Thermodynamic Model for Aqueous Electrolyte Solutions with Partial

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A Thermodynamic Model for Aqueous Electrolyte Solutions with Partial Ionization Ricardo Macías-Salinas, Juan Ramon Avendaño-Gómez, Fernando García-Sánchez, and Manuela Díaz-Cruz Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie400764t • Publication Date (Web): 04 Jun 2013 Downloaded from http://pubs.acs.org on June 12, 2013

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Industrial & Engineering Chemistry Research

A Thermodynamic Model for Aqueous Electrolyte Solutions with Partial Ionization Ricardo Macías-Salinas(1)*, Juan R. Avendaño-Gómez(1), Fernando García-Sánchez(2), and Manuela Díaz-Cruz(3)

(1) ESIQIE, Departamento de Ingeniería Química, Instituto Politécnico Nacional, Zacatenco, México, D.F. 07738, MEXICO. E-mail: [email protected], [email protected] (2) Laboratorio de Termodinámica, Programa de Investigación en Ingeniería Molecular, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, México, D.F., 07730, MEXICO. E-mail: [email protected] (3) ESIQIE, Departamento de Ingeniería Metalurgica y de Materiales, Instituto Politécnico Nacional, Zacatenco, México, D.F. 07738, MEXICO. E-mail: [email protected]

Ms. No. ie-2013-00764t Submitted for Publication to the Ind. Eng. Chem. Res. Second Version: May 2013

___________ *

Author to whom correspondence should be addressed, E-mail: [email protected]

ACS Paragon Plus Environment


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Abstract An equation of state has been developed to describe the thermodynamic properties of single electrolytes in water within a wide range of temperatures from 25 °C to near the critical point of the solvent. The new equation of state was obtained from an analytical expression of the Helmholtz free energy containing three major contributions: (1) a discrete-solvent term to account for short-range interactions between uncharged particles based on the Peng-Robinson equation of state, (2) an ion charging term described by the continuum-solvent model of Born, and (3) a charge-charge interaction term given by the explicit mean-spherical-approximation (MSA) expression. The thermodynamic model proposed here incorporates chemical equilibrium for the dissolved electrolyte allowing the calculation of the corresponding degree of dissociation of the salt at different temperatures. The present equation of state was applied to the representation of mean ionic activity coefficients, osmotic coefficients, standard free energies of hydration of ions, and densities for NaCl, CaCl2, K2SO4, and MgSO4 salts in water over a wide range of temperatures and salt molalities. The results indicated a good agreement between the experimental data and those calculated using the present equation of state.


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Introduction

A precise knowledge of the thermodynamic properties of electrolyte solutions is paramount in many industrial and natural processes. Some examples of such processes include precipitation and crystallization in geothermal-energy systems, partitioning in biochemical systems, desalination of water, salting-in and salting-out effects in distillation and extraction, food processing, etc. As a matter of fact, a huge amount of experimental thermodynamic properties for electrolyte solutions have appeared in the literature, predominately in the last forty years. As far as mathematical models for electrolyte solutions are concerned, the literature is also rich in them, however, much of these modeling efforts are not consistent because of the use of different standard states including not-well-defined extensions of thermodynamic frameworks usually used for non-electrolytes to contain also electrolytes.

Many theoretical and semi-empirical models have been developed for representing the thermodynamic properties of electrolyte solutions. There are various excellent reviews on electrolyte solution models available in the literature1-4 including the most recent one by Anderko et al.5. In general, models for electrolyte solutions fall into two main categories: (1) local-composition models based on the excess Gibbs energy6-14, and (2) equations of state based on the Helmholtz energy15-24. Electrolyte models based on the Helmholtz energy, unlike those based on the local-composition concept, can be applied over wider temperature and pressure ranges; they are also able to yield volumetric properties such as densities of the electrolyte solution.

