Thermodynamic Model for Aqueous Electrolyte Solutions with Partial

Article pubs.acs.org/IECR

Thermodynamic Model for Aqueous Electrolyte Solutions with Partial Ionization Ricardo Macías-Salinas,*,† Juan R. Avendaño-Gómez,† Fernando García-Sánchez,‡ and Manuela Díaz-Cruz§ †

ESIQIE, Departamento de Ingeniería Química, Instituto Politécnico Nacional, Zacatenco, México D.F. 07738, Mexico Laboratorio de Termodinámica, Programa de Investigación en Ingeniería Molecular, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, México D.F. 07730, Mexico § ESIQIE, Departamento de Ingeniería en Metalurgia y Materiales, Instituto Politécnico Nacional, Zacatenco, México D.F. 07738, Mexico ‡

ABSTRACT: An equation of state has been developed to describe the thermodynamic properties of single electrolytes in water over 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 the short-range interactions between uncharged particles based on the Peng−Robinson equation of state, (2) an ioncharging 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 the chemical equilibrium of the dissolved electrolyte, allowing for 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 the mean ionic activity coefficients, osmotic coefficients, standard free energies of ion hydration, and densities for NaCl, CaCl2, K2SO4, and MgSO4 salts in water over a wide range of temperatures and salt molalities. The results indicate good agreement between the experimental data and the calculations generated from the present equation of state.

■

INTRODUCTION A precise knowledge of the thermodynamic properties of electrolyte solutions is paramount to 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, and so forth. As a matter of fact, a large number of the experimental thermodynamic properties of electrolyte solutions have appeared in the literature, predominately over the last 40 years. As far as mathematical models for electrolyte solutions are concerned, the literature is also rich in them. However, many of these modeling efforts are not consistent because of the use of different standard states, including the ill-defined extension of the thermodynamic frameworks usually used for nonelectrolytes to electrolytes. Many theoretical and semiempirical models have been developed to represent the thermodynamic properties of electrolyte solutions. There are various excellent reviews on electrolyte-solution models available in the literature,1−4 including a recent one by Anderko et al.5 In general, the 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 energy.15−24 The electrolyte models based on the Helmholtz energy, unlike those based on the local-composition concept, can be applied over a wider range of temperatures and pressures, and they are also able to yield volumetric properties, such as the density of electrolyte solutions. © 2013 American Chemical Society

In the case of electrolyte models based on equations of state, the majority treat the electrolyte as being completely dissociated (or fully ionized),15−17,19,21,22 whereas some assume that the electrolyte is undissociated or completely associated.18,20,23,24 However, partial dissociation (or ionization) is an important component in the theory of electrolyte solutions, as, for example, in the chemistry of electrolyte solutions at high temperatures where its practical importance is found in the hydrothermal and geothermal processes involving acidic aqueous solutions containing high NaCl concentrations. Thus far, very few electrolyte models that assume partial dissociation or ionization of the salt in solution have been reported in the literature. 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 and do not take into account the effect of pressure, resulting in an inability to calculate the solution density. The purpose of the present work is therefore to develop a modeling approach that can correlate and/or predict the thermodynamic and volumetric properties of a partially ionized electrolyte in water in a wide range of temperatures, pressures, and salt molalities. Unlike the previous individual-ion and saltbased models, the present approach incorporates the chemical equilibrium for the dissolved electrolyte, thus making it possible Received: Revised: Accepted: Published: 8589

March 8, 2013 May 23, 2013 June 4, 2013 June 4, 2013 dx.doi.org/10.1021/ie400764t | Ind. Eng. Chem. Res. 2013, 52, 8589−8601

Industrial & Engineering Chemistry Research

Article

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

to estimate the extent of dissociation of the electrolyte at diverse temperatures and concentrations.

■

ΔAideal + ΔAPR = (A − Aref )PR = ΔAPR,r

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 the various contributions required to form the ionic solution at a pressure and density of interest. Figure 1 shows the isothermal path to forming a

(2)

which corresponds to the residual Helmholtz energy for the Peng−Robinson equation of state. Furthermore, the two Born contributions can be summed to yield ΔA εBorn + ΔAεBorn = ΔABorn vac

(3)

Hence, eq 1 reduces to A(T , V , n) = ΔABorn + ΔAPR,r + ΔAMSA

(4)

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

NAe0 2 ⎛ 1 ⎞ ⎜ − 1⎟∑ niZ i 2/σi ⎠ nt ⎝ ε ions

(5)

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

e0 2 =

Figure 1. Isothermal charging cycle to form the ionic solution.

