5102
Ind. Eng. Chem. Res. 2008, 47, 5102–5111
An Electrolyte Equation of State Based on a Hydrogen-Bonding Nonrandom Lattice Fluid Model for Concentrated Electrolyte Solutions Yong Soo Kim and Chul Soo Lee* Department of Chemical and Biological Engineering, Korea UniVersity, 5 Ga, Anam Dong, Sungbuk-Ku, Seoul 136-701, Korea
Equations of state that are based on lattice fluids have been in use for nonelectrolyte components and their mixtures with somewhat different characteristics from hard-sphere chain-based equations of state or cubic equations. In the present study, an electrolyte equation of state was developed, based on a hydrogen-bonding nonrandom lattice fluid theory, by adding the long-range contribution that is due to the mean spherical approximation. Hydrogen bonding of solvent molecules and solvation between solvent molecules and cations were explicitly included by association contribution to extend the applicability to highly concentrated electrolyte solutions. Segment numbers of ions were obtained from the Pauling diameter, using the previously developed relationship between lattice and off-lattice fluids. The remaining electrolyte parameterssnamely, interaction energy between electrolyte and solvent, hydrated ionic diameter, and hydration energy between cation and solvent moleculeswere fitted to osmotic coefficients and mean activity coefficients at 298.15 K and 1 bar. Good agreements were obtained between the experimental and calculated results over the wide range of compositions, up to a molality of 20, with average absolute deviations (AADs) of 1.0%, 1.1%, and 1.6% for the osmotic coefficients, the mean activity coefficients, and the densities of 94 aqueous electrolyte solutions, respectively. The equation of state was determined to be applicable to sodium chloride solutions in the temperature range of 273-373 K when these properties were calculated using a temperature-dependent binary interaction parameter. Examples were presented for 1:1 electrolytes to show that parameters of monovalent ions, rather than electrolyte parameters, can be used. 1. Introduction Electrolyte solutions are involved in a wide range of chemical processes. Zemaitis et al.1 reviewed leading excess Gibbs energy models, which are applied mostly to solutions with molalities up to 6. For highly concentrated solutions (molality of >6), hydration chemistry was invoked.2 Loehe and Donohue3 reviewed theoretical and engineering models on the thermodynamic properties of electrolyte solutions. A short summary4 is given for engineering equations of state based on the Helmholtz free energy, which are more convenient for describing the nonideality of the electrolyte solutions in high-pressure applications such as carbon dioxide ocean sequestration and enhanced oil recovery. Helmholtz free-energy or equation of state approaches can provide thermodynamic properties that excess Gibbs function models do not. Recently proposed Helmholtz free-energy models for electrolyte solutions generally yield a variety of thermodynamic properties with the inclusion of contributions from short-range physical interactions, associations, and long-range electrostatic interactions. For short-range physical interactions, cubic equations,5–9 the hard-sphere-based model,10 or hard-sphere chain-based models11–15 are commonly used. In some models, 5 the Born term is used for charging and solvation whereas, in some other models,6 the Born charging term and the ion solvation term are included separately. The Born term is not included in several models but the ion solvation term is used.11,12,14,15 A primitive or restrictive primitive mean spherical approximation (MSA) term with different approximations is used for long-range interactions in most models, whereas a nonprimitive MSA term is used in some other studies.10 Most applications are limited to solutions with molalities of up to 6. * To whom correspondence should be addressed. Tel.: +82-2-32903293. Fax: +82-2-926-6102. E-mail address:
[email protected].
