Equation of state for aqueous electrolyte systems based on the semirestricted non-primitive mean spherical approximation

Fluid Phase Equilibria 297 (2010) 23–33

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Equation of state for aqueous electrolyte systems based on the semirestricted non-primitive mean spherical approximation Stefanie Herzog a,1 , Joachim Gross b,∗ , Wolfgang Arlt a a b

Chair of Separation Science and Technology, Friedrich-Alexander University Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany Institut für Technische Thermodynamik und Thermische Verfahrenstechnik, Universität Stuttgart, D-70550 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 4 February 2010 Received in revised form 20 May 2010 Accepted 25 May 2010 Available online 1 June 2010 Keywords: Aqueous electrolyte systems Semirestricted non-primitive mean spherical approximation (npmsa) PC-SAFT equation of state

a b s t r a c t The semirestricted non-primitive mean spherical approximation (npmsa) is used in combination with the PC-SAFT equation of state to model completely dissociating aqueous alkali halide systems. The salts are described using ion-specific parameters from tables and correlations. Upon analyzing aqueous electrolyte systems at infinite dilution of the salt it was concluded that for the arithmetic mean ion diameter of anion and cation, the semirestricted npmsa contribution gives no reliable results. Therefore, this parameter is adjusted in this work. The model was applied to aqueous alkali halide systems up to the solubility limit at T = 298.15 K. Mean ionic activity coefficients and osmotic coefficients were correlated with good results. The model was subsequently applied to temperatures up to T = 373.15 K and compared to liquid densities and to system pressures up to the solubility limit of the salts in water. The agreement between experimental data and the proposed equation of state is satisfactory for the liquid densities and excellent in case of the system pressures. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Electrolyte systems are ubiquitous in nature and important for many industrial processes. Thermodynamic models for electrolyte systems usually comprise a conventional model for non-electrolyte systems and a theory which accounts for the long-ranged electrostatic interactions. Different approaches for modeling of electrolyte–solvent mixtures are used, viz. activity coefficient models and equations of states. Most of the equations of state proposed recently are based on theories from statistical mechanics. An important contribution was made by Wertheim who developed a perturbation theory for highly directional interactions, the thermodynamic perturbation theory of first and second order (TPT1 and TPT2) [1–4]. Chapman et al. [5,6]

Abbreviations:

AAD, average absolute deviation of calculated quantities

i,calc respective to their experimental values i,exp , AAD% = 1/N

N 

(˝i,exp −

i=1

˝i,calc )/(˝i,exp ) × 100; EOS, equation of state; (n)pmsa, (non-)primitive mean spherical approximation; PC-SAFT, perturbed-chain statistical association fluid theory; SAFT, statistical association fluid theory; SAFT-VR, statistical associating fluid theory for fluids with attractive potentials of variable range; TPT, thermodynamic perturbation theory. ∗ Corresponding author. Tel.: +49 711 685 66105; fax: +49 711 685 66140. E-mail addresses: [email protected] (S. Herzog), [email protected] (J. Gross), [email protected] (W. Arlt). 1 Present address: BASF SE, GCP/PD M 300, D-67056 Ludwigshafen, Germany. 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.05.024

have further developed the formalism to mixtures and derived simplified expressions for associating interactions of molecules and for the formation of chain molecules from spherical segments. This leads to the statistical associating fluid theory (SAFT) family of equations of state. A variant was proposed by Huang and Radosz [7,8] who applied a dispersion term proposed by Chen and Kreglewski [9]. Prominent further developments are the SAFT version for molecules with attractive potentials of variable range (SAFT-VR) [10,11], the Soft-SAFT model [12], and the perturbed chain SAFT (PC-SAFT) equation of state [13–15]. The theory of Debye and Hückel constitutes the first milestone for the description of electrolyte solutions. It is an implicit-solvent theory, where the structure and the electrostatic interactions of the solvent are not explicitly resolved; rather, the solvent is implicitly accounted for through a dielectric continuum. The Debye–Hückel theory can be used in equations of state to describe the electrostatic interactions between ions, as successfully shown by Cameretti et al. [16] and Held et al. [17]. More advanced implicit-solvent models for the description of electrostatic interactions are the primitive mean spherical approximation (pmsa) [18,19] and the primitive perturbation theory of Henderson [20]. The prefix ‘primitive’ is assigned to indicate that it is an implicit-solvent model. The pmsa was recently combined with several equations of state for aqueous electrolyte systems, e.g. by Galindo et al. [21] and by the group of Radosz [22–26], who used the so-called restricted pmsa, and by Gil-Villegas et al. [27] as well as by Liu et al. [28] who used the non-restricted pmsa. The term ‘restricted’ indicates that the hard-sphere ion diameters of anion and cation are equal so that an average value has to