A = ΔA

+

ΔA εBorn vac

PR

+ ΔA

+

ΔAεBorn

MSA

+ ΔA

(6)

where e is the unit electronic charge and ε0 is the permittivity 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 existing in a continuum having a uniform dielectric constant. Peng−Robinson Contribution. Short-range interactions among the uncharged ions, undissociated salt, and water are properly accounted for by the Peng−Robinson equation of state.26 This cubic equation of state was chosen because of its proven superiority at describing the volumetric properties and phase equilibria of pure substances and their mixtures as compared to that of other equations of state of the van der Waals type. The change in the Helmholtz energy in the Peng− Robinson equation of state (from the reference state (ideal gas) to the actual state) is given by

solution containing a partially dissolved electrolyte, which proceeds from the 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 the 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, and this change results in ΔAideal. (2) All ions are discharged, and the change in the Helmholtz energy for such a discharge is given by the Born equation25 using the dielectric constant of a vacuum, ΔAεBorn . (3) The mixture of uncharged particles is then vac compressed to system volume, Vsys, and an equation of state for nonelectrolytes is used to represent this transition. In this study, the Peng−Robinson cubic equation of state,26 ΔAPR, was used. (4) Finally, the ions are recharged at a constant volume, and the Born equation, ΔABorn ε , is again used to estimate this change. 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 long-range forces is conveniently represented by the mean spherical approximation (MSA) approach,27 ΔAMSA. The total change in the Helmholtz energy is therefore given by ideal

e2 4πε0

ΔAPR,r =

⎡ V + b(1 − a ln⎢ 2 2 bnt ⎣ V + b(1 + ⎛V − b⎞ V ⎟ − RT ln ln⎜ ⎝ V ⎠ V ref

2)⎤ ⎥ − RT 2)⎦ (7)

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

b=

∑ ∑ ninj aiaj (1 − ki , j) nt 2

∑ nibi nt

(8)

(9)

and ki,j (= kj,i) in eq 8 is the binary interaction parameter. The pure-component attraction and repulsion parameters (ai and bi) are determined as follows for ions

(1) 8590

dx.doi.org/10.1021/ie400764t | Ind. Eng. Chem. Res. 2013, 52, 8589−8601


Article

π bi = NA (σiP)3 6

(10)

ai = f 2 Rbi(ε /κ )i

(11)

ratios characteristic of most electrolytes. The expression for the Helmholtz energy is ΔAMSA = −

where σiP is the Pauling crystal diameter for the ion and f is a correction factor that will be discussed later. The dispersion theory by Mavroyannis and Stephen28 was used to estimate (ε/ k)i (ε /κ )i = 2.2784 × 10−11 αi 3ne /(σiP)6

(12)

κ2 =

(ε /κ )+ (ε /κ )−

EQUATION OF STATE AND DERIVED IONIC PROPERTIES A pressure-explicit PVT expression is obtained via

For water at subcritical temperatures (0 °C < T ≤ 0.99 Tc,w), the equations given by Xu and Sandler29 were used (15)

bw = (b0 + b1Tr + b2Tr 2)RTc,w /Pc,w

(16)

⎛ ∂n A ⎞ P = −⎜ t ⎟ ⎝ ∂V ⎠T , n ⎧ ntRT a − + Peng−Robinson ⎪ V (V + b) + b(V − b) ⎪V − b ⎪ 2 2 ⎪ NAe0 ⎜⎛ ∂ε ⎟⎞ ∑ niZ i + Born ⎪ ε 2 ⎝ ∂V ⎠T , n σi ⎪ i P=⎨ ⎪ 2RT 3⎡ ⎛1 ⎞ 3 Γ ⎢1 + σ Γ + 3V ⎜ + 2σ ⎟ MSA ⎪ ⎝Γ ⎠ 2 ⎪ 3πNA ⎣⎢ ⎪ ⎛ ⎞ ⎤ ⎪ ⎜ ∂Γ ⎟ ⎥ ⎪ ⎝ ∂V ⎠T , n ⎥⎦ ⎩

(22)

For a given T, P, and number of moles, n, the above equation yields different values of the 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 ⎞ P ln φi = ⎜ t ⎟ (RT )−1 − ln ref ⎝ ∂ni ⎠T , V , n (i) P j

(23)

From the resulting expression for the fugacity coefficient, other thermodynamic properties such as the activity coefficient of species i can be readily determined using the following equations. For water φ γi = i0 φi (24)

∑ions niσi ∑ions ni

(21)

The resulting equation of state is thus

where 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 the molar volume of pure water when compared to the use of a constant b value in the original Peng−Robinson equation of state. MSA Contribution. Ion−ion interactions can be described by the so-called primitive model in which the 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 the activity and osmotic coefficients of 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 the implicit solution of a complicated equation for the screening factor, Γ, that, to some extent, complicates the differentiation of the resulting expressions with respect to the 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 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

σ=

(20)

■

(14)

a w = (a0 + a1Tr + a 2Tr 2)R2Tc,w 2/Pc,w

4πe0 2NA 2 ∑ niZi 2 εRTV ions

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.