The conversion problem of the MSA term in the McMillansMeyer framework into that of Gibbs energy is included5 or referenced but neglected.6 The effect due to the conversion is not expected to be small for highly concentrated solutions. Lattice frameworks have traditionally been favored in polymer solutions. When compared to other leading models, the lattice fluid equation of state showed comparable accuracies.16 The GuggenheimsHugginssMiller approximation for athermal lattice chains was determined to be more accurate for fused hardsphere chains than the off-lattice tangentshard-sphere chain theory.17 A preliminary study indicated that the association contribution in lattice fluid models is simple and flexible, yet generally accurate. Thus, it is natural to develop an alternative electrolyte equation of state in the present study, based on a hydrogen-bonding nonrandom lattice fluid model for nonelectrolyte systems.18 The present approach is similar to previous approaches in that the Helmholtz free energy is represented by a sum of various contributions. The model is extensively applied to calculate osmotic coefficients, mean activity coefficients, and solution densities of aqueous electrolyte solutions over wide range of compositions (molalities up to 20). For applications up to highly concentrated solutions, the solvation between ions and solvent molecules and the association between solvent molecules are explicitly described. 2. Thermodynamic Model and Properties To obtain the Helmholtz energy representation of aqueous strong electrolyte solutions, we begin with a reference fluid mixture of physically interacting solvent molecules and uncharged bare ions that are not solvated. The extra effect of charging ions is considered in three steps: (1) In Step I, uncharged bare ions are charged.
10.1021/ie0711527 CCC: $40.75 2008 American Chemical Society Published on Web 12/12/2007
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5103
(2) In Step II, solvent molecules and charged bare ions are associated and solvated. (3) Finally, in step III, long-range electrostatic interactions between charged solvated ions are turned on. The present steps are similar to those of Wu and Prausnitz5 and Myers et al.6 In the reported work of Wu and Prausnitz,5 the association of solvent molecules is considered in the first step and all the remaining effects in the second step, and in Myers et al.6 solvation, is included in the Born term. The competition of association between solvent molecules and solvation of ions is clearly indicated in the present path. The total Helmholtz free energy is represented by a sum of these contributions, such that each additional contribution vanishes when the corresponding effect is not present: A ) Aphys + Achg + Aass + Alr
(1)
where the subscripts “phys”, “chg”, “ass”, and “lr” represent contributions from physical, charging, association, and longrange interactions, respectively. Physically interacting reference fluids of solvent and uncharged bare ions are described using the hydrogen-bonding nonrandom lattice fluid equation of state (NLF-HB),18 in which athermal chain interaction and residual physical interaction are given by the GuggenheimsHugginssMiller approximation and an expanded quasi-chemical approximation, respectively. We consider a mixture of N molecules and ions, in which the number of molecules or ions and mole fractions of species i are represented by Ni and xi, respectively. In the model, physical interactions of each species are characterized by segment number (ri), the surface area parameter (qi), and the interaction energy (ii). Moreover, there are N0 vacant sites whose segment number is unity (ri ) 1) and interaction energy is zero (ii ) 0). For open-chain molecules, ri and qi are related by zqi ) (z - 2)ri + 2
