correlation Function. J. Phys. Chem. 1977, 81, 1311.
(23) Harvey, A.; Copeman, T.; Prausnitz, J. Explicit Ap-
Born ) Born contribution to the equation of state proximation of the Mean Spherical Approximation for Electrolyte
IGM ) ideal gas mixture Systems with Unequal Ion Sizes. J. Phys. Chem. 1988, 92, 6432.
m ) hypothetical ideal 1 m standard state (24) Haynes, C. A.; Newman, J. On Converting from the
MSA ) mean spherical approximation contribution to the McMillan-Mayer Framework I. Single-Solvent System. Fluid
equation of state Phase Equilib. 1998, 145, 255.
PR ) Peng-Robinson contribution to the equation of state (25) Sandler, S. I. Chemical and Engineering Thermodynamics,
3rd ed.; Wiley: New York, 1999.
Subscripts (26) International Association for the Properties of Water and
chg ) charging process Steam. Release on the IAPWS formulation 1995 for the thermo-
dynamic properties of ordinary water substance for general
dis ) discharging process
and scientific use, Fredericia, Denmark, Sept 1996 (see www.
ij ) binary pair of i and j IAPWS.org for downloadable version).
r ) reduced quantity (e.g., temperature) (27) Uematsu, M.; Franck, E. U. Static Dielectric Constant of
( ) mean ionic property Water and Steam. J. Phys. Chem. Ref. Data 1980, 9, 1291.
(28) Fernández, D. P.; Goodwin, A. R. H.; Lemmon, E. W.;
Literature Cited Levelt Sengers, J. M. H.; Williams, R. C. A Formulation for the
Static Permittivity of Water and Steam at Temperatures from 238
(1) Liu, Y.; Watanasiri, S. Successfully Simulate Electrolyte to 873 K at Pressures up to 1200 MPa, Including Derivatives and
Systems. Chem. Eng. Prog. 1999, 10, 25. Debye-Hückel Coefficients. J. Phys. Chem. Ref. Data 1997, 26,
(2) Chen, C. C.; Britt, H. I.; Boston, J. F.; Evans, L. B. Local 1125.
Composition Model for Excess Gibbs Energy of Electrolyte Sys- (29) Friedman, H. L.; Krishnan, C. V. In Water, A Compre-
tems. AIChE J. 1982, 28, 588. hensive Treatise; Franks, F., Ed.; Plenum Press: New York, 1973;
(3) Chen, C. C.; Evans, L. B. A Local Composition Model for Vol. 3, Chapter 1.
the Excess Gibbs Energy of Aqueous Electrolyte Systems. AIChE (30) Harvey, A. H.; Prausnitz, J. M. Dielectric Constants of
J. 1986, 32, 444. Fluid Mixtures over a Wide Range of Temperature and Density.
(4) Mock, B.; Evans, L. B.; Chen, C. C. Thermodynamic J. Solution Chem. 1987, 16, 857.
Representation of Phase Equilibria of Mixed-Solvent Electrolyte (31) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions;
Systems. AIChE J. 1986, 32, 1655. Butterworth Scientific Publishing: London, 1959.
(5) Chen, C. C.; Mathias, P. M.; Orbey, H. Use of Hydration (32) Archer, D. Thermodynamic Properties of the NaCl + H2O
and Dissociation Chemistries with the Electrolyte-NRTL Model. System. 1. Thermodynamic Properties of NaCl(cr). J. Phys. Chem.
AIChE J. 1999, 45, 1576. Ref. Data 1992, 21, 1.
 Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3297

(33) Archer, D. Thermodynamic Properties of the NaCl + H2O (42) Flanagin, L. W.; Balbuena, P. B.; Johnston, K. P.; Rossky,
System. II. Thermodynamic Properties of the NaCl(aq), NaCl- P. J. Ion Solvation in Supercritical Water Based on an Adsorption
H2O(cr), and Phase Equilibria. J. Phys. Chem. Ref. Data 1992, Analogy. J. Phys Chem. B 1997, 101, 7998.
21, 793. (43) Gruszkiewicz, M. S.; Wood, R. H. Conductance of Dilute
(34) Archer, D. Thermodynamic Properties of the NaBr + H2O LiCl, NaCl, NaBr, and CsBr Solutions in Supercritical Water using
System. J. Phys. Chem. Ref. Data 1992, 20, 509. a Flow Conductance Cell. J. Phys. Chem. 1997, 101, 6549.
(35) Holmes, H. F.; Busey, R. H.; Simonson, J. M.; Mesmer, R. (44) Zimmerman, G. H.; Gruszkiewicz, M. S.; Wood, R. H. New
E. CaCl2(aq) at Elevated Temperatures. Enthalpies of dilution, Apparatus for Conductance Measurements at High Tempera-
Isopiestic Molalities, and Thermodynamic Properties. J. Chem. tures: Conductance of Aqueous Solutions of LiCl, NaCl, NaBr,
Thermodyn. 1994, 26, 271. and CsBr at 28 MPa and Water Densities from 700 to 260 kg/m3.
(36) Ananthaswamy, J.; Atkinson, G. Thermodynamics of J. Phys. Chem. 1995, 99, 11612.
Concentrated Electrolyte Mixtures. 5. A Review of the Thermo- (45) Wood, R. H.; Carter, R. W.; Quint, J. R.; Majer, V.;
dynamic Properties of Aqueous Calcium Chloride in the Temper- Thompson, P. T.; Boccio, J. R. Aqueous Electrolytes at High
ature Range 273.15-373.15 K. J. Chem. Eng. Data 1985, 30, 120. Temperatures: Comparison of Experiment with Simulation and
(37) Holmes, H. F.; Mesmer, R. E. Thermodynamics of Aqueous Continuum Models. J. Chem. Thermodyn. 1994, 26, 225.
Solutions of the Alkali Metal Sulfates. J. Solution Chem. 1986, (46) Robinson, R. A.; Wood, R. H.; Reilly, P. J. Calculations of
15, 495. Excess Gibbs Energies and Activity Coefficients from Isopiestic
(38) Pitzer, K. S.; Oakes, C. S. Thermodynamics of Calcium Measurements on Mixtures of Lithium and Sodium Salts. J. Chem.
Chloride in Concentrated Aqueous Solutions and in Crystals. J. Thermodyn. 1971, 3, 461.
Chem. Eng. Data 1994, 39, 553. (47) Holmes, H. F.; Baes, C. F.; Mesmer, R. E. Isopiestic Studies
(39) Jiang, S.; Pitzer, K. S. Phase Equilibria and Volumetric at Elevated Temperatures. III. {(1 - y)NaCl + yCaCl2} J. Chem.
Properties of Aqueous CaCl2 by an Equation of State. AIChE J. Thermodyn. 1981, 13, 101.
1996, 42, 585.
(40) Rard, J. A.; Archer, D. G. Isopiestic Investigation of the
Osmotic and Activity Coefficients of Aqueous NaBr and the
Solubility of NaBr‚2H2O(cr) at 298.15 K: Thermodynamic Proper- Received for review December 14, 2001
ties of the NaBr + H2O System over Wide Ranges of Temperature Revised manuscript received April 1, 2002
and Pressure. J. Chem. Eng. Data 1995, 40, 170. Accepted April 3, 2002
(41) Gates, J. A.; Wood, R. H. Density and Apparent Molar
Volume of Aqueous CaCl2 at 323-600 K. J. Chem. Eng. Data 1989,
34, 53. IE011016G