(13)

whereas as is calculated using eq 11 in which (ε /κ )s =

(18)

where Γ is the screening parameter of the MSA model that is used in the computation of the excess thermodynamic properties resulting from ion−ion interactions and is defined by 1 Γ= [ 1 + 2σκ − 1] (19) 2σ

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−

2Γ 3RTV (1 + 1.5σ Γ) 3πNAnt

For ions and undissociated salt φi γi = ∞ φi

(17)

According to Harvey et al.,30 the above mixing rule has been successfully tested against experimental data for the ion-size 8591

(25)



Article

where φi0 and φi∞ are the fugacity coefficients of the pure species and at infinite dilution, respectively. The above activitycoefficient expressions in turn serve to derive important ionic properties such as the mean ionic activity coefficient γ± = (γ+ν+γ−ν−)1/ ν

to the effect produced by the ions. Equation 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 the electrolyte is 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, the partial dissociation (or ionization) of an electrolyte Mv+Xv− can be represented by the following chemical reaction M ν +X ν − ⇔ ν+ M ν + + ν− X ν − (32)

(26)

and the osmotic coefficient (water activity) ϕ = −ln

(x wγw ) 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 to the aqueous solution. Friedman and Krishnan31 give tabulated values of standard free energies of hydration (ΔhG⊕ i ) for monatomic ions at 25 °C and 1 bar. The free energy of hydration can be computed from the present fugacity-coefficient expressions as follows: ⊕ ∞ Δh Gion = RT ln φion

in which the true molalities of ionic species (mi) and the true molality-based activity coefficients (γi,m) are related through the dissociation equilibrium equation Kd =

(28)

Density. The density of the ionic solution at a given temperature, pressure, and number of moles of each species can be calculated by means of 1 ρ = (n+M+ + n−M − + nsMs + n w M w ) (29) V However, the total volume, V, in the above equation cannot be readily calculated from the present equation of state because it requires an iterative procedure to obtain it. A convergence procedure for the simultaneous determination of V and the degree of dissociation will be described in detail later. 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 a decrease in 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 + (εw − 1)⎜ ⎟ ⎝ 1 + (ξ /2) ⎠

NAπ ∑ niσi 3 6V ions

(33)

where Kd is the dissociation constant on a molality scale. Expressing the above equation in terms of the degree of dissociation (α) Kd = mν − 1ν+ν+ν−ν−

(αγ±, 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 ⎛ 1 + M w νm ⎞ ⎟ ln γ±, m = ln γ± − ln⎜ ⎝ 1000 ⎠

(35)

where γ± is the true mole-fraction-based mean ionic activity coefficient given by eq 26. For a 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, the values of both V and α should be simultaneously determined by solving the following set of nonlinear equations derived from eqs 22 and 34:

(30)

where εw is the static dielectric constant of pure water. The above equation expresses ε in terms of ion concentration and size through ξ: ξ=

ν m+ν +m−ν − γ±, m mMX γMX, m

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

(36)

g2(V , α) = Kd(1 − α)γMX, m − mν − 1ν+ν+ν−ν−(αγ±, m)ν = 0 (37)

(31)

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

From eq 30, the partial derivatives of ε with respect to V and ni that appear in the equation of state (eq 22) and the fugacitycoefficient expression can be readily determined. However, the dielectric constants of pure water and its derivatives were obtained from the equation of Uematsu and Franck,33 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 solution,32 this effect is almost negligible when compared

■

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 other species (e.g., ions and water) are of the short-range type, which are accounted for in the Peng− Robinson part. However, specific ion−water interactions (i.e., ion hydration) are handled in an oversimplified manner by both 8592



Article

A least-squares 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 this purpose

the Peng−Robinson contribution through ki,w (= kw,i), and the Born contribution, where ⎛ ∂ε ⎞ ≠0 ⎜ ⎟ ⎝ ∂nw ⎠T , V , n (w)

⎧N ⎡ ⎤2 γ±eos ,m,i ⎥ ⎪ ⎢ at 25 °C and 1 bar ⎪∑ ⎢1 − γ exp ⎥ ±, m , i ⎦ ⎪ i=1 ⎣ 2 2 ⎡ ⎪ Δh Gi⊕,eos ⎤ ⎪ ⎢ ⎥ 1 + − ∑ min g = ⎨ Δh Gi⊕,exp ⎦ ⎪ i=1 ⎣ ⎪ 2 N ⎡ ⎪ γ eos ⎤ ⎢1 − ± , m , i ⎥ at T > 25 °C ⎪ ∑ ⎪ ⎢ ⎥ γ±exp ,m,i ⎦ i =1 ⎣ ⎩

(38)

j

This oversimplification 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. On the basis of 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 the ions and water k+,w and k−,w (from the Peng−Robinson contribution), and the correction factor, f, for the Peng− Robinson attraction parameter for all ionic species (eq 11). Of all these parameters, only Kd and f are considered to change with temperature. During a preliminary determination of the model parameters in which the experimental mean ionic activity coefficients at 25 °C measured by Hamer and Wu34 for various 1−1 electrolytes were correlated, it was found that the results were 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 because of the fact that the solvation of anions (i.e., anion−water interaction) is much lower than the solvation of cations.