(2)
where z is the coordination number (which is set to 10 in the present study). A binary interaction parameter (kij) is introduced for interactions between unlike species: ij ) (iijj) (1 - kij) 0.5
(3)
Equations for the Helmholtz free energy and thermodynamic properties are found in the cited reference.18 Equations for physical contributions to pressure and chemical potential are readily derived from the Helmholtz free energy. βPphys )
[ ( )]
qM zβ z 1 ( ln 1 + -1 F ln 1 - F) + θ2 2VH rM VH 2 M (4)
[
0 + ri ln βµphys,i ) βµphys,i
( )
ln 2
∑θ
k ik + β
[
( ) ]
]
1 + (qM ⁄ rM - 1)F + 1-F
[
ri θiP0 zβ qiMθ2 1 - + qiβVH 2 qi
( )
∑ ∑ ∑ θ θ θ ( j k l ij
ij + 2kl - 2jk - ik)
2
Mθ
]
(5)
where µphys, i is the chemical potential of a pure ideal gas at P0 ) 1 bar, VH is the constant lattice cell volume (9.75 cm3/mol), and
∑rxq
i i M)
∑qx
i i
1 θ2
qiNi N0 + qMN
(7a)
θ)
qMN N0 + qMN
(7b)
F)
r MN N0 + rMN
(8)
[∑ ∑ θ θ + ( β2 ) ∑ ∑ ∑ ∑ θ θ θ θ ( + i j ij
i j k l ij
(6)
ij
]
3kl - 2ik - 2jk)
(9)
Bare anions and cations in a solvent that is considered a dielectric continuum are charged in Step I. The free energy of charging cavities or uncharged ions, in the absence of interactions with other ions, is obtained by calculating the reversible work of charging isolated ions:19 βAchg )
(zke)2Nk 8π0DsσBk ions
∑
(10)
where zk, Nk, and σBk denote the charge number, the number and the radius of bare ionic species k. The Born model contribution is represented by the difference of charging free energies in solvent from that under vacuum. It includes the effect of ion solvation or polarization of solvent molecules and can be made to agree with experimental data with a radius greater than the ionic radius.20,21 Here, eq 10, with the ionic radius, is understood to represent the free energy of charging, excluding the solvation effect that is to be considered in the association contribution. The free energy is linear in the number of ions and independent of volume if the density dependence of solvent dielectric constant Ds is ignored. The the contribution to pressure then vanishes. βPchg ) 0
(11)
We still have the contribution to solute chemical potential, βµchg,i )
(zie)2 8πε0DsσBi
(12)
which is canceled out when the solute activity coefficients are calculated. The contribution to solvent chemical potential is obtained by taking the partial derivative of the dielectric constant, with respect to solvent mole number for mixed solvent systems. For pure solvent systems, the contribution vanishes. In Step II, contributions that are due to the solvation of ionic species and the association between solvent molecules are introduced by the association term in the hydrogen-bonding nonrandom lattice fluid lattice fluid equation of state.18 Water molecules become insufficient to fully solvate ions as ion concentrations increase and such an effect is considered explicitly in the association contribution. To evaluate the association contribution, the number of type k donors (dki), the number of type l acceptors in species i (ali), and the association free energy between the type k donor and the type l acceptor (Akl) are needed: Akl ) Ukl - TSkl