For the case of electrolyte models based on equations of state, the majority of them treat the electrolyte as completely dissociated (or fully ionized)15-17,19,21,22 while some of them assume that the electrolyte is undissociated or completely


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associated18,20,23,24. On the other hand, partial dissociation (or ionization) is an important component, for example, in the chemistry of electrolyte solutions at high temperatures where its practical importance can be found in hydrothermal and geothermal processes involving “acid” aqueous solutions containing for instance high NaCl concentrations. Very few electrolyte models have been so far reported in the literature that assume partial dissociation or ionization of the salt in solution. For example, Cruz and Renon6, and Mock et al.9 introduced partial dissociation into their modeling efforts, however, their electrolyte solution models are given in terms of the excess Gibbs free energy; they do not take into account the effect of pressure being thus unable to calculate the solution density.

The purpose of the present work is therefore to develop a modeling approach that can correlate and/or predict thermodynamic and volumetric properties of a partially ionized electrolyte in water within wide ranges of temperature, pressure, and salt molality. Unlike previous individual-ions and salt-based models, the present approach incorporates chemical equilibrium for the dissolved electrolyte thus making possible to estimate the extent of dissociation of the electrolyte at diverse temperatures and concentrations.

Description of Thermodynamic Model

The present equation of state is based on an analytical expression for the molar Helmholtz energy: A(T, V, n) where T is the temperature, V is the total volume, and n is the number of moles of each species in solution (ions, undissociated salt and solvent). This approach allows one to determine all thermodynamic properties (e.g. pressure, chemical potentials, fugacity coefficients, density, etc.) from a single expression. The molar Helmholtz energy is the sum of various contributions required to form the ionic


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solution at the pressure and density of interest. Figure 1 shows the isothermal path to form the solution containing the partially dissolved electrolyte from an ideal gas reference state to the actual state at system temperature and pressure. The isothermal path shown in Figure 1 actually represents a charging cycle to form the electrolyte solution; such a charging cycle was originally proposed by Simon et al.19. A similar charging cycle was also devised by Myers et al.23 during the development of their equation of state for electrolyte solutions.

The change in Helmholtz energy for this isothermal charging cycle is determined as follows: (1) charged ions, undissociated salt, and water in the ideal gas reference state (Pref and Vref) are mixed and expanded as ideal gases to a very low pressure (P→0 and V→∞). The pressure is so low that interactions between the particles can be neglected; this change results in ∆Aideal. (2) All ions are discharged; the change in Helmholtz energy for such a discharge is given by the Born equation25 using the dielectric constant of vacuum: ∆AεBorn . (3) The mixture of uncharged particles is then compressed to system vac . volume Vsys; an equation of state for nonelectrolytes is used to represent this transition. In this study, the Peng-Robinson cubic equation of state26 was used: ∆APR . (4) Finally, the ions are recharged at constant volume; the Born equation is again used to estimate this change: ∆AεBorn . One also has to take into account the interaction between the charged particles due to the electrostatic forces present in the mixture. The effect of these longrange forces are conveniently represented by the Mean Spherical Approximation (MSA) approach27: ∆AMSA .

The total change in Helmholtz energy is therefore given by:

A = ∆Aideal + ∆AεBorn + ∆APR + ∆AεBorn + ∆AMSA vac .


(1)

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The first and third terms of the above equation can be combined to give:

(

∆Aideal + ∆APR = A − Aref

)

PR

= ∆APR,r

(2)

which corresponds to the residual Helmholtz energy for the Peng-Robinson equation of state. Further, the two Born contributions can be summed to yield:

∆AεBorn + ∆AεBorn = ∆ABorn vac .

(3)

A(T,V,n ) = ∆ABorn + ∆APR, r + ∆AMSA

(4)

Eq. (1) hence reduces to:

Born Contribution The continuum hydration model developed by Born25 gives the change in Helmholtz energy when the initially uncharged species are given a permanent electric charge and become ions:

∆ABorn =

N a e02  1  ⋅  − 1 ⋅ ∑ ni Z i2 / σ i nt  ε  ions

(5)

here Na is the Avogadro’s number, nt is the total number of moles, Zi is the ionic charge, ε is the dielectric constant of the mixture which is a function of density, composition and temperature; its functionality will be further described in detail. Also, e0 is defined as follows:


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e2 4πε0

e02 =

(6)

where e is the unit electronic charge and ε0 is the permitivity in a vacuum. The ionic diameter σi in Eq. (5) is usually larger than the Pauling crystal diameter of the ion σiP and represents the solvation of the ion due to hydration effects. Ions in the Born equation are considered to be charged hard spheres in a continuum of uniform dielectric constant.