(39)

γexp ±,m

where N is the number of experimental points, and are the observed and calculated mean molal ionic activity coefficients, respectively, and ΔhG⊕,exp and ΔhG⊕,eos are the i i experimental and calculated standard free energies of hydration of the ions (cation, i = 1; anion, i = 2), respectively. The ⊕,eos calculated γeos values are obtained from eqs 35 ±,m and ΔhGi and 28, respectively. Prior to their fit, the experimental molalbased ΔhG⊕ i,m values were converted to values on a molefraction scale by means of Δh Gi⊕,exp = Δh Gi,⊕m,exp + RT ln(1000/M w )

RESULTS AND DISCUSSION The present modeling approach was first applied during the correlation of the experimental mean ionic activity coefficients (γ±,m) at different temperatures, including the free energies of hydration (ΔhG⊕ i ) at 25 °C and 1 bar of some representative electrolytes in water. The following electrolytes were chosen for this purpose because they have a large amount of experimental ionic property data available at 25 °C and also at elevated temperatures where salt dissociation effects become significant: NaCl, a 1−1 electrolyte; CaCl2, a 2−1 electrolyte; K2SO4, a 1− 2 electrolyte; and MgSO4, a 2−2 electrolyte. Table 1 shows the Table 1. Aqueous Electrolyte Solutions Considered in This Work σ+P (Å)

σ−P (Å)

T (°C)

mmax (mol/kg)

sources for γ± and ϕ

1−1: NaCl

1.90

3.62

25, 100, 200, 300

6.0

1.98

3.62

25, 100, 200, 300

3.5

2.66

4.45

25, 125, 225

2.0

1.30

4.45

25, 50, 80

2.0

Hamer and Wu34 Liu and Lindsay35 Staples and Nuttall36 Holmes et al.37 Holmes and Mesmer38 Pitzer39 Snipes et al.40

2−1: CaCl2 1−2: K2SO4 2−2: MgSO4

(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 the measured γ±,m and ΔhG⊕ i values yielded the following adjusted parameters: the 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 Peng−Robinson attraction parameter for the ionic species, and the molalitybased dissociation constant (Kd). The correlation 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 the calculated and experimental γ±,m and ΔhG⊕ i values. It can be seen from Table 2 that the ability of the present approach to represent the experimental data is quite good, with overall AAD values of 0.93% for γ±,m and 5.9% for ΔhG⊕ i . 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 SO4−2 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 the halide ions are lower than those of the alkali ions. Furthermore, the resulting adjusted value of the correction factor, f, is large enough to suggest that the use of the dispersion theory alone (eq 12) significantly underestimates the value of the short-range attraction parameter, ai, for the ionic species. As a matter of fact, on the basis of the optimum Kd values reported in Table 2, the present model correctly predicts an essentially complete dissociation for the four electrolyte solutions, with degrees of dissociation very close to unity even at their highest molalities (most strong electrolytes completely dissociate in water at ambient temperature). Figure 2 shows the excellent agreement between the calculated and observed γ±,m values at 25 °C within a molality

■

electrolyte

γeos ±,m

four electrolytes considered in this study along with their corresponding Pauling crystal ionic diameters, temperatures, and maximum molalities at which experimental data were measured as well as the sources for the experimental mean ionic activity coefficients and osmotic coefficients. In fact, the experimental ΔhG⊕ i and density data for all electrolytes were obtained from Friedman and Krishnan31 and Zaytsev and Aseyev,41 respectively. 8593



Article

Table 2. Optimized Model Parameters and AAD Values at 25 °C and 1 bar AAD (%)

a

electrolyte

N

σ+ (Å)

σ− (Å)

Kd (mol/kg)

k+,w

f

γ±

ΔhG⊕

1−1: NaCl 2−1: CaCl2 1−2: K2SO4 2−2: MgSO4 overall

29 33 9 13 84

3.24 3.65 3.35 3.10

4.28 4.55 3.30 3.86

3945 12 652 3.44 25.14

−0.325 −0.600 0.857 −0.104

7.1 14.0 5.9 2.1

0.28 1.28 1.19 1.29 0.93

0.37 1.56 13.5a 8.00a 5.90

Anion SO4−2 excluded.

Figure 2. Correlated mean ionic activity coefficients for the four aqueous electrolyte solutions at 25 °C and 1 bar.