24

S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

be used in practice, whereas for the ‘non-restricted’ pmsa the ion diameters can be different. The solvation of an ion is determined by the ion-solvent interactions so that various strategies have been applied with implicit-solvent models, e.g. lumping solvation-effects into the considered dielectric constant or by adjusting solvated ionic diameters [25] or by including it partly into a term which accounts for the dispersive interactions between solvent and ion [29]. The Born theory [30] is a continuum hydration model that was also used to calculate the solvation free energies of ions in water with an equation of state [31] thereby considering the solvation effect between ions and the solvent continuum. Explicit-solvent models consider the (usually dipolar) solvent molecules explicitly and thus account for electrostatic interactions between ions and solvent molecules. Statistical mechanics offers two important explicit-solvent approaches for the description of electrostatic interactions, namely perturbation theories and integral equation theories. Henderson et al. [32] developed a perturbation theory for a mixture of spherical dipolar molecules with hard repulsion representing the solvent, and hard-sphere molecules with point charges in the center representing the ions. The application of perturbation theories is hindered by the slow convergence of the perturbation series and by the fact that several terms in the perturbation series are divergent and need to be skillfully grouped in order to cancel. In the context of equations of state, the most important integral theory for explicit-solvent ionic systems is the non-primitive mean spherical approximation (npmsa) of Blum et al. [33–36]. Like in the perturbation theory of Henderson et al. [32] the theory accounts for interactions in a system which comprises hard-sphere molecules with point charges in the center representing the ions, and spherical dipolar hard-sphere molecules representing the solvent. The excess chemical potentials for the ions of the npmsa over a hard-sphere reference system were used several times to correlate mean ionic activity coefficients (m) ± , e.g. Lotfikian and Modarress [37], Liu et al. [38], Li et al. [39], Liu et al. [40], and Seyfkar et al. [41]. Zhao and McCabe recently combined the npmsa with the SAFT-VR equation of state [42,43] and the theory was compared to molecular simulation data with reduced dipole moments up to *2 = 2 for the solvent of the dipolar square-well monomer fluid, and up to *2 = 1 for the solvent in the ion-dipole system. The objective of this work is to implement the semirestricted npmsa with the PC-SAFT EOS and to apply the model to aqueous electrolyte systems. The EOS is applied to electrolyte systems, at infinite dilution of the salt and subsequently at finite salt concentrations up to the solubility limit of the salt. Fairly simple electrolyte systems (where the ions are usually treated as fully dissociated) are considered here, in order to evaluate the capability of the semirestricted npmsa to describe the ion–ion, ion–dipole and dipole–dipole interaction. 2. Thermodynamic model The equation of state comprises contributions of the PC-SAFT EOS accounting for repulsive, dispersive and associating interactions, as well as contributions of the semirestricted npmsa where the electrostatic interactions between the dipolar solvent molecules and the ions are considered. 2.1. PC-SAFT equation of state The PC-SAFT EOS [13,14] is used in this work to describe the non-electrostatic interactions between charged and non-charged molecules in aqueous electrolyte systems. The electrostatic contributions to the PC-SAFT equation of state proposed recently [44–47]

(accounting for dipolar, induced dipolar and quadrupolar interactions) are not considered here, because the dipolar contribution that stems from the npmsa shall here be evaluated. For details about the PC-SAFT EOS we refer to Refs. [14,15]. 2.2. Semirestricted non-primitive mean spherical approximation for electrostatic interactions The semirestricted non-primitive mean spherical approximation (npmsa) is an Ornstein-Zernike based integral equation theory. For a fluid with hard repulsion, the pair potential u(r) can be divided into a hard repulsive and a soft part usoft , with u(r) = ∞ u(r) = usoft (r)

for r ≤  for r > 

(1)

The soft part usoft is in the context of this work the electrostatic part of the potential. The msa closure which is needed for the solution of the Ornstein-Zernike equation reads as g(r) = 0 c(r) ≈ −

for r ≤  usoft (r) kT

for r > 

(2)

where g(r) is the radial pair distribution function and c(r) is the direct correlation function. The first condition in Eq. (2) is exact for fluids with a hard repulsion according to Eq. (1). The second condition in Eq. (2) is only exact at infinite intermolecular distance r. The contact theorem of fluids with hard repulsion enabled Blum et al. [33–36] to develop expressions for the thermodynamic properties of the msa. The system thereby considered consists of a mixture of hard-sphere particles, where some of the hard-sphere particles carry a point charge and the others a point dipole moment. The theory is called ‘non-primitive’ because the solvent is explicitly considered as a molecular species, and not just regarded by its dielectric constant, like in the primitive msa or in the Debye–Hückel theory. The theory delivers excess thermodynamic properties with respect to a hard-sphere repulsion part, viz. the excess internal npmsa energy Unpmsa , the excess chemical potential i , and the excess npmsa directly, without the need for prior deterHelmholtz energy A mination of the radial pair distribution functions. This contribution uses the so-called ‘semirestricted’ npmsa, due to its simplicity compared to the ‘non-restricted’ npmsa while drawing a more realistic picture than the ‘restricted’ npmsa. In the semirestricted npmsa the hard-sphere diameters of the ions are equal, while the hard-sphere diameter of the solvent can be different. The common hard-sphere ion diameter that is needed for the semirestricted npmsa is denoted as  ion throughout this work. The equations of the npmsa were initially presented in the work of Blum et al. [34–36]. Due to several misprints, however, all equations needed to solve the semirestricted npmsa and the equations for the Helmholtz energy are repeated below. To solve the npmsa, a set of three non-linear algebraic equations needs to be solved. The solution delivers three dimensionless energy parameters: b0 for the ion–ion interactions, b1 for the ion–dipole interactions, and b2 for the dipole–dipole interactions. A gradient based solver was here applied, where the partial derivatives for the determination of the Jacobian matrix were determined numerically. The set of equations that needs to be solved is 0 = a21 + a22 − d02 0 = a1 k10 − a2 (1 − k11 ) − d0 d2 2 + (1 − k )2 − y2 − d2 0 = k10 11 1 2