0
rM )
M )
θi )
(13)
Cations are considered to be proton donors. Water molecules provide donor and acceptor sites. Acceptor sites in solvent molecules are hydrogen-bonded to cations to form the solvation shell. Anion solvation and ion pairing can be included in the extension of the present framework but neglected. The number
5104 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
of donor or acceptor sites and the bonding free energy of ions and solvent molecules characterize association contributions. Association contributions to pressure and chemical potential are given by18 βPass ) βµass,i ) -
[
∑ ∑ (N
0 ij - Nij)
VHrMN
( )
∑d
i k
ln
k
N0k0
Nk0
-
]
F
( )
(15)
∑Nd -∑N N )∑Na -∑N i i k
i i l
kj 0l
i
j
i
jl
(16)
j
Nkl are obtained via the simultaneous solution of Nkl(N0 + rMN) ) rHNk0N0l exp(-βAkl)
0 kj
(18a)
∑Na -∑N
0 jl
(18b)
i i k
N00l )
j
i i l
i
j
Nkl are obtained by the simultaneous solution of N0kl(N0 + rMN) ) rHN0k0N00l
(19)
The contributions of the long-range electrostatic interaction in Step III are from the primitive MSA of Blum and Høye,23 as approximated and used by Ball et al.,24 in which ions are considered as the hard spheres in a dielectric medium of solvent but without short-range interactions. The Helmholtz energy for long-range contribution is given by βAlr )
Nkz2k Γ3V R2Γ 3π 4π ions 1 + Γσk
∑
(20)
where Nk is the number per kilogram of solvent, zk is the charge number, σk is the hydrated ionic diameter of ion species k, and the shielding parameter Γ is obtained from the solution of 4Γ2 )
z2k Nk R2 V ions (1 + Γσ )2
(21)
e2β 0D
(22)
∑
R2 )
3 k k
(24)
(25)
The MSA contribution is in the McMillansMeyer framework, in which the extra effect of turning on the long-range electrostatic interactions is taken into account at constant volume and solvent chemical potential. The chemical potential, after turning on the long-range interaction in the LewissRandall framework is obtained at a given temperature and pressure in two steps. The system pressure changes by Plr from the given pressure at constant volume in the turning-on process, which is then restored to the initial pressure P: µi(T, P, x) ) µi(T, P + Plr, x) - 〈Vi 〉Plr
(26)
where
∑Nd -∑N i
( 6Vπ )∑ N σ
V ) VH(N0 + rMN)
(17)
for all k and l. rH is a constant entropy factor (0.05) used to account for the fraction of a lattices cell in which a donor or an acceptor must be found for association.22 Nij0 is the number of association pairs with zero association energy and introduced for normalization.22 Similar equations are solved to obtain values for Nij0. N0k0 )
(23)
Volume is written in the lattice frame by
N00l 0l
l
]
ions
(14)
∑ a ln N i l
1 - ξ3 1 + (ξ3 ⁄ 2)
where ξ3 )
where Nij is the number of association pairs between the Type i donor and the Type j acceptor. Nk0 is the number of unpaired Type k donors and N0l is that of unpaired Type l acceptors. Nk0 )
[
D ) 1 + (DS - 1)
k
Equation 20 corresponds to the approximation for low-density solutions or for similar ionic diameters, as noted by Blum and Høye.23 Such an approximation is not essential but simplifies the present model. The iterative solution of eq 21 rapidly converges in the successive substitution. The dielectric constant (D), in the presence of electrolytes, is taken from Ball et al.,24 neglecting the second-order term:
∫
P+Plr
〈Vi 〉 )
P
Vi dP
(27)
Plr
Equations 26 and 27 are given by Breil and Mollerup25 for the process and are consistent with the work of Cabezas and O’Connell.26 An equation of state is solved to determine the volume at given temperature, pressure, and composition for the evaluation of chemical potential. For the chemical potential on the right-hand side, the new pressure P + Plr is identified as Pphys + Pass + Plr or P ) Pphys + Pass. Thus, the constant volume is ensured in the turning-on process by solving an equation of state that gives pressure as a sum of Pphys and Pass, in regard to determining the volume. For the evaluation of chemical potential on the left-hand side of eq 26, the volume is determined using an equation of state in which pressure is expressed by the sum Pphys + Pass + Plr. In both cases, the solute chemical potential is given by µi ) µphys,i + µass,i + µlr,i
(28)
Thus, we have the option of evaluating the right-hand side or the left-hand side of eq 26 after we have expressions for Plr and µlr, i. Evaluating the left-hand side is more convenient when an equation of state is available and avoids the determination of partial molar volumes. The solute activity coefficient evaluated from the left-hand side of eq 26 is consistent with that from the right-hand side, with deviations of 6 molality, they introduced an ion-pairing effect and parameters to improve fitting. In the Myers et al. equation of state model,6 there are two physical parameters for the PengsRobinson contribution and a size parameter for the MSA contribution. Deviations for concentrated solutions of >6 molality were not reported in these studies. The average absolute deviations (AADs) of activity coefficients for the present model, the electrolyte NRTL, and Myers et al. model are 1.0%, 1.5%, and 1.1%, respectively, at 298.15 K, as shown in Table 2. The AAD for osmotic coefficients of the present electrolyte NLFHB equation of state is 1.1%. The present model is comparable to these models, using the same number of adjustable parameters, up to very high electrolyte concentrations. The densities of the electrolyte solutions are sensitive to the segment numbers of constitutive species. In the present model, the segment number of ions was related to the crystal ionic diameter obtained from literature28 and estimated by eq 42. The AADs of calculated densities from experimental data36 are also shown in Table 2. The average of the AADs is obtained to be 1.6% without fitting the density data (mostly at 293.15 K) and in the concentration range of the mean activity coefficients and osmotic coefficients. Experimental data of the mean activity coefficients and osmotic coefficients were reported beyond a molality of 6 for concentrated electrolyte systems in the literature reports of Robinson and Stokes31 and Lobo.37 Abovsky et al.38 modified the electrolyte NRTL model by introducing the concentration dependency on interaction energy parameters, in which four adjustable parameters were used to fit the mean activity coefficients of electrolyte solutions. Their root-mean-square (rms) deviations are compared with the present model in Table 3. The average rms deviations of the present model and Abovsky et al.38 are 0.030 and 0.024, respectively. Chen et al.2 introduced ion-pair interactions and equilibrium constants of the partial dissociation for the electrolytes and obtained an AAD of 1.4% for LiCl solutions (molalities up to 20), which is smaller than the present value of 3.6% (given in Table 2). The present electrolyte NLF-HB equation of state was applied to 11