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

range of 0−6 mol/kg. The present approach also gives an accurate representation of the experimental γ±,m values in the dilute region (0 < m < 0.5 mol/kg), as demonstrated by Figure 1b that uses a logarithmic x axis. Moreover, Figure 3 depicts a comparison between the experimental and calculated ΔhG⊕ i values at 25 °C and 1 bar for the four alkali ions (Na+, K+, Ca2+, and Mg2+) and the only halide ion (Cl−) considered in this work. As shown by this figure, the present equation of state reasonably represents the observed ΔhG⊕ values for the i majority of the ions with the exception of the cation K+ for which the largest percent 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 ΔhG⊕ i . There is no contribution from the MSA term because it describes only ion− ion interactions. Furthermore, the major contribution to ΔhG⊕ i comes from the Born term. This indicates that the hydration behavior can be properly described using the Born equation by adjusting only 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 in the case of NaCl in water at 25 °C. This value is denoted by the line passing through the experimental points in Figure 4 and is obtained from the different contributions as follows:

(41)

It can be seen from Figure 4 that the MSA contribution meets the correct boundary condition (i.e., limiting Debye−Huckel law) near the dilute region of the electrolyte solution (for m < 0.1 mol/kg), which is better visualized in Figure 4b. As the salt concentration increases, the predictions from the MSA contribution deviate from the experimental data. However, as the concentration of ions in solution increases, the hydration becomes more important. This effect is properly accounted for by the Born term that along with the MSA contribution is able to represent the experimental data. At much higher concentrations (i.e., near saturation), the 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 obtained at 25 °C and 1 bar can also be used to predict the osmotic coefficient 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, whereas the model somewhat underpredicts the experimental osmotic coefficient for MgSO4, particularly 8594



Article

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 because of the decreasing salt dissociation with temperature as a result of the decrease in the dielectric constant of the solution. Furthermore, we chose to make the correction factor, f, and therefore all attraction parameters, ai, for the ionic species, temperature-dependent. This is consistent with the Peng−Robinson framework in which the van der Waals attraction parameters of all pure components have a temperature dependence. Table 3 shows the first set of the regressed model parameters for NaCl in water at 100, 200, and 300 °C. At each temperature, Table 3 also lists the calculated minimum percent of salt dissociation and the resulting percent of AAD values between the calculated and experimental γ±,m data. Figure 7 depicts these results in graphical form, revealing that there is excellent agreement between the experimental and correlated γ±,m data with an overall AAD value of 1.1%. Figures 8 and 9 show the corresponding model predictions for the osmotic coefficients and densities of aqueous NaCl, respectively. Although the model predictions for the osmotic coefficients look reasonable at the three isotherms, the predictions somewhat depart from the experimental data in certain molality regions (e.g., at molalities greater than 1 mol/ kg, particularly at 200 °C (Figure 8a), and in the dilute region at 300 °C (Figure 8b)). The density predictions are also in good agreement with the 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 3. Correlated standard free energies of hydration and the corresponding percent of deviation for the ions considered in this work.

near the dilute region. However, 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, where the model fairly underpredicted its experimental density data over the entire molality range considered.

Figure 4. Model contributions to the mean ionic activity coefficient of NaCl in water at 25 °C and 1 bar. 8595



Article

Figure 5. Predicted osmotic coefficients for the four aqueous electrolyte solutions at 25 °C and 1 bar.

Figure 10 shows the qualitatively correct variation of the dissociation constant with temperature for the NaCl solution (the 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 (150, 200, and 250 °C) correspond to the extrapolated data from the experimental dissociation constants obtained at elevated temperatures (i.e., higher than 250 °C). According to Figure 10, the agreement between the calculated and experimental dissociation constants is quite good, particularly at temperatures higher than 250 °C. The present approach also captures the correct variation in 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 slightly overestimates the experimental data of Helgeson42 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 (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 in this figure, the present model is also capable of correctly describing the decreasing variation in the dissociation constant with temperature for the aforementioned solutions.

Figure 6. Predicted mixture densities for the four aqueous electrolyte solutions at 25 °C and 1 bar.

Table 3. Regressed Model Parameters at Higher Temperatures (NaCl−Water) T (°C)

N

100 200 300 overall

17 17 17 51

Kd (mol/kg)

f

minimum dissociation (%)

AAD γ± (%)

18.6 1.77 0.17

15.2 24.8 28.3

84.5 65.7 57.9

1.01 0.76 1.52 1.10

8596



Article

Figure 7. Correlated mean ionic activity coefficients for NaCl in water at different temperatures and pressures (P = solvent Psat).

Figure 8. Predicted osmotic coefficients for NaCl−water at different temperatures and pressures (P = solvent Psat).

aqueous solutions of K2SO4 from measurements of electrical conductance at up to 400 °C. Figure 12 also shows four points (red circles) representing the 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 in a temperature range of 16−306 °C. As revealed by Figure 12, the agreement between the experimental dissociation constant for K2SO4 and

On the basis of these results, the lowest levels of dissociation were exhibited by aqueous K2SO4 at 61.1%, whereas the highest degree of dissociation was displayed by the MgSO4 solution at 99.4%; that is, it practically remained completely ionized over the whole temperature range considered (25−80 °C). Comparisons with the experimental dissociation data, however, are indeed needed to confirm these results. In this context, Sharygin et al.43 recently reported the dissociation constants in 8597



Article

Figure 9. Predicted mixture densities for NaCl−water at different temperatures and pressures (P = solvent Psat).