(3)

where the two parameters d02 and d22 are related to the ion–dipole system; d02 characterizes the charge density, and d22 characterizes

S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

the dipole density of the solvent, with 4 2 2  2 j zj e ion kT

d02 =

b1 = −b0 (2b2 ) (4)

j

4 2  d22 =  3kT D,solv solv

b2 3

ˇ6 = 1 −

b2 6

=

=

y1 =

ˇ3 ˇ6 b21 4

+ ˇ62

1 2DF2

ˇ6 (1 + b0 ) −

(8)

where  is the total number density,  = N/V. The packing fraction used within the PC-SAFT equation of state is defined in terms of a temperature-dependent effective hard-sphere diameter, rather than through the size parameter  i in Eq. (8). Since Eq. (8) is used only for generating initial values for b0 , b1 , and b2 , the precise definition does not have an effect on the results. Once the energy parameters b0 , b1 and b2 are iteratively determined from Eq. (3) for a given temperature T, density , and mole fractions xi , all excess thermodynamic quantities with respect to a hard-sphere repulsion part can be determined. The excess Helmholtz energy due to the ion–ion interaction ACC , due to the ion–dipole interaction ACD and due to the dipolar interaction ADD are

b21 solv



Qii = −a1 − 2 +



−2d0 d2 b1



ion − 1+ solv



ion

solv

  

ion

solv



 Qid

2 

(9)

( − 2ˇ6 DF )



−b1

 + DF ˇ3 ion 2 solv

2ˇ6 DF2



 Qid =

 = Qdd



ˇ3 − a2 b1

solv 2ion

 (5)

In order to find the physically meaningful solution for Eq. (3), appropriate starting values for the iterative solution procedure must be provided. Starting values for the solution of the semirestricted npmsa are suggested by Harvey [49], namely −2d0 (1 + d0 ) (4 + 8d0 + 3d02 ) 3d22 2 + d22

FH

FH0.5

b21 4ˇ62

b1 ˇ62 D 2

q = b2

solv b1 (1 + a1 ) · ion 2

 1

ˇ6 DF

12ion D=1+

k11 = 1 −

b2 =

  xi i3  6

The auxiliary quantities that appear in Eq. (9) are

2 ˇ12

ˇ6 solv

= 0.5(1 + b0 ) + 6ion

b0 =

3 =

ADD −1 2  2 = ([Qdd ] + 2(q ) ) 3 NkT 12sol v

ˇ6

k10 =

(7)

where xion equals the total concentration of ions in the solution and  3 is the packing fraction, given by

ion

DF = 0.5

a2 =

In Eq. (6) the quantity FH is defined as

1 ACD = NkT 12 3



a1 =

(6)

 2 ACC 1 (2d02 b0 − Qii ) = 3 NkT 12ion

b2 12

ˇ12 = 1 +

0.5

FH = 1 − 1.5xion xsolv 30.5

The index j in Eq. (4) indicates a summation over all ions in the system. The index ‘solv’ in Eq. (4) indicates that only the density and the dipole moment of the solvent are needed for the determination of d2 . The charge and dipole density d0 and d2 show that the dependent variables are the same as for any equation of state (N, V, T). Therefore, no conversion to another thermodynamic framework is needed, whereas the primitive msa is in the McMillan–Mayer framework, requiring a conversion irrespective of whether it is used as an activity coefficient model or as equation of state [48]. For solving the set of Eq. (3), some auxiliary parameters were defined, namely a1 , a2 , k10 , k11 and y1 , with ˇ3 = 1 +

25

[ˇ3 + a1 (3 − 2DF )]



ˇ32 −



1 − b2 /24 2 ˇ12

solv

2ion





b1 a2 (3 − 2DF ) − 2

(10)

Besides the equations for the excess Helmholtz energy ACC , ACD , and ADD , Blum et al. [33–36] provide equations for calculating the npmsa excess chemical potentials for the ions and the solvent i , and npmsa for the excess internal energy U . From the excess Helmholtz energy Anpmsa all other thermodynamic quantities, like the chemical potential, and the internal energy can be determined by partial derivation with respect to the canonical variables. The equations for the Helmholtz energies were confirmed to be consistent with expressions for the excess internal energy Unpmsa , npmsa the excess chemical potentials i and the compressibility factor Znpmsa . In order to further validate the equations summarized above, the model was solved for conditions where results of Monte Carlo simulations for charged and dipolar hard-sphere are available [50], as well as previous msa studies [39,51].

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S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

The PC-SAFT EOS with the semirestricted npmsa can be written in terms of the residual Helmholtz energy Ares as Ares Ahc Adisp Aassoc Anpmsa = + + + NkT NkT NkT NkT NkT

Ion

(11)

Anpmsa

stands for the excess Helmholtz energy from the where semirestricted npmsa. Ahc , Adisp and Aassoc are the hard chain, the dispersion part and the association term of the Helmholtz energy, respectively. Anpmsa comprises ACC , ACD , and ADD according to ACC ACD ADD Anpmsa = + + NkT NkT NkT NkT

(12)

The reference fluid for the semirestricted npmsa is a hard-sphere fluid so that the term ‘excess’ in the excess Helmholtz energy refers to a system of hard-spheres. The total Helmholtz energy is finally given as the sum of residual Helmholtz energy and the ideal gas contribution, as A Ares Aideal = + NkT NkT NkT

(13)

During the phase equilibrium iteration the fugacity coefficients ϕi for the individual ions as well as for the solvent are determined. The fugacity coefficients ϕi are obtained via the chemical potentials i by numerical differentiation of the residual Helmholtz energy Ares (as summarized in the appendix of Ref. [14]). The mean ionic (m) activity coefficient ± and the osmotic coefficient (m) can be calculated from the fugacity coefficients ϕi for the individual ions and the solvent, according to [31] ±m =



×

(m)

˚

1 1 +  · m · Msolv

  ϕ (T, p, x ) + / + + ϕ+ (T, p, x+ = 0)

 ϕ (T, p, x ) − / − − ϕ− (T, p, x− = 0)

Table 1 Ion parameters for the alkali and halide ions used in this work.