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5107 Table 2. Parameters of Electrolyte Solutions for Electrolyte NLF-HB Equation of State and Comparisons of Calculated Results by Electrolyte NRTL, Myers et al.,6 and the Present Model electrolyte
i/k [K]
σi [Å]
A/R [K]
LiCl LiBr LiI LiClO4 LiNO3 LiOH Li2SO4 NaCl NaBr NaI NaClO4 NaNO3 NaOH Na2SO4 NaClO3 NaF NaBrO3 NaCNS Na2CrO4 KCl KBr KI KNO3 KOH K2SO4 KClO3 KF KBrO3 KCNS K2CrO4 RbCl RbBr RbI RbNO3 Rb2SO4 CsCl CsBr CsI CsNO3 CsOH Cs2SO4 NH4Cl NH4NO3 (NH4)2SO4 AgNO3 TlClO4 TlNO3 MgCl2 MgBr2 MgI2 Mg(ClO4)2 Mg(NO3)2 MgSO4 CaCl2 CaBr2 CaI2 Ca(ClO4)2 Ca(NO3)2 SrCl2 SrBr2 SrI2 Sr(ClO4)2 Sr(NO3)2 BaCl2 BaBr2 BaI2 Ba(ClO4)2 Ba(NO3)2 MnCl2 MnSO4 FeCl2 CoCl2
86.61 90.15 152.2 85.58 74.36 66.50 37.28 47.35 59.80 73.27 71.08 70.49 24.98 27.33 60.31 67.89 39.94 71.23 46.32 57.09 63.28 71.99 40.26 73.29 46.14 53.23 32.28 34.81 64.88 55.60 57.71 63.06 67.41 43.19 50.44 59.81 61.80 67.23 40.89 90.11 59.93 61.95 50.23 41. 13 33.68 61.23 30.94 68.20 86.61 90.39 86.30 78.86 37.75 63.10 72.45 81.57 82.35 37.23 52.70 64.38 76.89 79.44 37.23 55.07 62.10 75.98 76.27 36.52 63.09 61.76 57.23 70.66
3.10 2.85 4.54 4.39 3.82 2.67 4.05 4.15 4.12 4.33 4.20 3.53 3.25 3.98 3.80 3.54 3.68 4.35 4.43 3.69 3.86 4.14 3.10 3.76 3.66 4.05 3.87 4.19 3.79 4.10 3.50 3.53 3.56 3.00 3.90 3.00 3.20 3.34 3.15 4.00 3.95 3.78 3.60 3.55 3.10 2.95 1.69 4.21 4.30 4.38 4.46 4.75 3.30 4.13 4.30 4.81 4.23 4.67 4.61 4.76 4.85 4.45 4.69 4.55 4.77 5.00 4.91 3.45 4.58 4.23 4.60 4.40
2007 2201 2550 2687 1855 1260 1876 2143 2146 2130 1742 2005 2100 1838 1575 1234 1675 1843 1998 1675 1694 1695 1502 2146 1600 1345 2095 1615 1576 1733 1667 1623 1657 1512 1670 1640 1670 1624 1885 1782 1624 1631 1595 1610 1560 1546 1564 2311 2369 2582 3130 2345 1590 2079 2293 2651 2123 2016 2365 2741 3113 2020 1910 1910 2196 2723 1900 1513 2010 2001 2452 2045
AAD in Mean Activity Coefficient (%) Myers et al. 0.25a 0.37a 1.29 4.26 1.55a 1.01 0.27 0.08 0.09 0.19 0.07 0.11 0.67a 0.16 0.09 0.02 0.08 0.36 0.41 0.08 0.09 0.20 0.13 0.60a 0.28 0.08 0.08a 0.08 0.13 0.25 0.07 0.07 0.12 0.19 0.14 0.28a 0.31 0.17 0.15 0.46 0.22 0.29 0.16a 0.25 0.20a 0.39 0.27 0.50 0.48 0.34 2.17 0.71 2.11 0.18 3.90 0.21 0.30 0.16 0.23 6.45 0.23 0.17 0.16 0.30 4.75 0.11 6.80 1.78 0.96 2.65
6
AAD, Using the Electrolyte NLF-HB
electrolyte NRTL
γ(
φ
F
0.88a 0.95a 1.78 1.32 0.35a 3.00 0.81 0.44 0.36 0.57 0.46 0.12 0.63a 0.26 0.30 0.02 0.10 1.22 2.23 0.23
3.61 3.51 1.44 0.75 1.88 0.44 0.34 0.12 0.22 0.22 0.24 0.68 1.65 0.29 0.43 0.10 0.14 0.43 0.65 0.23 0.14 0.13 0.16 0.97 0.12 0.07 0.07 0.07 0.16 0.33 0.06 0.06 0.10 0.31 1.8 0.20 0.34 0.28 0.14 0.12 0.33 0.10 1.38 0.19 0.39 0.06 0.26 2.11 2.30 3.33 3.04 0.90 2.05 3.12 2.67 1.46 3.24 1.39 0.84 0.86 1.17 1.59 0.58 0.27 0.19 0.41 0.38 0.09 1.02 0.55 0.57 1.38
1.57 1.48 1.16 0.46 1.29 0.63 0.39 0.13 0.15 0.21 0.21 1.34 1.07 0.57 0.36 0.30 0.19 0.33 0.97 0.19 0.15 0.12 0.26 0.55 0.16 0.12 0.23 0.17 0.17 0.28 0.09 0.09 0.11 0.38 0.24 0.24 0.36 0.35 0.20 0.49 0.27 0.14 1.93 0.80 0.91 0.18 0.45 1.41 1.40 2.01 1.91 0.54 6.75 1.76 1.37 1.16 1.98 1.23 0.63 0.75 1.01 0.84 0.71 0.35 0.19 0.49 0.23 0.37 0.79 2.09 0.53 1.03