Peng-Robinson Contribution Short-range interactions between uncharged ions, undissociated salt, and water are properly accounted for by the Peng-Robinson equation of state26. This cubic equation of state was chosen because of its proven superiority in describing volumetric properties and phase equilibria of pure substances and their mixtures as compared to other equations of state of the van der Waals type. The change of Helmholtz energy for the Peng-Robinson equation of state from the reference state (ideal gas) to the actual state is given by:

∆APR,r =

V + b( 1 − 2 )  a V V − b  ln   − RTln ref  − RTln V 2 2bnt V + b( 1+ 2 )   V 

(7)

where the mixture parameters a and b are defined using simple van der Waals mixing rules:

a=

∑∑ n n i

j

ai a j (1 − ki, j )

(8)

nt2 nb b= ∑ i i nt


(9)

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and ki,j (= kj,i) in Eq. (8) is the binary interaction parameter. Pure-component attractive and repulsion parameters (ai and bi) are determined as follows:

For ions:

bi = N a ⋅

( )

π P σi 6

3

(10)

ai = f ⋅ 2 Rbi (ε / κ )i

(11)

where σiP is the Pauling crystal diameter for the ion and f is a correction factor which will be discussed later. Dispersion theory by Mavroyannis and Stephen28 was used to estimate (ε / κ )i

:

(ε / κ )i = 2.2784 ×10−11

( )

αi3 ⋅ ne / σ iP

6

(12)

where α is the polarizability and ne is the number of electrons in the ion.

For the undissociated salt, its excluded volume bs is the sum of the co-volume parameters of the cation (+) and anion (-):

bs = b+ + b−

(13)

while as is calculated using Eq. (11) with

(ε / κ )s =

(ε / κ)+ (ε / κ)−


(14)

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For water at subcritical temperatures ( 0°C < T ≤ 0.99Tc,w ), the equations given by Xu and Sandler29 were used:

(

)

(15)

(

)

(16)

aw = a0 + a1Tr + a2Tr2 ⋅ R 2Tc,2w / Pc,w

bw = b0 + b1Tr + b2Tr2 ⋅ RTc,w / Pc,w

with

a0 = 0.85802239,

a1 = -0.60228193,

a2 = 0.13341056,

b0 = 0.11334278,

b1 = -0.007292674, and b2 = 0.0221809. It is important to note that the use of a temperature-dependent bw parameter yielded significantly improved estimations of molar volumes of pure water, compared to the use of a constant b value in the original PengRobinson equation of state.

MSA Contribution Ion-ion interactions can be described by the so-called primitive model (ions are charged hard spheres in a medium of uniform dielectric constant). The mean spherical approximation (MSA)27 has been applied to the primitive model and has been used to predict ionic properties such as activity and osmotic coefficients in aqueous electrolyte solutions. The solution of the MSA approach is analytical. Unlike the Debye-Huckel approach, the MSA model gives reasonable results for activity coefficients up to moderate ion concentrations, provided that the ionic diameters σi are adjusted to fit the experimental data. The original form of the MSA model, however, involves an implicit solution of a complicated equation for the screening factor Γ that complicates to some extent differentiation of the resulting expressions with respect to volume and number of moles. For efficient and reliable calculations involving an equation of state, it is desirable to use analytical rather than numerical derivatives of the Helmholtz energy. Therefore, in


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this study, a Γ-explicit approximation of the MSA by Harvey et al.30 was used. It is mathematically simple and provides similar results to those obtained from the full MSA. This explicit approximation assumes that all ions in the mixture have the same effective diameter σ given by:

∑n σ σ= ∑n

i i

ions

(17)

i

ions

According to Harvey et al.30, the above mixing rule has been successfully tested against experimental data for ion-size ratios characteristic of most electrolytes. The expression for the Helmholtz energy is the following:

∆AMSA = −

2Γ 3 RTV ⋅ (1+1.5σΓ ) 3π N a nt

(18)

where Γ is the screening parameter of the MSA model which is used in the computation of the excess thermodynamic properties due to ion-ion interactions and is defined by:

Γ=

κ2 =

[ 1+ 2σκ − 1]

(19)

4πe02 N a2 ⋅ ∑ ni Z i2 εRTV ions

(20)

1 2σ

e0 in the above equation is given by Eq. (6) whereas κ is the reciprocal Debye screening length. As a matter of fact, the MSA model approaches the Debye-Huckel expression in the limit σ→0.


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Equation of State and Derived Ionic Properties

A pressure-explicit PVT expression is obtained via:

 ∂n ⋅ A P = − t  ∂V

  T,n

(21)

The resulting equation of state is thus:

 a  nt RT  V − b − V (V + b) + b(V − b) +   N e2  ∂ ε  n Z2  ∑i i i + P =  a2 0  σi  ε  ∂ V T , n  1  ∂ Γ    2RT 3  3   σ σ Γ 1 + Γ + 3 V + 2     3π N 2 ∂ V Γ     T ,n  a  

Peng-Robinson Born

(22)

MSA

For given T, P and number of moles n, the above equation yields different values of total volume V. The lowest value of V corresponds to the liquid phase, whereas the largest value of V is the correct solution for the vapor phase.

Fugacity and Activity Coefficients For any component i in the mixture, the fugacity coefficient can be obtained from the following expression:

 ∂n ⋅ A lnφi =  t  ∂ni

 P  ⋅ (RT)−1 − ln ref P T,V,n j (i )


(23)

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From the resulting expression for the fugacity coefficient, other thermodynamic properties can be readily determined such as the activity coefficient of species i using the following equations:

For water: γi =

φi φi0

(24)

γi =

φi φi∞

(25)

For ions and undissociated salt:

where φi0 and φi∞ are the fugacity coefficients of pure species and at infinite dilution, respectively. The above activity-coefficient expressions serve in turn to derive important ionic properties such as the mean ionic activity coefficient:

(

ν

)

ν 1/ ν

γ± = γ++ ⋅ γ- -

(26)

and the osmotic coefficient (water activity):

φ = −ln

( xw ⋅ γ w ) νmM w

(27)

where ν+ is the number of cations, ν- is the number of anions, ν = ν+ + ν-, m is the molality, and Mw is the molar mass of the solvent (water).

Standard Free Energy of Hydration

This variable is a fundamental quantity used in ion solvation and represents the free energy change required to transfer an infinitesimal amount of the ion from the gas phase


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to the aqueous solution. Friedman and Krishnan31 give tabulated values of standard free energies of hydration ( ∆ hG⊕ i ) for monoatomic ions at 25 °C and 1 bar. The free energy of hydration can be computed from the present fugacity-coefficient expressions as follows: ∞ ∆ hG⊕ ion = RT ⋅ lnφion

(28)

Density

The density of the ionic solution at given temperature, pressure, and number of moles of each species can be calculated by means of:

ρ=

1 ⋅ (n+M + + n− M − + ns M s + nw M w ) V

(29)

However, the total volume V appearing in the above equation cannot be readily calculated from the present equation of state since it requires an iterative procedure to obtain it; a convergence procedure will be later described in detail for the simultaneous determination of V and the degree of dissociation.