(14)

+

Li Na+ K+ Rb+ Cs+ F− Cl− Br− I−

zi

 (Å) Ref. [54]

˛ (Å3 ) Ref. [56]

ε/k (K)

+1 +1 +1 +1 +1 −1 −1 −1 −1

1.2 1.9 2.66 2.96 3.38 2.72 3.62 3.9 4.32

0.080 0.210 0.870 1.810 2.790 0.99 3.020 4.170 6.480

244.17 147.38 221.14 494.94 523.28 175.25 225.44 330.82 424.90

Individual parameters for cations and anions are nonetheless desirable because different ion combinations can be described with the same set of parameters. Two routes for the parameter determination are feasible. On the one hand, ion-specific parameters can simultaneously be adjusted to a large set of experimental data. This approach should be successful if the molecular model captures the essential characteristics of the considered systems and the model parameters are thus physically based. On the other hand, ion-specific parameters can be taken from tables and correlations. The latter strategy is applied here. In this work, the Pauling crystal diameter [54] is used as the ion size parameter  i . The charge number zi is ±1 for alkali and halide ions, respectively. Dispersive interactions between ions are not considered so that the dispersion energy parameter for the individual ions εi is only needed to describe the ion/water dispersion εij using the Lorentz–Berthelot combining rules. For the ion dispersion energy parameter εi the relationship of Mavroyannis and Stephen [55] is used, which uses the size of the ion  i , the number of electrons ne,i , and the polarizability ˛i of the ion, and reads as 3/2 1/2

εi 3/2 ˛i ne,i = 356 K · Å k i6

ln(xsolv (ϕsolv (T, p, xsolv )/ϕsolv (T, p, xsolv = 1))) (T, p, ni ) = −  · m · Msolv (15)

where + and − are the number of moles of cations or anions, respectively, which are formed from 1 mol of salt, and  is defined as  = + + − .

(16)

In Eq. (16) the values for the polarizability of the ions ˛i were taken from Jin and Donohue [56], and for the ion diameter  i the values of Pauling [54] were used. Table 1 summarizes the ion parameters which were used to model the ions in this work. 3. Results and discussion

2.3. Model parameters 3.1. Results for pure water In the PC-SAFT EOS water is described using five parameters: the number of segments mi , the segment size parameter  i , the dispersion energy parameter εi /k, and two association parameters εAiBi /k, and AiBi (association energy, and effective association volume) [15]. The association in the SAFT-formalism is a coarse-grained representation of reality and we have adopted two association sites for water, although other studies show evidence that four association sites give better results for description of vapor liquid equilibria [52,53]. In order to describe the dipolar character of the solvent within the semirestricted npmsa theory, a further parameter, the dipole moment D,i is required. With a predefined number of association sites of 2 six pure component parameters for water could have been adjusted. Because water is a small molecule and the segment number is close to unity in the PC-SAFT EOS [15], its value is fixed to mi = 1 in this work. Therefore, in total five parameters ( i , εi /k, εAiBi /k, AiBi , and D,i ) were adjusted to liquid density data and to vapor pressure data. The salt constituents require a different strategy for identifying the pure component parameters. The reason is twofold: (1) inorganic salts have no measurable vapor pressure at moderate temperatures and (2) cations and anions cannot trivially be isolated and consequently no individual bulk ion properties are measurable.

The pure component parameters for water (the molecular diameter  i , the dispersion energy εi /k, the dipole moment D,i , and the two association parameters εAiBi /k and AiBi ) were simultaneously adjusted to liquid density and vapor pressure data of the VDI steam tables [57]. The data cover a temperature range from 273.2 to 613.15 K (equivalent to 0.95Tc ). The adjusted parameters are summarized in the Table 2. If the adjusted parameters of Table 2 are compared to those of the original PC-SAFT EOS [15] it shows that the dispersion energy parameter εi /k is adjusted to a lower value, because the dipole–dipole interactions are individually accounted for. Table 2 Pure component parameters for water with the npmsa + PC-SAFT EOS. Parameter Segment number Segment diameter Dispersion energy parameter Association parameters Dipole moment

m=1  = 2.98 Å ε/k = 199.74 K εAiBi /k = 1937.52 K AiBi = 0.0654 D,i = 2.28 D

S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

Fig. 1. Coexisting densities of water for the complete vapor–liquid region. Comparison of the npmsa + PC-SAFT EOS (line) with data from VDI steam tables [57] (symbols).