Figure 11. Variation of the degree of dissociation with salt molality and temperature for NaCl−water.

Figure 10. Variation of the dissociation constant with temperature for NaCl−water.

Figure 12. Variation in the dissociation constant with temperature for three aqueous electrolyte solutions.

that calculated by the model is quite good at 100 °C; however, at higher 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 in the dilute region (i.e., m ≪ 0.1 mol/kg), whereas in the present approach, a much larger K2SO4 concentration was considered (up to 2 mol/kg), thus yielding lower dissociation constants and decreased degrees of dissociation. Finally, the corresponding density predictions for the three aqueous solutions using the present model are shown in Figure 13 at selected temperatures (above 25 °C). The present equation of state accurately predicts the experimental

densities of CaCl2 and MgSO4 solutions at 100 and 50 °C, respectively, but it largely overpredicts the observed density of aqueous K2SO4 at 125 °C as its maximum molality is reached. The major contribution for the density calculations in the present model comes from the Peng−Robinson equation. It is very likely that this equation is not predicting the correct trend in density in some instances.

■

CONCLUSIONS A versatile equation of state was developed here for partially ionized aqueous electrolyte solutions and is applicable over a 8598



■

Article

ACKNOWLEDGMENTS R.M.-S. gratefully acknowledges the Instituto Politécnico Nacional and CONACyT for providing financial support for this work.

■

LIST OF SYMBOLS a Attraction parameter of the mixture in the Peng− Robinson contribution ai Attraction parameter of species i in the Peng−Robinson contribution A Helmholtz free energy b Co-volume parameter of the mixture in the Peng− Robinson contribution bi Co-volume parameter of species i in the Peng−Robinson contribution e Unit electronic charge e0 As defined in eq 6 f Correction factor for the attraction parameter of ionic species in the Peng−Robinson contribution g As defined in eq 39 g1 As defined in eq 36 g2 As defined in eq 37 G⊕ Gibbs free energy of hydration ki,j Binary interaction parameter in the Peng−Robinson contribution Kd Dissociation constant m Salt molality M Molar mass; cation in partial dissociation chemistry MX Undissociated salt in partial dissociation chemistry n Vector containing the number of moles of each species in solution ne Number of electrons in the ion ni Number of moles of species i nt Total number of moles N Number of data points NA Avogadro’s number Pressure P R Universal gas constant T Temperature V Total volume x Liquid mole fraction X Anion in partial dissociation chemistry Zi Ionic charge

Figure 13. Predicted mixture densities for three aqueous electrolyte solutions at different temperatures.

wide range of temperatures, pressures, and salt molalities. The present equation of state was successfully applied to the representation of mean ionic activity coefficients, osmotic coefficients, standard free energies of hydration of ions, and densities of aqueous solutions containing a single electrolyte. The following conclusions can be drawn from this work: (1) The present modeling approach correlated quite well the experimental mean ionic activity coefficients and standard energies of hydration for aqueous NaCl, CaCl2 and some alkalimetal sulfate solutions over a wide range of temperatures, pressures, and salt molalities. (2) The new equation of state was also accurate in predicting the observed osmotic coefficient and mixture density data of the aforementioned electrolyte solutions. (3) The use of partial dissociation chemistry in the present modeling work provides a comprehensive and relatively simple thermodynamic framework that was proven here to have good correlative and predictive capabilities in representing well the nonidealities of aqueous electrolyte solutions. (4) Because of its relative simplicity, the present approach may be well suited for use in engineering calculations for solutions of strong and weak electrolytes where partial dissociation plays a significant role in their thermodynamic properties. (5) Because of the semidiscrete nature of the present modeling approach in which the solvent is not treated as a discrete molecule within the MSA and Born primitive equations, our current model is not recommended for application at very high electrolyte concentrations.

Greek Letters

*E-mail: [email protected].