Dielectric Constant

Ions in solution are considered to be dielectric holes, polarizing the surrounding solvent molecules in inverse proportion to their ionic diameters σi. This leads to the decrease of the dielectric constant of the solvent as the ionic concentration is increased. The expression used for the dielectric constant of the solution (ε) was developed by Giese et al.32 and is given by:

 1− ξ  ε = 1+ (ε w − 1) ⋅    1+ ξ / 2 


(30)

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where εw is the static dielectric constant of pure water. The above equation expresses ε in terms of concentrations and sizes of the ions through ξ:

ξ=

N aπ ⋅ ∑ ni σ i3 6V ions

(31)

From Eq. (30), the partial derivatives of ε with respect to V and ni which appear in the equation of state (Eq. 22) and the fugacity-coefficient expression, respectively, can be readily determined. On the other hand, the dielectric constant of pure water and its derivatives were obtained from the equation of Uematsu and Franck33 which gives εw as a function of temperature and density over a wide range of temperatures and pressures. It is important to note that, according to Eq. (30), ε is insensitive to the concentration of the undissociated salt. Although it has been established that the presence of molecular salt indeed affects the dielectric constant of the solution32, this effect is almost negligible compared to the effect produced by the ions. Eq. (30) is therefore a good approximation of ε for any degree of dissociation.

Partial Dissociation Chemistry

An electrolyte may dissociate partially or completely in solution. When completely dissociated, the mixture contains only three species: cations, anions and solvent molecules. In the case of partial dissociation, however, molecular salt (undissociated electrolyte) is also present. In addition, cations and anions may associate to form ion pairs. Although, the molecular salt and ion pairs may have different properties, for simplicity both species are treated as a single component in this study. Accordingly,


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partial dissociation (or ionization) of an electrolyte M ν+ X ν − can be represented by the following chemical reaction:

M ν+ X ν − ⇔ ν+M ν+ + ν− X ν −

(32)

for which the true molalities of ionic species (mi) and the true molality-based activity coefficients (γi,m) are related through the dissociation equilibrium equation: ν m+ν+m−ν− γ±,m Kd = ⋅ mMX γMX,m

(33)

where Kd is the dissociation constant in a molality scale. Expressing the above equation in terms of the degree of dissociation (α):

ν

ν

K d = m ν −1ν++ ν−- ⋅

(αα±,m )ν ( 1 − α)γMX,m

(34)

where m is the stoichiometric (total) molality of the salt. The mean molal ionic activity coefficient (γ±,m) is obtained from: lnγ±,m = lnγ± − ln (1+ M wνm / 1000 )

(35)

here γ± is the true mole-fraction-based mean ionic activity coefficient given by Eq. (26). For given temperature, pressure, and salt molality, Eq. (34) must be solved iteratively to determine the true composition of all species, namely the degree of dissociation (α). To achieve this, the dissociation constant Kd should be treated as a temperature-dependent adjustable parameter. As mentioned earlier, both V and α values should be simultaneously


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determined by solving the following set of non-linear equations derived from Eqs. (22) and (34):

g1(V,α) = P − P PR − P Born − P MSA = 0 ν

ν

g 2 (V ,α) = K d ⋅ ( 1 − α)γMX,m − m ν −1ν++ ν−- ⋅ (αα±,m )ν = 0

(36)

(37)

Accounting for dissociation in ionic solutions is necessary in order to provide an adequate representation of ionic properties, particularly at elevated temperatures, and high concentrations of the electrolyte.

Adjustable Parameters of the Model

From the present equation of state (Eq. 22), it can readily deduced that the only molecular interactions between the undissociated salt and the other species (ions and water) are of the short-range type, accounted for in the Peng-Robinson part. On the other hand, specific ion-water interactions (ion hydration) are handled in an oversimplified manner by both the Peng-Robinson contribution through ki,w (= kw,i), and the Born contribution, since

 ∂ε    ≠0  ∂nw T,V,n j ( w )

(38)

This simplification is the result of a major shortcoming of the primitive-based models (Born and MSA) in that the discrete nature of the solvent is neglected.