The average deviation for liquid densities is 0.7%, and for the vapor pressures it is 0.33%. All parameters are in the expected range of values. The adjusted dipole moment of D,i = 2.28 D is between the value for the gas phase and for the liquid phase at ambient conditions, which are D,i = 1.855 D and approximately D,i = 2.9 D, respectively [58]. Figs. 1 and 2 show the excellent agreement between the equation of state and the data from the VDI steam tables. 3.2. Aqueous electrolyte systems at infinite dilution The objective of this section is to evaluate the accuracy of the semirestricted npmsa + PC-SAFT EOS at infinite dilution of alkali halide salts in water. At the infinite dilution limit the ion–ion interactions are negligible so that the effect of ion–dipole interaction is singled out. For evaluating the performance of the model at infinite dilution of the salt, the Gibbs energy of solvation was determined using the fugacity coefficients of the individual ions at infinite dilution, and at given temperature and pressure from Gis (T, P) = RT ln ϕi (T, P, ni = 0)

(17)

27

Fig. 2. Vapor pressure of water for the complete vapor–liquid region. Comparison of the npmsa + PC-SAFT EOS (line) with data from VDI steam tables [57] (symbols).

In order to compare the results to literature data [59], the Gibbs energy of solvation was determined at T = 298.15 K and p = 1 bar. The single ion diameter  ion for the semirestricted npmsa contribution was determined by arithmetically averaging the Pauling diameters of the anion and the cation in each alkali halide/water system, as ion =

1 + anion ) ( 2 cation

(18)

The single ion diameter  ion was solely used for the semirestricted npmsa term whereas individual ion size parameters  i are handled in the PC-SAFT EOS. Table 3 summarizes the values for the Gibbs energy of solvation as calculated with the semirestricted npmsa + PC-SAFT EOS, the literature data compiled by Fawcett [59], and the deviations between the two. From these results the following conclusions were drawn: 1. The deviation between the Gibbs energy of solvation given by Fawcett and the value calculated from the semirestricted npmsa + PC-SAFT EOS is pronounced for most ions of the aqueous alkali halide systems. 2. Due to the use of a single ion diameter for the semirestricted npmsa term  ion in the EOS the calculated value for the Gibbs

Table 3 Gibbs energy of solvation at T = 298.15 K and p = 1 bar for individual ions in aqueous alkali halide systems at infinite dilution calculated with the semirestricted npmsa + PC-SAFT EOS and data from Fawcett [59]. Salt

LiF NaF KF RbF CsF LiCl NaCl KCl RbCl CsCl LiBr NaBr KBr RbBr CsBr LiI NaI KI RbI CsI

 mean (Å)

1.96 2.31 2.69 2.84 3.05 2.41 2.76 3.14 3.29 3.5 2.55 2.9 3.28 3.43 3.64 2.76 3.11 3.46 3.64 3.85

Calculated

Fawcett [59]

Deviation (%)

s Gcation (kJ/mol)

s Ganion (kJ/mol)

s Gcation (kJ/mol)

s Ganion (kJ/mol)

Cation

Anion

−432.6 −383.1 −343.0 −338.5 −321.9 −374.9 −336.6 −304.6 −302.7 −289.9 −359.9 −324.2 −294.3 −293.1 −281.1 −339.8 −307.3 −279.8 −280.0 −268.6

−427.1 −380.9 −340.9 −327.1 −309.1 −366.4 −331.3 −299.5 −288.4 −274.0 −355.9 −323.4 −293.6 −283.2 −270.1 −339.6 −310.3 −282.8 −273.7 −261.4

−529.4 −423.7 −351.9 −329.3 −306.1 −529.4 −423.7 −351.9 −329.3 −306.1 −529.4 −423.7 −351.9 −329.3 −306.1 −529.4 −423.7 −351.9 −329.3 −306.1

−429.1 −429.1 −429.1 −429.1 −429.1 −304.0 −304.0 −304.0 −304.0 −304.0 −277.7 −277.7 −277.7 −277.7 −277.7 −242.6 −242.6 −242.6 −242.6 −242.6

−18.3 −9.6 −2.5 2.8 5.2 −29.2 −20.6 −13.4 −8.1 −5.3 −32.0 −23.5 −16.4 −11.0 −8.2 −35.8 −27.5 −20.5 −15.0 −12.3

0.8 −11.2 −20.6 −23.8 −28.0 20.5 9.0 −1.5 −5.1 −9.9 28.2 16.5 5.7 2.0 −2.7 40.0 27.9 16.6 12.8 7.7

28

S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

Table 4 Relevant parameters for the semirestricted npmsa + PC-SAFT EOS for modeling of aqueous alkali halide systems. Index i for {cation, anion} and index j for water. Parameter Ion–ion Ion–water Water–water a

Repulsion a

i  ij b jd

Dispersion

Association

semirestricted npmsa

– εij /kc εj /kd

– εAiBj /ke , f AjBj d , εAjBj /kd

 ion e , zi = ± 1  ion e , zi = ± 1, D,j d  j d , D,j d

b

Pauling diameter, Ref. [54].  ij = 1/2( i+  j ).

c

εij /k = ((εi /k)·(εj /k))0.5 , with εi /k = 356 K · Å

d

Adjusted to vapor pressure and liquid density data of water between 273.2 and 613.15 K. (m) Adjusted to mean ionic activity coefficient ± and osmotic coefficient (m) data at 298.15 K. Ion–water association only between cations with AiBi = 0.03 and water.

e f

3/2

3/2 1/2

(˛i ne,i /i6 ) (for the ions).