2.25 0.83 0.34 1.77 3.41 0.18 1.27 0.17 0.41 0.89 2.60 1.85 2.72 1.55 1.25 0.16 1.03 2.35 5.34 0.35 0.22 0.56 1.14 1.70 0.68 0.20 0.42 0.30 2.23 1.35 0.52 0.18 0.51 1.26 0.37 0.69 0.19 0.67 0.31 0.28 1.26 1.44 5.98 1.42 0.14
0.38 0.58 0.48a 0.24 0.36a 0.14 0.13 1.37 0.15 0.16 0.18 0.85 0.47 0.52a 0.52 0.52 0.24 0.47 0.06 0.93a 0.92 0.77a 0.22 0.55 3.65 4.16
1.45 3.06 5.72 2.46 3.67 1.84 2.73 2.35 2.57 2.76 2.00 1.33 1.63 2.02 3.22 0.07 1.90 1.50 1.08 1.94
1.42 1.28 2.08 4.14 0.29 5.55 1.25 1.45 2.11 4.78 1.98 1.50 1.03 2.11 3.92 0.66 0.97 0.93 2.29 4.34 0.06 1.17 2.79
maximum molality 20.0 20.0 3.0 4.0 20.0 4.0 4.0 6.0 4.0 12.0 6.0 11.0 20.0 4.0 3.5 1.0 2.5 4.0 4.0 4.8 5.0 4.5 3.5 16.0 0.7 0.7 17.5 0.5 5.0 3.5 5.0 5.0 5.0 4.5 0.78 11.0 5.0 3.0 1.4 1.0 1.8 6.0 20.0 5.5 13.0 0.5 0.4 5.0 5.0 5.0 4.0 5.0 3.0 10.0 9.2 2.0 6.0 6.0 4.0 2.0 2.0 6.0 4.0 1.8 2.0 2.0 5.5 0.4 6.0 4.0 2.0 4.0
5108 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 2. Continued electrolyte
i/k [K]
σi [Å]
A/R [K]
CoBr2 CoI2 Co(NO3)2 NiCl2 NiSO4 CuCl2 Cu(NO3)2 CuSO4 Cd(NO3)2 CdSO4 Pb(ClO4)2 Pb(NO3)2 BeSO4 AlCl3 ScCl3 CrCl3 YCl3 LaCl3 CeCl3 Cr(NO3)3 Al2(SO4)3 Cr2(SO4)3
89.13 96.93 61.11 71.18 58.94 46.62 55.76 91.38 56.84 27.83 74.65 63.13 28.72 65.50 57.23 65.73 48.13 45.87 43.52 56.84 17.14 16.59
4.56 4.20 4.55 4.30 4.25 4.87 4.65 3.09 4.90 3.39 4.83 2.46 3.40 4.70 4.80 4.95 4.88 4.99 4.93 5.11 4.60 5.45
2230 2113 2146 2058 2131 2170 2150 1130 2100 1670 2020 1120 1738 3010 2532 2596 2863 2750 2705 2950 1835 2080
a
AAD in Mean Activity Coefficient (%)
AAD, Using the Electrolyte NLF-HB
Myers et al.6
electrolyte NRTL
γ(
φ
4.70 18.92 0.73 3.89 1.76 4.59 1.71 1.56 0.52 1.53 2.68 0.19 1.86 0.61 0.29 0.49 0.29 0.16 0.44 0.21 2.81 0.52
2.45
3.15 7.81 0.75 2.18 0.42 2.78 0.67 0.97 0.21 1.47 1.18 0.46 1.86 2.66 0.99 1.21 0.80 0.81 0.60 0.46 1.22 0.59
2.10 3.23 0.62 1.44 2.07 1.23 0.37 4.93 0.20 5.47 0.67 0.77 5.97 2.25 1.14 1.39 0.85 0.83 0.78 0.60 6.11 2.04
2.39 2.85 1.26 2.62 1.33 1.94 2.95 1.45 1.83 4.57 3.46 2.69 4.07 3.96 3.97 2.88
F
1.35 0.31 3.25
1.95 1.00
1.03 0.96 0.34 5.03 3.02
maximum molality 5.5 6.0 5.0 5.0 2.5 6.0 6.0 1.4 2.5 3.5 6.0 2.0 4.0 1.8 1.8 1.2 2.0 2.0 2.0 1.4 1.0 1.2
Calculated for molalities up to 6.
Table 3. Root-Mean-Square (rms) Errors of the Present Model and from Abovsky et al.38 for Mean Activity Coefficients of Concentrate Electrolyte Solutions at 298.15 K electrolyte KF LiBr LiCl LiNO3 NaI NaNO3 NaOH CaBr2 CaCl2
rms Error present model Abovsky et al.38 0.017 0.058 0.045 0.023 0.016 0.024 0.020 0.034 0.037
0.007 0.045 0.024 0.013 0.003 0.007 0.022 0.063 0.036
maximum molality 17.5 20.0 20.0 12.0 11.0 11.0 20.0 9.2 10.0
energy change of bringing the Na ion from vacuum to an aqueous solution is -375 kJ/mol at 298.15 K.31 The calculated value of the Born contribution, using the diameter given in Table 2, is -722 kJ/mol. The difference of -347 kJ/mol of ion may be explained by the ion solvation with -50 kJ/mol of an association bond, which is larger in absolute value than the model free energy of approximately -17 kJ/mol of association bond in Table 2. With a free energy of -50 kJ, the interaction energy and hydrated ionic diameter can be fitted to activity and osmotic coefficient data with deviations of