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Based on the above, six adjustable parameters in the model were initially proposed: the dissociation constant Kd, the effective ionic diameters σ+ and σ- (both appearing in the Born and MSA terms), the binary interaction parameters between ions and water k+,w and k-,w (from the Peng-Robinson contribution), and the correction factor f for the Peng-Robinson attraction parameter of all ionic species (Eq. 11). Of all these parameters, only Kd, and f are considered to change with temperature. During a preliminary determination of model parameters by correlating experimental mean ionic activity coefficients at 25 °C measured by Hamer and Wu34 for various 1-1 electrolytes, it was found that the results are not very sensitive to the parameter k-,w, and that its value is close to zero. This parameter was therefore excluded as an adjustable parameter in the model. In most cases, this is acceptable due to the fact that the solvation of anions (anionwater interaction) is much lower than the solvation of cations.

Results and Discussion

The present modeling approach was applied to the correlation of experimental mean ionic activity coefficients (γ±,m) at different temperatures including free energies of hydration ( ∆ G⊕) at 25 ºC and 1 bar of some representative electrolytes in water. The h

i

following electrolytes were chosen for this purpose because of their large amount of experimental ionic properties available (not only at 25 °C but also at elevated temperatures where salt dissociation effects become significant): NaCl an 1-1 electrolyte, CaCl2 an 2-1 electrolyte, K2SO4 an 1-2 electrolyte, and MgSO4 an 2-2 electrolyte. Table 1 gives the four electrolytes considered in this study along with their corresponding Pauling crystal ionic diameters, temperatures and maximum molalities at which experimental data were measured, and sources of experimental mean ionic activity coefficients and osmotic


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coefficients. As a matter fact, experimental ∆ hG⊕ i , and density data for all electrolytes were obtained from Friedman and Krishnan31, and Zaytsev and Aseyev41, respectively. A least-square fit based on the Levenberg-Marquadt method was performed to obtain the model parameters at different temperatures. The minimization of the following objective functions served for this purpose:

 N  γ EoS  2 2  ∆ G ⊕ ,EoS  2 , m ,i h i  ∑ 1 − ±exp  + ∑ 1 − ⊕ ,exp  γ ∆  i =1  i =1  ± , m ,i  h Gi   min g =   N 2 EoS   γ ± , m ,i   ∑ 1 − γ exp  ± , m ,i   i =1 

at 25 ºC & 1 bar (39)

at T > 25 ºC

EoS where N is the number of experimental points, γ±exp ,m and γ± ,m are the observed and

calculated mean molal ionic activity coefficients, respectively, whereas ∆ hGi⊕ ,exp and

∆ hGi⊕ ,EoS are the experimental and calculated standard free energies of hydration of the ions (cation: i=1, anion: i=2 ), respectively. On the other hand, calculated γ±EoS ,m and

∆ hGi⊕ ,EoS values are obtained from Eqs. (35) and (28), respectively. Prior to their fit, experimental molal-based ∆ hGi,⊕m values were converted to values in a mole fraction scale by means of:

∆hGi⊕ ,exp = ∆hGi,⊕m,exp + RTln (1000 / M w )

(40)

The fit of the experimental data was first carried out at 25 °C and 1 bar. According to Eq. (39), the simultaneous fitting of measured γ± ,m and ∆ hGi⊕ values yielded the following adjusted parameters: hydrated diameters of the cation and anion (σ+ and σ-), the short-range cation-water interaction parameter (k+,w), the factor (f) appearing in the


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Peng-Robinson attraction parameter for the ionic species, and the molality-based dissociation constant (Kd). The correlating results for the four electrolyte solutions are summarized in Table 2 which shows the five adjusted parameters, and the resulting percent of absolute average deviations (AAD) between calculated and experimental γ± ,m and ∆ hGi⊕ values. It can be seen from Table 2 that the ability of the present approach in representing the experimental data is quite good with overall AAD values of 0.93% for γ± ,m , and 5.9% for ∆ hGi⊕ .