energy of solvation for each ion depends on the counterion in the system. 3. The deviation in the Gibbs energy of solvation is strongly influenced by the ratio of the individual ion diameters. As can be seen in Table 3, the deviation for the cations approaches 0 for approximately equal diameters of anion and cation. The cation and anion diameter of the systems KF/H2 O, RbF/H2 O, and CsCl/H2 O are s for the cation is small, with about equal and the error of Gion values of 3–5%. Only if the cation diameter is larger than the anion diameter (RbF and CsF) the solvation for the cations is overestimated. Otherwise, if the ratio  cation / anion < 1, the solvation of the cations is consistently underestimated. For the anions the picture is less sharp; the solvation is underestimated for some and overestimated for other systems. Summarizing it can be stated that the semirestricted npmsa + PC-SAFT EOS does not describe the solvation behavior of the ions in aqueous alkali halide systems well. This seems to be an indication for applying the non-restricted npmsa where the anion and cation diameter are individually treated. Because there is no reason to use arithmetic mean values of the individual Pauling diameters for the semirestricted npmsa contribution, the salt specific diameter  ion can be made adjustable. We found, that even if a salt-specific diameter  ion is adjusted, the solvation for most cations will be still underestimated. Therefore, some systems require a further cation solvation term to account for the ion–dipole interactions, which are not only long ranged, but also directional and very strong at short range. 3.3. Results for salt solutions up to their solubility limit From the analysis of the previous section, we drew two conclusions:

The cation–water association is implemented by assigning association sites to the cations, which can only bond to water molecules. To adjust as few parameters as possible, the number of association sites for the cations is defined based on molecular simulation results at infinite dilution of the salt [61]. Following the results of Zhou et al. [61], four association sites are assigned to Li+ and Na+ cations, and three association sites to the K+ cation. The Rb+ and Cs+ cations are not considered in the work of Zhou et al. [61]. Since these cations have a smaller charge density compared to the other alkali cations, they should have fewer association sites. In this work two association sites are assigned to Rb+ and Cs+ cations. The association volume AiBj for the cation–water association was predefined to a representative value of AiBj = 0.03 [15] in order to minimize the number of adjustable parameters. Therefore, to describe the cation–water association only the association energy εAiBj /k needed to be adjusted. These salt-specific parameters (the salt-specific ion diameter  ion and association energy εAiBj /k for the cation–water associa(m) tion) are adjusted to mean ionic activity coefficient data ± and (m) osmotic coefficient data at T = 298.15 K over the full solubility range of the salt. There is other experimental data (liquid densities [62] and system pressures [63] at different temperatures) that can be used to validate the model. No binary interaction parameter kij was introduced. The dispersive interactions between the ions were neglected because their contribution is small compared to the Coulombic interactions [16,25], and unexpectedly the representation of experimental data is improved without ion dispersion interaction. For clarity, Table 4 summarizes the parameters which are used in the semirestricted npmsa + PC-SAFT EOS. Correlation results for the KF/H2 O solution are presented in Figs. 3–5, for LiCl/H2 O-system in Fig. 3, and Figs. 6–8, for LiBraqueous solution in Fig. 3, and Figs. 9–11, and for NaBr in water in Figs. 12–15. For each of the systems the experimental data

1. Instead of using the arithmetic mean value of the individual ion diameters according to Eq. (18) for the semirestricted npmsa contribution, a salt-specific ion diameter  ion can be adjusted to macroscopic properties of the mixture. 2. For some systems a further ion–dipole contribution is needed in order to describe the cation solvation sufficiently. We suspect that an additional contribution to the cation solvation is needed because the ion–dipole term does not suffice in representing the highly structured solvation shells around cations (at infinite dilution). In order to mimic a higher specific ion–dipole interaction, we here take a pragmatic approach, employing an association term of Wertheim for additional cation–water interactions. This approach was suggested by Gil-Villegas et al. for the SAFT-VRE EOS [27] and it was later adopted by Behzadi et al. [29] and by Liu et al. [60]. The Wertheim association is used only for cations because the Gibbs energy of solvation is more distinctly underestimated for cations.

Fig. 3. Osmotic coefficient (m) for the system KF/H2 O (), LiCl/H2 O (♦), and LiBr/H2 O () at T = 298.15 K for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (line) to experimental data of Hamer and Wu [70] (symbols).

S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

29

(m)

Fig. 4. Mean ionic activity coefficient ± for the system KF/H2 O at T = 298.15 K for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (line) to experimental data of Hamer and Wu [70] (symbols).

Fig. 7. System pressure p for the system LiCl/H2 O at T = 303.15 K (䊉), 313.15 K (), 323.15 K (♦), 333.15 K (), and 343.15 K () for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (lines) to experimental data of Patil et al. [63] (symbols).

Fig. 5. Liquid densities  for the system KF/H2 O at T = 273.15 K (♦), at 293.15 K () and at 313.15 K () over the full solubility range. Comparison of the npmsa + PCSAFT EOS (lines) to experimental data [62] (symbols).

for the osmotic coefficient (m) and for the mean ionic activ(m) ity coefficient ± is compared with results of the semirestricted npmsa + PC-SAFT EOS for the full solubility range at T = 298.15 K (Figs. 3, 4, 6, 9, 12 and 13). The data is very well correlated by the

(m)

Fig. 6. Mean ionic activity coefficient ± for the system LiCl/H2 O at T = 298.15 K for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (line) to experimental data of Hamer and Wu [70] (symbols).