α γi Γ ε ε0 (ε/k) Δ k ξ v vi ρ σ σi ϕ φi

Notes

Subscripts

The authors declare no competing financial interest.

c

■

AUTHOR INFORMATION

Corresponding Author

8599

Ion polarizability Activity coefficient of species i Screening factor in the MSA contribution Dielectric constant of the mixture Permittivity in vacuum Ion dispersion energy Change in energy Reciprocal Debye screening length As defined by eq 31 Total number of dissociated ions Number of dissociated ions Density of the mixture Effective ionic diameter Diameter of the ionic species Osmotic coefficient Fugacity coefficient of species i

Critical property dx.doi.org/10.1021/ie400764t | Ind. Eng. Chem. Res. 2013, 52, 8589−8601

Industrial & Engineering Chemistry Research h m r s w ε εvac + − ±

Article

(14) Haghtalab, A.; Mazloumi, S. H. A Nonelectrolyte Local Composition Model and Its Application in the Correlation of the Mean Activity Coefficient of Aqueous Electrolyte Solutions. Fluid Phase Equilib. 2009, 275, 70. (15) Ball, F. X.; Planche, H.; Furst, W.; Renon, H. Representation of Deviation from Ideality in Concentrated Aqueous Solutions of Electrolytes Using the Mean Spherical Approximation Molecular Model. AIChE J. 1985, 31, 1233. (16) Copeman, T. W.; Stein, F. P. A Perturbed Hard-Sphere Equation of State for Solutions Containing an Electrolyte. Fluid Phase Equilib. 1987, 35, 165. (17) Harvey, A. H.; Prausnitz, J. M. Thermodynamics of HighPressure Aqueous Systems Containing Gases and Salts. AIChE J. 1989, 35, 635. (18) Aasberg-Petersen, K.; Stenby, S.; Fredenslund, A. Prediction of High Pressure Gas Solubilities in Aqueous Mixtures of Electrolytes. Ind. Eng. Chem. Res. 1991, 30, 2180. (19) Simon, H. G.; Kistenmacher, H.; Prausnitz, J. M.; Vortmeyer, D. An Equation of State for Systems Containing Electrolytes and NonElectrolytes. Chem. Eng. Process. 1991, 29, 139. (20) Anderko, A.; Pitzer, K. S. Equation-of-State Representation of Phase Equilibria and Volumetric Properties of the System NaCl-H2O above 573 K. Geochim. Cosmochim. Acta 1993, 57, 1657. (21) Furst, W.; Renon, H. Representation of Excess Properties of Electrolyte Solutions Using a New Equation of State. AIChE J. 1993, 39, 335. (22) Wu, J.; Prausnitz, J. M. Phase Equilibria for Systems Containing Hydrocarbons, Water, and Salt: An Extended Peng-Robinson Equation of State. Ind. Eng. Chem. Res. 1998, 37, 1634. (23) Myers, J. A.; Sandler, S. I.; Woods, R. H. An Equation of State for Electrolyte Solutions Covering Wide Ranges of Temperature, Pressure, and Composition. Ind. Eng. Chem. Res. 2002, 41, 3282. (24) Tan, S. P.; Adidharma, H.; Radosz, M. Statistical Associating Fluid Theory Coupled with Restricted Primitive Model to Represent Aqueous Strong Electrolytes. Ind. Eng. Chem. Res. 2005, 44, 4442. (25) Born, M. Volumen und Hydrationswarme der Ionen. Z. Phys. 1920, 1, 45. (26) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (27) Blum, L. Mean Spherical Model for Asymmetric Electrolytes. I. Method of Solution. Mol. Phys. 1975, 30, 1529. (28) Mavroyannis, C.; Stephen, M. J. Dispersion Forces. Mol. Phys. 1962, 5, 629. (29) Xu, Z.; Sandler, S. I. Temperature-Dependent Parameters and the Peng-Robinson Equation of State. Ind. Eng. Chem. Res. 1987, 26, 601. (30) Harvey, A. H.; Copeman, T. W.; Prausnitz, J. M. Explicit Approximation to the Mean Spherical Approximation for Electrolyte Systems with Unequal Ion Sizes. J. Phys. Chem. 1988, 92, 6432. (31) Friedman, H. L.; Krishnan, C. V. In Water: A Comprehensive Treatise; Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 3, Chapter 1. (32) Giese, K.; Kaatz, V.; Pottel, R. Permittivity and Dielectric and Proton Magnetic Relaxation of Aqueous Solutions of the Alkali Halides. J. Phys. Chem. 1970, 74, 178. (33) Uematsu, M.; Franck, E. U. Static Dielectric Constant of Water and Steam. J. Phys. Chem. Ref. Data 1980, 9, 1291. (34) Hamer, W. J.; Wu, Y. C. Osmotic Coefficients and Mean Activity Coefficients of Uni-Univalent Electrolytes in Water at 25 °C. J. Phys. Chem. Ref. Data 1972, 1, 1047. (35) Liu, C. T.; Lindsay, W. T., Jr. Thermodynamics of Sodium Chloride Solutions at High Temperatures. J. Solution Chem. 1972, 1, 45. (36) Staples, B. R.; Nuttall, R. L. The Activity and Osmotic Coefficients of Aqueous Calcium Chloride at 298.15 K. J. Phys. Chem. Ref. Data 1977, 6, 385. (37) Holmes, H. F.; Baes, C. F., Jr.; Mesmer, M. E. Isopiestic Studies of Aqueous Solutions at Elevated Temperatures I. KCl, CaCl2, and MgCl2. J. Chem. Thermodyn. 1978, 10, 983.