It was also found from Table 2 that the adjusted values of the ionic diameters (σ+ and σ-) are higher than their crystallographic values (except for the SO4-- anion), presumably reflecting the effect of hydration on the ionic diameter. The increase in ionic diameter was much larger for the cations than for the anions, with average values of 1.8 and 1.01 times their Pauling crystal diameters, respectively. This is in agreement with the experimental evidence in that the hydration numbers of halide ions are lower than those of alkali ions. Further, the resulting adjusted value of the correction factor f is large enough to suggest that the use of dispersion theory alone (Eq. 12) underestimates significantly the value of the short-range attractive parameter (ai) for the ionic species.

As a matter of fact, based on the optimum Kd values reported in Table 2, the present model correctly predicts essentially complete dissociation for the four electrolyte solutions (with degrees of dissociation very close to unity even at their highest molalities); most strong electrolyte completely dissociates in water at ambient temperature.

Figure 2 graphically shows the excellent agreement between calculated and observed γ± ,m values at 25 °C within a molality range of 0-6 mol/kg. The present approach also gives an accurate representation of experimental γ± ,m values within the


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dilute region ( 0 < m < 0.5 mol/kg) as demonstrated by Figure 1(b) which uses a logarithmic x-axis. Moreover, Figure 3 depicts a comparison between experimental and calculated ∆ hGi⊕ values at 25 °C and 1 bar for the four alkali ions (Na+, K+, Ca++, and Mg++), and the only halide ion (Cl-) considered in this work. As shown by this figure, the present equation of state reasonably represents the observed ∆ hGi⊕ values for the majority of the ions except for the cation K+ for which the largest percent of deviation was obtained (13.5%). It can be readily deduced from the present model that at infinite dilution only two terms of the model, the Peng-Robinson and Born terms, contribute to the value of ∆ hGi⊕ ; there is no contribution from the MSA term since it only describes ion-ion interactions. Further, the major contribution to ∆ hGi⊕ comes from the Born term. This indicates that the hydration behavior can be properly described using the Born equation by only adjusting the effective diameter of each ion.

Each term appearing in the present model contributes in a distinctive manner to the total value of γ± ,m , as shown in Figure 4 for the case of NaCl in water at 25 °C. This value (line passing through the experimental points in Figure 4) is obtained from the different contributions as follows:

PR MSA γ± ,m = γ±Born ,m ⋅ γ± ,m ⋅ γ± ,m

(41)

It can be seen from Figure 4 that the MSA contribution meets the correct boundary condition (limiting Debye-Huckel law) near the dilute region of the electrolyte solution (for m < 0.1 mol/kg), which is better visualized in Figure 4(b). As the salt concentration increases, predictions from the MSA contribution deviates from the experimental data, however, as the concentration of ions in solution increases, hydration becomes more important. This effect is properly accounted for by the Born term that along with the MSA contribution are able to represent the experimental data. At much higher


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concentrations (near saturation), short-range intermolecular forces become important as well. Accordingly, as shown in Figure 4, the Peng-Robinson term captures this behavior with a small contribution within the range of moderate to high salt molalities.

The model parameters so far obtained at 25 °C and 1 bar can also be used to predict the osmotic coefficients and the solution density at the same T and P using Eqs. (27) and (29), respectively. Figure 5 demonstrates that the predicted osmotic coefficients of aqueous NaCl, CaCl2, and K2SO4 agree quite well with the experimental data; for MgSO4, however, the model somewhat underpredicts its experimental osmotic coefficients, particularly near the dilute region. On the other hand, the predicted mass densities of aqueous NaCl, K2SO4, and MgSO4, are in good agreement with the experimental values, as shown in Figure 6. This is not the case for CaCl2 for which the model fairly underpredicted its experimental density data over the entire molality range considered.

The present model was also used to correlate γ± ,m values at temperatures higher than 25 °C (up to 300 °C). For this purpose, the second objective function given in Eq. (39) was minimized by adjusting only two temperature-dependent parameters: Kd and f; the other model parameters were set equal to those obtained at 25 °C. The choice of Kd as an adjustable parameter is justified due to the decreasing salt dissociation with temperature as a result of the decrease in the dielectric constant of the solution. Further, we choose to make the correction factor f, and therefore all attractive parameters (ai) for the ionic species, temperature dependent. This is consistent with the Peng-Robinson framework, in which the van der Waals attractive parameters of all pure components have a temperature dependency.