Fig. 8. Liquid density  for the system LiCl/H2 O at T = 293.15 K () and 373.15 K () for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (lines) to experimental data [62] (symbols).

(m)

Fig. 9. Mean ionic activity coefficient ± for the system LiBr/H2 O at T = 298.15 K for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (line) to experimental data of Hamer and Wu [70] (symbols).

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S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

(m)

Fig. 10. System pressure p for the system LiBr/H2 O at T = 303.15 K (䊉), 313.15 K (), 323.15 K (♦), 333.15 K (), and 343.15 K () for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (lines) to experimental data of Patil et al. [63] (symbols).

Fig. 11. Liquid density  for the system LiBr/H2 O at T = 293.15 K () and 373.15 K () for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (lines) to experimental data [62] (symbols).

Fig. 12. Osmotic coefficient (m) for the system NaBr/H2 O at T = 298.15 K for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (line) to experimental data of Hamer and Wu [70] (symbols).

Fig. 13. Mean ionic activity coefficient ± for the system NaBr/H2 O at T = 298.15 K for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (line) to experimental data of Hamer and Wu [70] (symbols).

Fig. 14. System pressure p for the system NaBr/H2 O at T = 303.15 K (䊉), 313.15 K (), 323.15 K (♦), 333.15 K (), and 343.15 K () for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (lines) to experimental data of Patil et al. [63] (symbols).

Fig. 15. Liquid densities  for the system NaBr/H2 O at T = 273.15 K () and 373.15 K () for the full solubility range. Comparison of the npmsa + PC-SAFT EOS (lines) to experimental data [62] (symbols).

S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

31

Table 5 Correlation results for the semirestricted npmsa + PC-SAFT EOS. System

LiF/H2 O NaF/H2 O KF/H2 O RbF/H2 O CsF/H2 O LiCl/H2 O NaCl/H2 O KCl/H2 O RbCl/H2 O CsCl/H2 O LiBr/H2 O NaBr/H2 O KBr/H2 O RbBr/H2 O CsBr/H2 O LiI/H2 O NaI/H2 O KI/H2 O RbI/H2 O CsI/H2 O

EOS parameter

AAD (%)

 ion (Å)

εAcat ,BH2 O /k (K)

(m)

(m) ±

No data available 3.234 3.658 3.976 3.66 3.302 3.036 3.140 3.356 3.010 3.073 2.958 3.487 3.022 2.730 3.450 2.706 2.987 2.679 2.499

4500 3680 2000 2000 4920 3900 3000 0 0 5380 3300 0 0 0 1960 3130 0 0 0

5.85 2.71 2.35 1.63 2.08 2.69 3.63 4.26 7.67 2.01 1.26 1.10 4.66 7.36 2.03 3.47 3.10 7.16 4.40

5.51 5.32 3.00 3.11 8.43 5.00 3.35 4.78 7.94 4.41 2.36 3.78 4.20 7.39 3.47 4.98 2.85 9.17 6.81

model, certainly in view of the wide concentration range for these systems. The adjusted parameters and the correlation results for (m) the mean ion activity coefficient ± and the osmotic coefficient (m)

are summarized in Table 5. The salt concentration considered in our calculations covers the full solubility range and is reported in Table 5. It needs to be recognized that because the cation–water association energy parameter εAiBj /k is fitted to the water-salt systems (salt-specific parameter) its value depends on the salt and not on the cation. Further, it is seen that the cation–water association term is not needed for 8 of the 19 considered aqueous alkali halide systems. For these systems only the salt-specific ion diameter  ion (m) was adjusted to data for the mean ionic activity coefficient ± and for the osmotic coefficient (m) . The results of Table 5 show that for 11 systems the AAD for the osmotic coefficient (m) and the mean ionic activity coeffi(m) cient ± is less than 5%, whereby the experimental data cover the full solubility range. For another four systems the AAD for the osmotic coefficient (m) is less than 5%, and the AAD for the (m) mean ionic activity coefficient ± is less than 10%. For another (m)

four systems the AAD for the mean ionic activity coefficient ± and osmotic coefficient (m) is less than 10%. This is slightly more than in earlier reported literature, but most of these studies only consider salt concentrations up to 3 mol/kg [38,39,41], or up to 6 mol/kg [28,31,56,60,64,65]. Here, we describe aqueous electrolyte systems over the full solubility range whereby, at the same time, the number of adjustable parameters is small compared to many of the earlier studies. However, this does not mean that the theory cannot be improved in terms of precise representation of data. Following the idea of Liu et al. [60] the EOS might be improved by taking into account dipolar interactions using a perturbation theory, compare [45,46], and by exchanging the ion–ion term by a primitive model that uses the solvent’s or solution’s dielectric constant. The semirestricted npmsa + PCSAFT EOS requires for 8 of the considered 19 systems in Table 5 only one salt-specific parameter ( ion ) and for the other 11 systems only two salt-specific parameters ( ion and εAiBj /k), in order to describe aqueous alkali halide systems over the complete solubility range. We like to conclude from this, that despite the model’s simplicity (where ion dimerization [66–69] for example is not explicitly accounted for), much of the required characteristics of electrolyte solutions are covered. We want to substantiate this point by extrapolating the model to temperatures other than

Molality (mol kg−1 )

Ref.