Hydration condition Mean molal condition Reduced property Undissociated salt Water Dielectric constant of the mixture Dielectric constant of a vacuum Cation Anion Mean ionic condition

Superscripts

Born exp eos ideal MSA P PR r ref 0 ∞

■

Born contribution Experimental value Equation of state Ideal-gas condition Mean spherical-approximation contribution Pauling crystal condition Peng−Robinson contribution Residual property Ideal-gas reference state Pure species Infinite dilution

REFERENCES

(1) Zemaitis, J. F., Jr.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; Design Institute for Physical Property Data sponsored by the American Institute of Chemical Engineers: New York, 1986. (2) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, FL, 1991. (3) Rafal, M.; Berthold, J. W.; Scrivner, N. V.; Grise, S. L. Models for Electrolyte Solutions. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S. I., Ed.; Marcell Decker: New York, 1994. (4) Loehe, J. R.; Donohue, M. D. Recent Advances in Modeling Thermodynamic Properties of Aqueous Strong Electrolyte Systems. AIChE J. 1997, 43, 180. (5) Anderko, A.; Wang, P.; Rafal, M. Electrolyte Solutions: From Thermodynamic and Transport Property Models to the Simulation of Industrial Processes. Fluid Phase Equilib. 2002, 194−197, 123. (6) Cruz, J. L.; Renon, H. A New Thermodynamic Representation of Binary Electrolyte Solutions Nonideality in the Whole Range of Concentrations. AIChE J. 1978, 24, 817. (7) Chen, C. C.; Britt, H. L.; Boston, J. F.; Evans, L. B. Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. AIChE J. 1982, 28, 588. (8) Chen, C. C.; Evans, L. B. Local Composition Model for Excess Gibbs Energy of Aqueous Electrolyte Systems. AIChE J. 1986, 32, 444. (9) Mock, B.; Evans, L. B.; Chen, C. C. Thermodynamic Representation of Phase Equilibria of Mixed-Solvent Electrolyte Systems. AIChE J. 1986, 32, 1655. (10) Liu, Y.; Gren, U.; Wimby, M. A Systematic Comparison of Nine Models Formed by Various Combinations of Three Debye-Hückel Expressions and Three Local-Composition Expressions for Representation of Nonideality in Electrolyte Solutions. Fluid Phase Equilib. 1989, 53, 269. (11) Macedo, E. A.; Skovborg, P.; Rasmussen, P. Calculation of Phase Equilibria for Solutions of Strong Electrolytes in Solvent-Water Mixtures. Chem. Eng. Sci. 1990, 45, 875. (12) Jaretun, A.; Aly, G. New Local Composition Model for Electrolyte Solutions: Single Solvent, Single Electrolyte Systems. Fluid Phase Equilib. 1999, 163, 175. (13) Simonin, J. P.; Krebs, S.; Kunz, W. Inclusion of Ionic Hydration and Association in the MSA-NRTL Model for a Description of the Thermodynamic Properties of Aqueous Ionic Solutions: Application to Solutions of Associating Acids. Ind. Eng. Chem. Res. 2006, 45, 4345. 8600



Article

(38) Holmes, H. F.; Mesmer, M. E. Thermodynamics of Aqueous Solutions of the Alkali Metal Sulfates. J. Solution Chem. 1986, 15, 495. (39) Pitzer, K. S. Thermodynamic Properties of Aqueous Solutions of Bivalent Sulphates. J. Chem. Soc., Faraday Trans. 2 1972, 68, 101. (40) Snipes, H. P.; Manly, C.; Ensor, D. D. Heats of Dilution of Aqueous Electrolytes: Temperature Dependence. J. Chem. Eng. Data 1975, 20, 287. (41) Properties of Aqueous Solutions of Electrolytes; Zaytsev, I. D., Aseyev, G. G., Eds.; CRC Press: Boca Raton, FL, 1992. (42) Helgeson, H. C. Thermodynamics of Hydrothermal Systems at Elevated Temperatures and Pressures. Am. J. Sci. 1969, 267, 729. (43) Sharygin, A. V.; Grafton, B. K.; Xiao, C.; Wood, R. H.; Balashov, V. N. Dissociation Constants and Speciation in Aqueous Li2SO4 and K2SO4 from Measurements of Electrical Conductance to 673 K and 29 MPa. Geochim. Cosmochim. Acta 2006, 70, 5169. (44) Noyes, A. A. The Electrical Conductivity of Aqueous Solutions; Carnegie Institute of Washington: Washington, D.C., 1907; No. 63, pp 71−111.

8601


Thermodynamic Model for Aqueous Electrolyte Solutions with Partial

Recommend Documents