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Table 3 shows the first set of regressed model parameters for the case of NaCl in water at 100, 200, 300 °C. At each temperature, Table 3 also lists the calculated minimum percent of salt dissociation, and the resulting percent of AAD values between calculated and experimental γ± ,m data. Figure 7 depicts these results in graphical form revealing that there is an excellent agreement between experimental and correlated γ± ,m data with an overall AAD value of 1.1 %.

On the other hand, Figures 8 and 9 show the corresponding model predictions for osmotic coefficients, and densities of aqueous NaCl, respectively. Although the model predictions for osmotic coefficients look reasonable at the three isotherms, predictions somewhat depart from the experimental data within certain molality regions; e.g. at molalities greater than 1 mol/kg, particularly at 200 °C (see Figure 8a), and the dilute region at 300 °C (see Figure 8b). Density predictions are also in good agreement with experimental data as shown by Figure 9, however, at 300 °C the present approach increasingly underpredicts the experimental density as the molality is increased.

Figure 10 shows the qualitatively correct variation of the dissociation constant with temperature for the NaCl solution (dissociation constant should decrease with temperature). For comparison, the dissociation constants given by Helgeson42 for aqueous NaCl at five temperatures are also depicted in Figure 10 as filled circles. The first three circles (at 150, 200, and 250 °C) correspond to extrapolated data from experimental dissociation constants obtained at elevated temperatures (greater than 250 °C). According to Figure 10, the agreement between calculated and experimental dissociation constants is quite good, particularly at temperatures larger than 250 °C. The present approach also captures the correct variation of the degree of dissociation with salt molality and temperature for the NaCl solution as shown in Figure 11; however, it largely underestimates the experimental degrees of dissociation of Helgeson42 at 200 °C, and it


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slightly overestimates the experimental data of the same author at 300 °C. As shown in Figure 11, the lowest degrees of salt dissociation occur at the highest values of both temperature and molality. This is because, as the temperature increases, the dielectric constant of the solution decreases thus causing lower levels of salt dissociation. Furthermore, as the salt concentration increases, ions may associate to form ion pairs (presently, ion pairs are treated as undissociated salt). Quantified values of the aforementioned minimum degrees of dissociation are listed in Table 3. It was also found that the correction factor f increased with temperature (see Table 3), thus suggesting that both ions and undissociated salt exert larger effective attraction forces as the temperature increases.

Figure 12 shows the variation of the dissociation constant with temperature for the other three electrolyte solutions considered in this work (CaCl2, K2SO4, and MgSO4). As depicted by this figure, the present model is also capable of correctly describing the decreasing variation of the dissociation constant with temperature for the aforementioned solutions. Based on these results, the lowest levels of dissociation were exhibited by aqueous K2SO4 with 61.1%, whereas the highest degree of dissociation was displayed by the MgSO4 solution with 99.4%, that is, it practically remained completely ionized over the whole temperature range considered (25-80 °C). Comparisons with experimental dissociation data, however, are indeed needed to confirm these results. In this context, Sharygin et al.43 recently reported dissociation constants in aqueous solutions of K2SO4 from measurements of electrical conductance up to 400 °C. Figure 12 also shows four points (in red circles) representing dissociation constants for aqueous K2SO4 obtained by Sharygin et al.43 when fitting their conductance model to the experimental electrical conductivities of Noyes and Melcher44 within a temperature range of 16-306 °C. As revealed by Figure 12, the agreement between the experimental dissociation constant for K2SO4 and that calculated by the model is quite good at 100 °C; however, at higher


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temperatures, the present model largely underestimates the measured dissociation constants. Such discrepancies may be due to the fact that Noyes and Melcher44 measured their conductivity data within the dilute region (at m

Thermodynamic Model for Aqueous Electrolyte Solutions with Partial

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