0.2–0.983 0.2–17.5 0.2–3.5 0.1–3.5 0.2–19.219 0.2–6.144 0.2–4.803 0.2–7.8 0.3–11 0.2–20 0.2–9 0.2–5.5 0.2–5 0.3–5 0.2–3 0.3–12 0.3–4.5 0.3–5 0.2–3

[70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70] [70]

the 298.15 K for which the one or two model parameters were identified. The salt-specific parameters in this work are temperature independent. The model is applied to describe system pressures p for varying temperature up to T = 343.15 K as shown in Figs. 7, 10 and 14. Very good results are found in comparison to the experimental data. Liquid densities  in various temperature ranges between 273.15 and 373.15 K are calculated with satisfactory agreement to the experimental data (Figs. 5, 8, 11 and 15). 4. Conclusions The semirestricted npmsa together with the PC-SAFT EOS was applied to aqueous alkali halide systems at infinite dilution of the salt, and at finite salt concentrations up to its solubility limit. By applying the model at the infinite dilution limit, it was shown that the single ion diameter for the semirestricted npmsa contribution  ion should be an adjustable parameter and that for cations the solvation is often underestimated. In order to model aqueous alkali halide systems up to the solubility limit the single ion diameter  ion for the semirestricted npmsa term for all systems, and in addition a cation solvation energy parameter for 11 out of 19 (m) systems were adjusted to mean ionic activity coefficient ± and osmotic coefficient data (m) at T = 298.15 K. The average absolute (m) deviations for the mean ionic activity coefficient ± and osmotic (m) coefficient are always within 10%, and for 11 out of the 19 systems within 5%. Using these parameters the system pressures p and liquid densities  in the temperature range of T = 273.15–373.15 K were described with good results without any further temperaturedependent adjustable parameter. List of symbols A Helmholtz energy Eq. (13) quantity for determination of starting values for the soluFH tion of the semirestricted npmsa Eq. (7) Gis Gibbs energy of solvation Eq. (17) molar mass of the solvent, here Msolv = MH2 O = Msolv 18.01 kg/kmol Eqs. (14) and (15) N number of molecules Eqs. (9), (11), (12) and (13) T temperature Eqs. (2), (4), (9), (11)–(17) a1 , a2 auxiliary parameters for the solution of the semirestricted npmsa Eqs. (3) and (5)

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S. Herzog et al. / Fluid Phase Equilibria 297 (2010) 23–33

b0 , b1 , b2 energy parameters for the semirestricted npmsa (solution of the semirestricted npmsa) Eqs. (5), (6), (9) and (10) c(r) direct correlation function Eq. (2) d02 , d22 charge density of the ions, and dipole density of the solvent in the semirestricted npmsa (These two parameters describe the properties of the hard-sphere ion–dipole system within the npmsa.) Eqs. (3), (4), (6) and (9) e electron/elementary charge Eq. (4) g(r) radial distribution function Eq. (2) k Boltzmann constant, k = 1.38066 × 10−23 J K−1 Eqs. (4), (9), (11)–(13) k10 , k11 auxiliary parameters for the solution of the semirestricted npmsa Eqs. (3) and (5) m salt molality; and number of segments in PC-SAFT EOS, Eqs. (14) and (15) ne,ion number of electrons of an ion Eq. (16) p pressure Eqs. (14), (15) and (17) r radial distance between molecules or molecular sites Eqs. (1) and (2) u pair potential Eqs. (1) and (2) xi mole fraction of i Eqs. (14) and (15) y1 auxiliary parameters for the solution of the semirestricted npmsa Eqs. (3) and (5) zi valence/charge number of ions Eq. (4) Greek symbols ion polarizability Eq. (16) ˛ion (m) ± mean ionic activity coefficient Eq. (14) ε dispersion parameter for the PC-SAFT EOS Table 2 εAiBi pure component parameter for the PC-SAFT EOS (potential depth for association) association energy between cations and water to describe εAiBj the additional solvation in the semirestricted npmsa + PCSAFT EOS Table 2 AiBi pure component parameter for the PC-SAFT EOS (effective association volume) Table 2 D,i dipole moment of i Table 2 i ,  number of moles of ion i from one mole of salt/total number of moles of ions from one mole of salt ( = + + − ) Eq. (15)  3 3 packing fraction for spherical molecules, 3 = 6  xi i (The packing fraction used within the PC-SAFT equation of state is defined in terms of a temperature-dependent effective hard-sphere diameter.) Eqs. (7) and (8)  (number) density (i = Ni /V) Eqs. (4), (8) and (9)  segment diameter in PC-SAFT equation of state hardsphere diameter in the msa Eqs. (1), (2), (4), (5), (8)–(10), (16)  ion ion diameter for the semirestricted npmsa term (adjustable parameter in the semirestricted npmsa + PCSAFT EOS) Eq. (18) ϕi fugacity coefficient Eq. (14), (15) and (17) molality based osmotic coefficient Eq. (15)

(m) Subscripts anion refers to the anion cation refers to the cation ion refers to ion(s) soft contrary to repulsive solv refers to the solvent (in this work H2 O)

CD DD disp hc

Acknowledgements The authors would like to thank the Deutsche Forschungsgemeinschaft for supporting this work under grant Ar 236/24-1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

Superscripts assoc association CC Coulomb–Coulomb

Coulomb–dipole dipole–dipole dispersive hard chain

[55] [56] [57]

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